0.00/0.04	% 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   : n120.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 05:50:55 CDT 2018
0.03/0.23	% CPUTime    : 
285.13/285.45	% SZS status Theorem
285.13/285.45	
285.65/285.93	% SZS output start Proof
285.65/285.93	Take the following subset of the input axioms:
286.14/286.49	  fof(additive_associativity, axiom,
286.14/286.49	      ![A, B, C]:
286.14/286.49	        addition(A, addition(B, C))=addition(addition(A, B), C)).
286.14/286.49	  fof(additive_commutativity, axiom,
286.14/286.49	      ![A, B]: addition(B, A)=addition(A, B)).
286.14/286.49	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
286.14/286.49	  fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
286.14/286.49	  fof(backward_box, axiom,
286.14/286.49	      ![X0, X1]: c(backward_diamond(X0, c(X1)))=backward_box(X0, X1)).
286.14/286.49	  fof(backward_diamond, axiom,
286.14/286.49	      ![X0, X1]:
286.14/286.49	        codomain(multiplication(codomain(X1), X0))=backward_diamond(X0,
286.14/286.49	                                                                    X1)).
286.14/286.49	  fof(codomain1, axiom,
286.14/286.49	      ![X0]: multiplication(X0, coantidomain(X0))=zero).
286.14/286.49	  fof(codomain2, axiom,
286.14/286.49	      ![X0, X1]:
286.14/286.49	        addition(coantidomain(multiplication(X0, X1)),
286.14/286.49	                 coantidomain(multiplication(coantidomain(coantidomain(X0)),
286.14/286.49	                                             X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0)),
286.14/286.49	                                                                               X1))).
286.14/286.49	  fof(codomain3, axiom,
286.14/286.49	      ![X0]:
286.14/286.49	        one=addition(coantidomain(coantidomain(X0)), coantidomain(X0))).
286.14/286.49	  fof(codomain4, axiom,
286.14/286.49	      ![X0]: codomain(X0)=coantidomain(coantidomain(X0))).
286.14/286.49	  fof(complement, axiom, ![X0]: antidomain(domain(X0))=c(X0)).
286.14/286.49	  fof(domain1, axiom,
286.14/286.49	      ![X0]: zero=multiplication(antidomain(X0), X0)).
286.14/286.49	  fof(domain2, axiom,
286.14/286.49	      ![X0, X1]:
286.14/286.49	        antidomain(multiplication(X0,
286.14/286.49	                                  antidomain(antidomain(X1))))=addition(antidomain(multiplication(X0,
286.14/286.49	                                                                                                  X1)),
286.14/286.49	                                                                        antidomain(multiplication(X0,
286.14/286.49	                                                                                                  antidomain(antidomain(X1)))))).
286.14/286.49	  fof(domain3, axiom,
286.14/286.49	      ![X0]: one=addition(antidomain(antidomain(X0)), antidomain(X0))).
286.14/286.49	  fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))).
286.14/286.49	  fof(domain_difference, axiom,
286.14/286.49	      ![X0, X1]:
286.14/286.49	        domain_difference(X0, X1)=multiplication(domain(X0),
286.14/286.49	                                                 antidomain(X1))).
286.14/286.49	  fof(forward_box, axiom,
286.14/286.49	      ![X0, X1]: forward_box(X0, X1)=c(forward_diamond(X0, c(X1)))).
286.14/286.49	  fof(forward_diamond, axiom,
286.14/286.49	      ![X0, X1]:
286.14/286.49	        forward_diamond(X0, X1)=domain(multiplication(X0, domain(X1)))).
286.14/286.49	  fof(goals, conjecture,
286.14/286.49	      ![X0, X1, X2]:
286.14/286.49	        (zero=multiplication(forward_diamond(X0, domain(X1)), domain(X2))
286.14/286.50	           => zero=multiplication(domain(X1),
286.14/286.50	                                  backward_diamond(X0, domain(X2))))).
286.14/286.50	  fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)).
286.14/286.50	  fof(left_distributivity, axiom,
286.14/286.50	      ![A, B, C]:
286.14/286.50	        addition(multiplication(A, C),
286.14/286.50	                 multiplication(B, C))=multiplication(addition(A, B), C)).
286.14/286.50	  fof(multiplicative_associativity, axiom,
286.14/286.50	      ![A, B, C]:
286.14/286.50	        multiplication(multiplication(A, B), C)=multiplication(A,
286.14/286.50	                                                               multiplication(B, C))).
286.14/286.50	  fof(multiplicative_left_identity, axiom,
286.14/286.50	      ![A]: A=multiplication(one, A)).
286.14/286.50	  fof(multiplicative_right_identity, axiom,
286.14/286.50	      ![A]: A=multiplication(A, one)).
286.14/286.50	  fof(order, axiom, ![A, B]: (leq(A, B) <=> B=addition(A, B))).
286.14/286.50	  fof(right_annihilation, axiom, ![A]: multiplication(A, zero)=zero).
286.14/286.50	  fof(right_distributivity, axiom,
286.14/286.50	      ![A, B, C]:
286.14/286.50	        addition(multiplication(A, B),
286.14/286.50	                 multiplication(A, C))=multiplication(A, addition(B, C))).
286.14/286.50	
286.14/286.50	Now clausify the problem and encode Horn clauses using encoding 3 of
286.14/286.50	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
286.14/286.50	We repeatedly replace C & s=t => u=v by the two clauses:
286.14/286.50	  $$fresh(y, y, x1...xn) = u
286.14/286.50	  C => $$fresh(s, t, x1...xn) = v
286.14/286.50	where $$fresh is a fresh function symbol and x1..xn are the free
286.14/286.50	variables of u and v.
286.14/286.50	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
286.14/286.50	input problem has no model of domain size 1).
286.14/286.50	
286.14/286.50	The encoding turns the above axioms into the following unit equations and goals:
286.14/286.50	
286.14/286.50	Axiom 1 (order): $$fresh2(X, X, Y, Z) = $$true.
286.14/286.50	Axiom 2 (order_1): $$fresh(X, X, Y, Z) = Z.
286.14/286.50	Axiom 3 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
286.14/286.50	Axiom 4 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
286.14/286.50	Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
286.14/286.50	Axiom 6 (multiplicative_left_identity): X = multiplication(one, X).
286.14/286.50	Axiom 7 (multiplicative_right_identity): X = multiplication(X, one).
286.14/286.50	Axiom 8 (left_annihilation): zero = multiplication(zero, X).
286.14/286.50	Axiom 9 (additive_identity): addition(X, zero) = X.
286.14/286.50	Axiom 10 (right_annihilation): multiplication(X, zero) = zero.
286.14/286.50	Axiom 11 (multiplicative_associativity): multiplication(multiplication(X, Y), Z) = multiplication(X, multiplication(Y, Z)).
286.14/286.50	Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
286.14/286.50	Axiom 13 (order_1): $$fresh(leq(X, Y), $$true, X, Y) = addition(X, Y).
286.14/286.50	Axiom 14 (order): $$fresh2(X, addition(Y, X), Y, X) = leq(Y, X).
286.14/286.50	Axiom 15 (additive_idempotence): X = addition(X, X).
286.14/286.50	Axiom 16 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
286.14/286.50	Axiom 17 (codomain3): one = addition(coantidomain(coantidomain(X)), coantidomain(X)).
286.14/286.50	Axiom 18 (domain2): antidomain(multiplication(X, antidomain(antidomain(Y)))) = addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))).
286.14/286.50	Axiom 19 (codomain1): multiplication(X, coantidomain(X)) = zero.
286.14/286.50	Axiom 20 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
286.14/286.50	Axiom 21 (domain1): zero = multiplication(antidomain(X), X).
286.14/286.50	Axiom 22 (domain3): one = addition(antidomain(antidomain(X)), antidomain(X)).
286.14/286.50	Axiom 23 (domain4): domain(X) = antidomain(antidomain(X)).
286.14/286.50	Axiom 24 (backward_diamond): codomain(multiplication(codomain(X), Y)) = backward_diamond(Y, X).
286.14/286.50	Axiom 25 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
286.14/286.50	Axiom 26 (backward_box): c(backward_diamond(X, c(Y))) = backward_box(X, Y).
286.14/286.50	Axiom 27 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
286.14/286.50	Axiom 28 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
286.14/286.50	Axiom 29 (complement): antidomain(domain(X)) = c(X).
286.14/286.50	Axiom 30 (goals): zero = multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)).
286.14/286.50	
286.14/286.50	Lemma 31: addition(coantidomain(X), coantidomain(coantidomain(X))) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(coantidomain(X), coantidomain(coantidomain(X)))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(coantidomain(coantidomain(X)), coantidomain(X))
286.14/286.50	= { by axiom 17 (codomain3) }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 32: addition(antidomain(X), antidomain(antidomain(X))) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(antidomain(X), antidomain(antidomain(X)))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(antidomain(antidomain(X)), antidomain(X))
286.14/286.50	= { by axiom 22 (domain3) }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 33: coantidomain(one) = zero.
286.14/286.50	Proof:
286.14/286.50	  coantidomain(one)
286.14/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.14/286.50	  multiplication(one, coantidomain(one))
286.14/286.50	= { by axiom 19 (codomain1) }
286.14/286.50	  zero
286.14/286.50	
286.14/286.50	Lemma 34: antidomain(one) = zero.
286.14/286.50	Proof:
286.14/286.50	  antidomain(one)
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  multiplication(antidomain(one), one)
286.14/286.50	= { by axiom 21 (domain1) }
286.14/286.50	  zero
286.14/286.50	
286.14/286.50	Lemma 35: addition(zero, X) = X.
286.14/286.50	Proof:
286.14/286.50	  addition(zero, X)
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(X, zero)
286.14/286.50	= { by axiom 9 (additive_identity) }
286.14/286.50	  X
286.14/286.50	
286.14/286.50	Lemma 36: codomain(one) = coantidomain(zero).
286.14/286.50	Proof:
286.14/286.50	  codomain(one)
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  coantidomain(coantidomain(one))
286.14/286.50	= { by lemma 33 }
286.14/286.50	  coantidomain(zero)
286.14/286.50	
286.14/286.50	Lemma 37: domain(one) = antidomain(zero).
286.14/286.50	Proof:
286.14/286.50	  domain(one)
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  antidomain(antidomain(one))
286.14/286.50	= { by lemma 34 }
286.14/286.50	  antidomain(zero)
286.14/286.50	
286.14/286.50	Lemma 38: codomain(coantidomain(X)) = coantidomain(codomain(X)).
286.14/286.50	Proof:
286.14/286.50	  codomain(coantidomain(X))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  coantidomain(coantidomain(coantidomain(X)))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  coantidomain(codomain(X))
286.14/286.50	
286.14/286.50	Lemma 39: domain(antidomain(X)) = c(X).
286.14/286.50	Proof:
286.14/286.50	  domain(antidomain(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  antidomain(antidomain(antidomain(X)))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  antidomain(domain(X))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  c(X)
286.14/286.50	
286.14/286.50	Lemma 40: backward_diamond(one, X) = codomain(codomain(X)).
286.14/286.50	Proof:
286.14/286.50	  backward_diamond(one, X)
286.14/286.50	= { by axiom 24 (backward_diamond) }
286.14/286.50	  codomain(multiplication(codomain(X), one))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  codomain(codomain(X))
286.14/286.50	
286.14/286.50	Lemma 41: forward_diamond(one, X) = domain(domain(X)).
286.14/286.50	Proof:
286.14/286.50	  forward_diamond(one, X)
286.14/286.50	= { by axiom 27 (forward_diamond) }
286.14/286.50	  domain(multiplication(one, domain(X)))
286.14/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.14/286.50	  domain(domain(X))
286.14/286.50	
286.14/286.50	Lemma 42: domain(domain(X)) = antidomain(c(X)).
286.14/286.50	Proof:
286.14/286.50	  domain(domain(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  antidomain(antidomain(domain(X)))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  antidomain(c(X))
286.14/286.50	
286.14/286.50	Lemma 43: c(domain(X)) = domain(c(X)).
286.14/286.50	Proof:
286.14/286.50	  c(domain(X))
286.14/286.50	= { by lemma 39 }
286.14/286.50	  domain(antidomain(domain(X)))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  domain(c(X))
286.14/286.50	
286.14/286.50	Lemma 44: addition(coantidomain(X), codomain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(coantidomain(X), codomain(X))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  addition(coantidomain(X), coantidomain(coantidomain(X)))
286.14/286.50	= { by lemma 31 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 45: coantidomain(zero) = one.
286.14/286.50	Proof:
286.14/286.50	  coantidomain(zero)
286.14/286.50	= { by lemma 36 }
286.14/286.50	  codomain(one)
286.14/286.50	= { by lemma 35 }
286.14/286.50	  addition(zero, codomain(one))
286.14/286.50	= { by lemma 33 }
286.14/286.50	  addition(coantidomain(one), codomain(one))
286.14/286.50	= { by lemma 44 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 46: domain_difference(X, X) = zero.
286.14/286.50	Proof:
286.14/286.50	  domain_difference(X, X)
286.14/286.50	= { by axiom 28 (domain_difference) }
286.14/286.50	  multiplication(domain(X), antidomain(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  multiplication(antidomain(antidomain(X)), antidomain(X))
286.14/286.50	= { by axiom 21 (domain1) }
286.14/286.50	  zero
286.14/286.50	
286.14/286.50	Lemma 47: addition(antidomain(X), domain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(antidomain(X), domain(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  addition(antidomain(X), antidomain(antidomain(X)))
286.14/286.50	= { by lemma 32 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 48: antidomain(zero) = one.
286.14/286.50	Proof:
286.14/286.50	  antidomain(zero)
286.14/286.50	= { by lemma 37 }
286.14/286.50	  domain(one)
286.14/286.50	= { by lemma 35 }
286.14/286.50	  addition(zero, domain(one))
286.14/286.50	= { by lemma 34 }
286.14/286.50	  addition(antidomain(one), domain(one))
286.14/286.50	= { by lemma 47 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 49: c(zero) = one.
286.14/286.50	Proof:
286.14/286.50	  c(zero)
286.14/286.50	= { by lemma 39 }
286.14/286.50	  domain(antidomain(zero))
286.14/286.50	= { by lemma 48 }
286.14/286.50	  domain(one)
286.14/286.50	= { by lemma 37 }
286.14/286.50	  antidomain(zero)
286.14/286.50	= { by lemma 48 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 50: domain_difference(X, zero) = domain(X).
286.14/286.50	Proof:
286.14/286.50	  domain_difference(X, zero)
286.14/286.50	= { by axiom 28 (domain_difference) }
286.14/286.50	  multiplication(domain(X), antidomain(zero))
286.14/286.50	= { by lemma 48 }
286.14/286.50	  multiplication(domain(X), one)
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  domain(X)
286.14/286.50	
286.14/286.50	Lemma 51: addition(X, addition(X, Y)) = addition(X, Y).
286.14/286.50	Proof:
286.14/286.50	  addition(X, addition(X, Y))
286.14/286.50	= { by axiom 12 (additive_associativity) }
286.14/286.50	  addition(addition(X, X), Y)
286.14/286.50	= { by axiom 15 (additive_idempotence) }
286.14/286.50	  addition(X, Y)
286.14/286.50	
286.14/286.50	Lemma 52: addition(one, coantidomain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(one, coantidomain(X))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(coantidomain(X), one)
286.14/286.50	= { by lemma 31 }
286.14/286.50	  addition(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
286.14/286.50	= { by lemma 51 }
286.14/286.50	  addition(coantidomain(X), coantidomain(coantidomain(X)))
286.14/286.50	= { by lemma 31 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 53: addition(one, antidomain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(one, antidomain(X))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(antidomain(X), one)
286.14/286.50	= { by lemma 32 }
286.14/286.50	  addition(antidomain(X), addition(antidomain(X), antidomain(antidomain(X))))
286.14/286.50	= { by lemma 51 }
286.14/286.50	  addition(antidomain(X), antidomain(antidomain(X)))
286.14/286.50	= { by lemma 32 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 54: addition(one, codomain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(one, codomain(X))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  addition(one, coantidomain(coantidomain(X)))
286.14/286.50	= { by lemma 52 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 55: addition(one, domain(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(one, domain(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  addition(one, antidomain(antidomain(X)))
286.14/286.50	= { by lemma 53 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 56: addition(domain(X), c(X)) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(domain(X), c(X))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  addition(antidomain(antidomain(X)), c(X))
286.14/286.50	= { by lemma 39 }
286.14/286.50	  addition(antidomain(antidomain(X)), domain(antidomain(X)))
286.14/286.50	= { by lemma 47 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 57: multiplication(antidomain(X), multiplication(X, Y)) = zero.
286.14/286.50	Proof:
286.14/286.50	  multiplication(antidomain(X), multiplication(X, Y))
286.14/286.50	= { by axiom 11 (multiplicative_associativity) }
286.14/286.50	  multiplication(multiplication(antidomain(X), X), Y)
286.14/286.50	= { by axiom 21 (domain1) }
286.14/286.50	  multiplication(zero, Y)
286.14/286.50	= { by axiom 8 (left_annihilation) }
286.14/286.50	  zero
286.14/286.50	
286.14/286.50	Lemma 58: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
286.14/286.50	Proof:
286.14/286.50	  multiplication(domain(X), domain(Y))
286.14/286.50	= { by axiom 23 (domain4) }
286.14/286.50	  multiplication(domain(X), antidomain(antidomain(Y)))
286.14/286.50	= { by axiom 28 (domain_difference) }
286.14/286.50	  domain_difference(X, antidomain(Y))
286.14/286.50	
286.14/286.50	Lemma 59: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
286.14/286.50	Proof:
286.14/286.50	  c(multiplication(X, domain(Y)))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  antidomain(domain(multiplication(X, domain(Y))))
286.14/286.50	= { by axiom 27 (forward_diamond) }
286.14/286.50	  antidomain(forward_diamond(X, Y))
286.14/286.50	
286.14/286.50	Lemma 60: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
286.14/286.50	Proof:
286.14/286.50	  multiplication(domain(X), c(Y))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  multiplication(domain(X), antidomain(domain(Y)))
286.14/286.50	= { by axiom 28 (domain_difference) }
286.14/286.50	  domain_difference(X, domain(Y))
286.14/286.50	
286.14/286.50	Lemma 61: addition(codomain(X), coantidomain(codomain(X))) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(codomain(X), coantidomain(codomain(X)))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  addition(coantidomain(coantidomain(X)), coantidomain(codomain(X)))
286.14/286.50	= { by lemma 38 }
286.14/286.50	  addition(coantidomain(coantidomain(X)), codomain(coantidomain(X)))
286.14/286.50	= { by lemma 44 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 62: addition(c(X), antidomain(c(X))) = one.
286.14/286.50	Proof:
286.14/286.50	  addition(c(X), antidomain(c(X)))
286.14/286.50	= { by axiom 29 (complement) }
286.14/286.50	  addition(antidomain(domain(X)), antidomain(c(X)))
286.14/286.50	= { by lemma 42 }
286.14/286.50	  addition(antidomain(domain(X)), domain(domain(X)))
286.14/286.50	= { by lemma 47 }
286.14/286.50	  one
286.14/286.50	
286.14/286.50	Lemma 64: addition(X, addition(Y, Z)) = addition(Z, addition(X, Y)).
286.14/286.50	Proof:
286.14/286.50	  addition(X, addition(Y, Z))
286.14/286.50	= { by axiom 12 (additive_associativity) }
286.14/286.50	  addition(addition(X, Y), Z)
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(Z, addition(X, Y))
286.14/286.50	
286.14/286.50	Lemma 64: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
286.14/286.50	Proof:
286.14/286.50	  addition(Z, addition(X, Y))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(addition(X, Y), Z)
286.14/286.50	= { by axiom 12 (additive_associativity) }
286.14/286.50	  addition(X, addition(Y, Z))
286.14/286.50	
286.14/286.50	Lemma 65: addition(X, addition(Y, Z)) = addition(Y, addition(X, Z)).
286.14/286.50	Proof:
286.14/286.50	  addition(X, addition(Y, Z))
286.14/286.50	= { by axiom 12 (additive_associativity) }
286.14/286.50	  addition(addition(X, Y), Z)
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  addition(addition(Y, X), Z)
286.14/286.50	= { by axiom 12 (additive_associativity) }
286.14/286.50	  addition(Y, addition(X, Z))
286.14/286.50	
286.14/286.50	Lemma 66: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y).
286.14/286.50	Proof:
286.14/286.50	  addition(Y, multiplication(X, Y))
286.14/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.14/286.50	  addition(multiplication(one, Y), multiplication(X, Y))
286.14/286.50	= { by axiom 4 (left_distributivity) }
286.14/286.50	  multiplication(addition(one, X), Y)
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  multiplication(addition(X, one), Y)
286.14/286.50	
286.14/286.50	Lemma 67: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
286.14/286.50	Proof:
286.14/286.50	  addition(X, multiplication(X, Y))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  addition(multiplication(X, one), multiplication(X, Y))
286.14/286.50	= { by axiom 3 (right_distributivity) }
286.14/286.50	  multiplication(X, addition(one, Y))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  multiplication(X, addition(Y, one))
286.14/286.50	
286.14/286.50	Lemma 68: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
286.14/286.50	Proof:
286.14/286.50	  multiplication(X, addition(Y, coantidomain(X)))
286.14/286.50	= { by axiom 3 (right_distributivity) }
286.14/286.50	  addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
286.14/286.50	= { by axiom 19 (codomain1) }
286.14/286.50	  addition(multiplication(X, Y), zero)
286.14/286.50	= { by axiom 9 (additive_identity) }
286.14/286.50	  multiplication(X, Y)
286.14/286.50	
286.14/286.50	Lemma 69: multiplication(coantidomain(X), coantidomain(X)) = coantidomain(X).
286.14/286.50	Proof:
286.14/286.50	  multiplication(coantidomain(X), coantidomain(X))
286.14/286.50	= { by lemma 68 }
286.14/286.50	  multiplication(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
286.14/286.50	= { by lemma 31 }
286.14/286.50	  multiplication(coantidomain(X), one)
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  coantidomain(X)
286.14/286.50	
286.14/286.50	Lemma 70: multiplication(codomain(X), codomain(X)) = codomain(X).
286.14/286.50	Proof:
286.14/286.50	  multiplication(codomain(X), codomain(X))
286.14/286.50	= { by lemma 68 }
286.14/286.50	  multiplication(codomain(X), addition(codomain(X), coantidomain(codomain(X))))
286.14/286.50	= { by lemma 61 }
286.14/286.50	  multiplication(codomain(X), one)
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  codomain(X)
286.14/286.50	
286.14/286.50	Lemma 71: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
286.14/286.50	Proof:
286.14/286.50	  multiplication(addition(X, antidomain(Y)), Y)
286.14/286.50	= { by axiom 4 (left_distributivity) }
286.14/286.50	  addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
286.14/286.50	= { by axiom 21 (domain1) }
286.14/286.50	  addition(multiplication(X, Y), zero)
286.14/286.50	= { by axiom 9 (additive_identity) }
286.14/286.50	  multiplication(X, Y)
286.14/286.50	
286.14/286.50	Lemma 72: multiplication(antidomain(X), antidomain(X)) = antidomain(X).
286.14/286.50	Proof:
286.14/286.50	  multiplication(antidomain(X), antidomain(X))
286.14/286.50	= { by lemma 71 }
286.14/286.50	  multiplication(addition(antidomain(X), antidomain(antidomain(X))), antidomain(X))
286.14/286.50	= { by lemma 32 }
286.14/286.50	  multiplication(one, antidomain(X))
286.14/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.14/286.50	  antidomain(X)
286.14/286.50	
286.14/286.50	Lemma 73: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
286.14/286.50	Proof:
286.14/286.50	  domain(multiplication(X, c(Y)))
286.14/286.50	= { by lemma 39 }
286.14/286.50	  domain(multiplication(X, domain(antidomain(Y))))
286.14/286.50	= { by axiom 27 (forward_diamond) }
286.14/286.50	  forward_diamond(X, antidomain(Y))
286.14/286.50	
286.14/286.50	Lemma 74: forward_diamond(X, one) = domain(X).
286.14/286.50	Proof:
286.14/286.50	  forward_diamond(X, one)
286.14/286.50	= { by lemma 48 }
286.14/286.50	  forward_diamond(X, antidomain(zero))
286.14/286.50	= { by lemma 73 }
286.14/286.50	  domain(multiplication(X, c(zero)))
286.14/286.50	= { by lemma 49 }
286.14/286.50	  domain(multiplication(X, one))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  domain(X)
286.14/286.50	
286.14/286.50	Lemma 75: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
286.14/286.50	Proof:
286.14/286.50	  multiplication(c(X), antidomain(Y))
286.14/286.50	= { by lemma 39 }
286.14/286.50	  multiplication(domain(antidomain(X)), antidomain(Y))
286.14/286.50	= { by axiom 28 (domain_difference) }
286.14/286.50	  domain_difference(antidomain(X), Y)
286.14/286.50	
286.14/286.50	Lemma 76: domain_difference(one, X) = antidomain(X).
286.14/286.50	Proof:
286.14/286.50	  domain_difference(one, X)
286.14/286.50	= { by lemma 48 }
286.14/286.50	  domain_difference(antidomain(zero), X)
286.14/286.50	= { by lemma 75 }
286.14/286.50	  multiplication(c(zero), antidomain(X))
286.14/286.50	= { by lemma 49 }
286.14/286.50	  multiplication(one, antidomain(X))
286.14/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.14/286.50	  antidomain(X)
286.14/286.50	
286.14/286.50	Lemma 77: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y).
286.14/286.50	Proof:
286.14/286.50	  domain(domain_difference(X, antidomain(Y)))
286.14/286.50	= { by lemma 58 }
286.14/286.50	  domain(multiplication(domain(X), domain(Y)))
286.14/286.50	= { by axiom 27 (forward_diamond) }
286.14/286.50	  forward_diamond(domain(X), Y)
286.14/286.50	
286.14/286.50	Lemma 78: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
286.14/286.50	Proof:
286.14/286.50	  multiplication(addition(X, Y), coantidomain(X))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  multiplication(addition(Y, X), coantidomain(X))
286.14/286.50	= { by axiom 4 (left_distributivity) }
286.14/286.50	  addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
286.14/286.50	= { by axiom 19 (codomain1) }
286.14/286.50	  addition(multiplication(Y, coantidomain(X)), zero)
286.14/286.50	= { by axiom 9 (additive_identity) }
286.14/286.50	  multiplication(Y, coantidomain(X))
286.14/286.50	
286.14/286.50	Lemma 79: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
286.14/286.50	Proof:
286.14/286.50	  multiplication(antidomain(X), addition(X, Y))
286.14/286.50	= { by axiom 5 (additive_commutativity) }
286.14/286.50	  multiplication(antidomain(X), addition(Y, X))
286.14/286.50	= { by axiom 3 (right_distributivity) }
286.14/286.50	  addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
286.14/286.50	= { by axiom 21 (domain1) }
286.14/286.50	  addition(multiplication(antidomain(X), Y), zero)
286.14/286.50	= { by axiom 9 (additive_identity) }
286.14/286.50	  multiplication(antidomain(X), Y)
286.14/286.50	
286.14/286.50	Lemma 80: multiplication(codomain(X), coantidomain(X)) = zero.
286.14/286.50	Proof:
286.14/286.50	  multiplication(codomain(X), coantidomain(X))
286.14/286.50	= { by lemma 68 }
286.14/286.50	  multiplication(codomain(X), addition(coantidomain(X), coantidomain(codomain(X))))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(codomain(X))))
286.14/286.50	= { by lemma 38 }
286.14/286.50	  multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), codomain(coantidomain(X))))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(coantidomain(coantidomain(X)))))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  multiplication(codomain(X), addition(coantidomain(multiplication(X, one)), coantidomain(multiplication(coantidomain(coantidomain(X)), one))))
286.14/286.50	= { by axiom 16 (codomain2) }
286.14/286.50	  multiplication(codomain(X), coantidomain(multiplication(coantidomain(coantidomain(X)), one)))
286.14/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.14/286.50	  multiplication(codomain(X), coantidomain(coantidomain(coantidomain(X))))
286.14/286.50	= { by axiom 20 (codomain4) }
286.14/286.50	  multiplication(codomain(X), codomain(coantidomain(X)))
286.14/286.50	= { by lemma 38 }
286.14/286.50	  multiplication(codomain(X), coantidomain(codomain(X)))
286.14/286.50	= { by axiom 19 (codomain1) }
286.14/286.50	  zero
286.14/286.50	
286.14/286.50	Lemma 81: multiplication(X, codomain(X)) = X.
286.14/286.50	Proof:
286.14/286.50	  multiplication(X, codomain(X))
286.14/286.50	= { by axiom 20 (codomain4) }
286.24/286.50	  multiplication(X, coantidomain(coantidomain(X)))
286.24/286.50	= { by lemma 68 }
286.24/286.50	  multiplication(X, addition(coantidomain(coantidomain(X)), coantidomain(X)))
286.24/286.50	= { by axiom 5 (additive_commutativity) }
286.24/286.50	  multiplication(X, addition(coantidomain(X), coantidomain(coantidomain(X))))
286.24/286.50	= { by lemma 31 }
286.24/286.50	  multiplication(X, one)
286.24/286.50	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.50	  X
286.24/286.50	
286.24/286.50	Lemma 82: multiplication(addition(antidomain(Y), X), Y) = multiplication(X, Y).
286.24/286.50	Proof:
286.24/286.50	  multiplication(addition(antidomain(Y), X), Y)
286.24/286.50	= { by axiom 5 (additive_commutativity) }
286.24/286.50	  multiplication(addition(X, antidomain(Y)), Y)
286.24/286.50	= { by lemma 71 }
286.24/286.50	  multiplication(X, Y)
286.24/286.50	
286.24/286.50	Lemma 83: multiplication(domain(X), X) = X.
286.24/286.50	Proof:
286.24/286.50	  multiplication(domain(X), X)
286.24/286.50	= { by axiom 23 (domain4) }
286.24/286.50	  multiplication(antidomain(antidomain(X)), X)
286.24/286.50	= { by lemma 82 }
286.24/286.50	  multiplication(addition(antidomain(X), antidomain(antidomain(X))), X)
286.24/286.50	= { by lemma 32 }
286.24/286.50	  multiplication(one, X)
286.24/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.50	  X
286.24/286.50	
286.24/286.50	Lemma 84: domain_difference(antidomain(X), X) = antidomain(X).
286.24/286.50	Proof:
286.24/286.50	  domain_difference(antidomain(X), X)
286.24/286.50	= { by axiom 28 (domain_difference) }
286.24/286.50	  multiplication(domain(antidomain(X)), antidomain(X))
286.24/286.50	= { by lemma 83 }
286.24/286.50	  antidomain(X)
286.24/286.50	
286.24/286.50	Lemma 85: backward_diamond(X, one) = codomain(X).
286.24/286.50	Proof:
286.24/286.50	  backward_diamond(X, one)
286.24/286.50	= { by axiom 24 (backward_diamond) }
286.24/286.50	  codomain(multiplication(codomain(one), X))
286.24/286.50	= { by lemma 36 }
286.24/286.50	  codomain(multiplication(coantidomain(zero), X))
286.24/286.50	= { by lemma 45 }
286.24/286.50	  codomain(multiplication(one, X))
286.24/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.50	  codomain(X)
286.24/286.50	
286.24/286.50	Lemma 86: forward_box(X, backward_diamond(Y, c(Z))) = c(forward_diamond(X, backward_box(Y, Z))).
286.24/286.50	Proof:
286.24/286.50	  forward_box(X, backward_diamond(Y, c(Z)))
286.24/286.50	= { by axiom 25 (forward_box) }
286.24/286.50	  c(forward_diamond(X, c(backward_diamond(Y, c(Z)))))
286.24/286.50	= { by axiom 26 (backward_box) }
286.24/286.50	  c(forward_diamond(X, backward_box(Y, Z)))
286.24/286.50	
286.24/286.50	Lemma 87: domain_difference(multiplication(X, domain(Y)), Z) = multiplication(forward_diamond(X, Y), antidomain(Z)).
286.24/286.50	Proof:
286.24/286.50	  domain_difference(multiplication(X, domain(Y)), Z)
286.24/286.50	= { by axiom 28 (domain_difference) }
286.24/286.50	  multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
286.24/286.50	= { by axiom 27 (forward_diamond) }
286.24/286.50	  multiplication(forward_diamond(X, Y), antidomain(Z))
286.24/286.50	
286.24/286.50	Lemma 88: multiplication(domain(X), coantidomain(antidomain(X))) = coantidomain(antidomain(X)).
286.24/286.50	Proof:
286.24/286.50	  multiplication(domain(X), coantidomain(antidomain(X)))
286.24/286.50	= { by axiom 23 (domain4) }
286.24/286.50	  multiplication(antidomain(antidomain(X)), coantidomain(antidomain(X)))
286.24/286.50	= { by lemma 78 }
286.24/286.50	  multiplication(addition(antidomain(X), antidomain(antidomain(X))), coantidomain(antidomain(X)))
286.24/286.50	= { by lemma 32 }
286.24/286.50	  multiplication(one, coantidomain(antidomain(X)))
286.24/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.50	  coantidomain(antidomain(X))
286.24/286.50	
286.24/286.50	Lemma 89: addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))) = one.
286.24/286.50	Proof:
286.24/286.50	  addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y)))
286.24/286.50	= { by axiom 27 (forward_diamond) }
286.24/286.50	  addition(domain(multiplication(X, domain(Y))), antidomain(forward_diamond(X, Y)))
286.24/286.50	= { by lemma 59 }
286.24/286.50	  addition(domain(multiplication(X, domain(Y))), c(multiplication(X, domain(Y))))
286.24/286.50	= { by lemma 56 }
286.24/286.50	  one
286.24/286.50	
286.24/286.50	Lemma 90: addition(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y))) = one.
286.24/286.50	Proof:
286.24/286.50	  addition(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y)))
286.24/286.50	= { by axiom 24 (backward_diamond) }
286.24/286.50	  addition(codomain(multiplication(codomain(Y), X)), coantidomain(backward_diamond(X, Y)))
286.24/286.50	= { by axiom 24 (backward_diamond) }
286.24/286.50	  addition(codomain(multiplication(codomain(Y), X)), coantidomain(codomain(multiplication(codomain(Y), X))))
286.24/286.50	= { by lemma 61 }
286.24/286.50	  one
286.24/286.50	
286.24/286.50	Lemma 91: addition(backward_box(X, Y), antidomain(backward_box(X, Y))) = one.
286.24/286.50	Proof:
286.24/286.50	  addition(backward_box(X, Y), antidomain(backward_box(X, Y)))
286.24/286.50	= { by axiom 26 (backward_box) }
286.24/286.50	  addition(c(backward_diamond(X, c(Y))), antidomain(backward_box(X, Y)))
286.24/286.50	= { by axiom 26 (backward_box) }
286.24/286.50	  addition(c(backward_diamond(X, c(Y))), antidomain(c(backward_diamond(X, c(Y)))))
286.24/286.50	= { by lemma 62 }
286.24/286.50	  one
286.24/286.50	
286.24/286.50	Lemma 92: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
286.24/286.50	Proof:
286.24/286.50	  multiplication(addition(X, Y), coantidomain(Y))
286.24/286.50	= { by axiom 5 (additive_commutativity) }
286.24/286.50	  multiplication(addition(Y, X), coantidomain(Y))
286.24/286.50	= { by lemma 78 }
286.24/286.50	  multiplication(X, coantidomain(Y))
286.24/286.50	
286.24/286.50	Lemma 93: coantidomain(codomain(X)) = coantidomain(X).
286.24/286.50	Proof:
286.24/286.50	  coantidomain(codomain(X))
286.24/286.50	= { by lemma 38 }
286.24/286.50	  codomain(coantidomain(X))
286.24/286.50	= { by axiom 20 (codomain4) }
286.24/286.50	  coantidomain(coantidomain(coantidomain(X)))
286.24/286.50	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.50	  multiplication(one, coantidomain(coantidomain(coantidomain(X))))
286.24/286.50	= { by lemma 31 }
286.24/286.50	  multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(X))))
286.24/286.50	= { by lemma 92 }
286.24/286.50	  multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X))))
286.24/286.50	= { by axiom 20 (codomain4) }
286.24/286.50	  multiplication(coantidomain(X), codomain(coantidomain(X)))
286.24/286.50	= { by lemma 81 }
286.24/286.51	  coantidomain(X)
286.24/286.51	
286.24/286.51	Lemma 94: codomain(codomain(X)) = codomain(X).
286.24/286.51	Proof:
286.24/286.51	  codomain(codomain(X))
286.24/286.51	= { by axiom 20 (codomain4) }
286.24/286.51	  coantidomain(coantidomain(codomain(X)))
286.24/286.51	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.51	  multiplication(one, coantidomain(coantidomain(codomain(X))))
286.24/286.51	= { by lemma 61 }
286.24/286.51	  multiplication(addition(codomain(X), coantidomain(codomain(X))), coantidomain(coantidomain(codomain(X))))
286.24/286.51	= { by lemma 92 }
286.24/286.51	  multiplication(codomain(X), coantidomain(coantidomain(codomain(X))))
286.24/286.51	= { by axiom 20 (codomain4) }
286.24/286.51	  multiplication(codomain(X), codomain(codomain(X)))
286.24/286.51	= { by lemma 81 }
286.24/286.51	  codomain(X)
286.24/286.51	
286.24/286.51	Lemma 95: backward_diamond(X, codomain(Y)) = backward_diamond(X, Y).
286.24/286.51	Proof:
286.24/286.51	  backward_diamond(X, codomain(Y))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  codomain(multiplication(codomain(codomain(Y)), X))
286.24/286.51	= { by lemma 94 }
286.24/286.51	  codomain(multiplication(codomain(Y), X))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  backward_diamond(X, Y)
286.24/286.51	
286.24/286.51	Lemma 96: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(antidomain(X), addition(Y, X))
286.24/286.51	= { by axiom 5 (additive_commutativity) }
286.24/286.51	  multiplication(antidomain(X), addition(X, Y))
286.24/286.51	= { by lemma 79 }
286.24/286.51	  multiplication(antidomain(X), Y)
286.24/286.51	
286.24/286.51	Lemma 97: c(X) = antidomain(X).
286.24/286.51	Proof:
286.24/286.51	  c(X)
286.24/286.51	= { by lemma 39 }
286.24/286.51	  domain(antidomain(X))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(antidomain(antidomain(X)))
286.24/286.51	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.51	  multiplication(antidomain(antidomain(antidomain(X))), one)
286.24/286.51	= { by lemma 32 }
286.24/286.51	  multiplication(antidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))
286.24/286.51	= { by lemma 96 }
286.24/286.51	  multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  multiplication(domain(antidomain(X)), antidomain(X))
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  domain_difference(antidomain(X), X)
286.24/286.51	= { by lemma 84 }
286.24/286.51	  antidomain(X)
286.24/286.51	
286.24/286.51	Lemma 98: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
286.24/286.51	Proof:
286.24/286.51	  domain(forward_diamond(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(antidomain(forward_diamond(X, Y)))
286.24/286.51	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.51	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), one)
286.24/286.51	= { by lemma 89 }
286.24/286.51	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))))
286.24/286.51	= { by lemma 96 }
286.24/286.51	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  multiplication(domain(forward_diamond(X, Y)), forward_diamond(X, Y))
286.24/286.51	= { by lemma 83 }
286.24/286.51	  forward_diamond(X, Y)
286.24/286.51	
286.24/286.51	Lemma 99: domain(forward_box(X, Y)) = forward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  domain(forward_box(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(antidomain(forward_box(X, Y)))
286.24/286.51	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.51	  multiplication(antidomain(antidomain(forward_box(X, Y))), one)
286.24/286.51	= { by lemma 62 }
286.24/286.51	  multiplication(antidomain(antidomain(forward_box(X, Y))), addition(c(forward_diamond(X, c(Y))), antidomain(c(forward_diamond(X, c(Y))))))
286.24/286.51	= { by axiom 25 (forward_box) }
286.24/286.51	  multiplication(antidomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), antidomain(c(forward_diamond(X, c(Y))))))
286.24/286.51	= { by axiom 25 (forward_box) }
286.24/286.51	  multiplication(antidomain(antidomain(forward_box(X, Y))), addition(forward_box(X, Y), antidomain(forward_box(X, Y))))
286.24/286.51	= { by lemma 96 }
286.24/286.51	  multiplication(antidomain(antidomain(forward_box(X, Y))), forward_box(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  multiplication(domain(forward_box(X, Y)), forward_box(X, Y))
286.24/286.51	= { by lemma 83 }
286.24/286.51	  forward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 100: domain(backward_box(X, Y)) = backward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  domain(backward_box(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(antidomain(backward_box(X, Y)))
286.24/286.51	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.51	  multiplication(antidomain(antidomain(backward_box(X, Y))), one)
286.24/286.51	= { by lemma 91 }
286.24/286.51	  multiplication(antidomain(antidomain(backward_box(X, Y))), addition(backward_box(X, Y), antidomain(backward_box(X, Y))))
286.24/286.51	= { by lemma 96 }
286.24/286.51	  multiplication(antidomain(antidomain(backward_box(X, Y))), backward_box(X, Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  multiplication(domain(backward_box(X, Y)), backward_box(X, Y))
286.24/286.51	= { by lemma 83 }
286.24/286.51	  backward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 101: domain_difference(X, domain(Y)) = domain_difference(X, Y).
286.24/286.51	Proof:
286.24/286.51	  domain_difference(X, domain(Y))
286.24/286.51	= { by lemma 60 }
286.24/286.51	  multiplication(domain(X), c(Y))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  multiplication(domain(X), antidomain(Y))
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  domain_difference(X, Y)
286.24/286.51	
286.24/286.51	Lemma 102: codomain(multiplication(coantidomain(Y), X)) = backward_diamond(X, coantidomain(Y)).
286.24/286.51	Proof:
286.24/286.51	  codomain(multiplication(coantidomain(Y), X))
286.24/286.51	= { by lemma 93 }
286.24/286.51	  codomain(multiplication(coantidomain(codomain(Y)), X))
286.24/286.51	= { by lemma 38 }
286.24/286.51	  codomain(multiplication(codomain(coantidomain(Y)), X))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  backward_diamond(X, coantidomain(Y))
286.24/286.51	
286.24/286.51	Lemma 103: backward_diamond(coantidomain(X), coantidomain(X)) = coantidomain(X).
286.24/286.51	Proof:
286.24/286.51	  backward_diamond(coantidomain(X), coantidomain(X))
286.24/286.51	= { by lemma 102 }
286.24/286.51	  codomain(multiplication(coantidomain(X), coantidomain(X)))
286.24/286.51	= { by lemma 69 }
286.24/286.51	  codomain(coantidomain(X))
286.24/286.51	= { by lemma 38 }
286.24/286.51	  coantidomain(codomain(X))
286.24/286.51	= { by lemma 93 }
286.24/286.51	  coantidomain(X)
286.24/286.51	
286.24/286.51	Lemma 104: forward_diamond(X, domain(Y)) = forward_diamond(X, Y).
286.24/286.51	Proof:
286.24/286.51	  forward_diamond(X, domain(Y))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  domain(multiplication(X, domain(domain(Y))))
286.24/286.51	= { by lemma 42 }
286.24/286.51	  domain(multiplication(X, antidomain(c(Y))))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  domain(multiplication(X, antidomain(antidomain(Y))))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  domain(multiplication(X, domain(Y)))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  forward_diamond(X, Y)
286.24/286.51	
286.24/286.51	Lemma 105: domain_difference(domain(X), Y) = domain_difference(X, Y).
286.24/286.51	Proof:
286.24/286.51	  domain_difference(domain(X), Y)
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  multiplication(domain(domain(X)), antidomain(Y))
286.24/286.51	= { by lemma 42 }
286.24/286.51	  multiplication(antidomain(c(X)), antidomain(Y))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  multiplication(antidomain(antidomain(X)), antidomain(Y))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  multiplication(domain(X), antidomain(Y))
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  domain_difference(X, Y)
286.24/286.51	
286.24/286.51	Lemma 106: multiplication(X, multiplication(Y, coantidomain(multiplication(X, Y)))) = zero.
286.24/286.51	Proof:
286.24/286.51	  multiplication(X, multiplication(Y, coantidomain(multiplication(X, Y))))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  multiplication(multiplication(X, Y), coantidomain(multiplication(X, Y)))
286.24/286.51	= { by axiom 19 (codomain1) }
286.24/286.51	  zero
286.24/286.51	
286.24/286.51	Lemma 107: forward_diamond(domain(X), antidomain(Y)) = domain(domain_difference(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  forward_diamond(domain(X), antidomain(Y))
286.24/286.51	= { by lemma 73 }
286.24/286.51	  domain(multiplication(domain(X), c(Y)))
286.24/286.51	= { by lemma 60 }
286.24/286.51	  domain(domain_difference(X, domain(Y)))
286.24/286.51	= { by lemma 101 }
286.24/286.51	  domain(domain_difference(X, Y))
286.24/286.51	
286.24/286.51	Lemma 108: multiplication(backward_diamond(X, Y), backward_diamond(X, Y)) = backward_diamond(X, Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(backward_diamond(X, Y), backward_diamond(X, Y))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  multiplication(codomain(multiplication(codomain(Y), X)), backward_diamond(X, Y))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  multiplication(codomain(multiplication(codomain(Y), X)), codomain(multiplication(codomain(Y), X)))
286.24/286.51	= { by lemma 70 }
286.24/286.51	  codomain(multiplication(codomain(Y), X))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  backward_diamond(X, Y)
286.24/286.51	
286.24/286.51	Lemma 109: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  antidomain(forward_diamond(X, antidomain(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(forward_diamond(X, c(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  c(forward_diamond(X, c(Y)))
286.24/286.51	= { by axiom 25 (forward_box) }
286.24/286.51	  forward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 110: antidomain(backward_diamond(X, antidomain(Y))) = backward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  antidomain(backward_diamond(X, antidomain(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(backward_diamond(X, c(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  c(backward_diamond(X, c(Y)))
286.24/286.51	= { by axiom 26 (backward_box) }
286.24/286.51	  backward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 111: backward_box(X, domain(Y)) = backward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  backward_box(X, domain(Y))
286.24/286.51	= { by lemma 110 }
286.24/286.51	  antidomain(backward_diamond(X, antidomain(domain(Y))))
286.24/286.51	= { by axiom 29 (complement) }
286.24/286.51	  antidomain(backward_diamond(X, c(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(backward_diamond(X, antidomain(Y)))
286.24/286.51	= { by lemma 110 }
286.24/286.51	  backward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 112: forward_diamond(addition(X, antidomain(Y)), Y) = forward_diamond(X, Y).
286.24/286.51	Proof:
286.24/286.51	  forward_diamond(addition(X, antidomain(Y)), Y)
286.24/286.51	= { by axiom 5 (additive_commutativity) }
286.24/286.51	  forward_diamond(addition(antidomain(Y), X), Y)
286.24/286.51	= { by lemma 97 }
286.24/286.51	  forward_diamond(addition(c(Y), X), Y)
286.24/286.51	= { by axiom 29 (complement) }
286.24/286.51	  forward_diamond(addition(antidomain(domain(Y)), X), Y)
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  domain(multiplication(addition(antidomain(domain(Y)), X), domain(Y)))
286.24/286.51	= { by lemma 82 }
286.24/286.51	  domain(multiplication(X, domain(Y)))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  forward_diamond(X, Y)
286.24/286.51	
286.24/286.51	Lemma 113: forward_diamond(X, antidomain(Y)) = antidomain(forward_box(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  forward_diamond(X, antidomain(Y))
286.24/286.51	= { by lemma 98 }
286.24/286.51	  domain(forward_diamond(X, antidomain(Y)))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(antidomain(forward_diamond(X, antidomain(Y))))
286.24/286.51	= { by lemma 109 }
286.24/286.51	  antidomain(forward_box(X, Y))
286.24/286.51	
286.24/286.51	Lemma 114: forward_box(X, antidomain(Y)) = antidomain(forward_diamond(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  forward_box(X, antidomain(Y))
286.24/286.51	= { by lemma 109 }
286.24/286.51	  antidomain(forward_diamond(X, antidomain(antidomain(Y))))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(forward_diamond(X, domain(Y)))
286.24/286.51	= { by lemma 104 }
286.24/286.51	  antidomain(forward_diamond(X, Y))
286.24/286.51	
286.24/286.51	Lemma 115: forward_box(domain(X), Y) = antidomain(domain_difference(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  forward_box(domain(X), Y)
286.24/286.51	= { by lemma 109 }
286.24/286.51	  antidomain(forward_diamond(domain(X), antidomain(Y)))
286.24/286.51	= { by lemma 107 }
286.24/286.51	  antidomain(domain(domain_difference(X, Y)))
286.24/286.51	= { by axiom 29 (complement) }
286.24/286.51	  c(domain_difference(X, Y))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(domain_difference(X, Y))
286.24/286.51	
286.24/286.51	Lemma 116: antidomain(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  antidomain(multiplication(X, domain(Y)))
286.24/286.51	= { by axiom 23 (domain4) }
286.24/286.51	  antidomain(multiplication(X, antidomain(antidomain(Y))))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(multiplication(X, antidomain(c(Y))))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  c(multiplication(X, antidomain(c(Y))))
286.24/286.51	= { by lemma 42 }
286.24/286.51	  c(multiplication(X, domain(domain(Y))))
286.24/286.51	= { by lemma 59 }
286.24/286.51	  antidomain(forward_diamond(X, domain(Y)))
286.24/286.51	= { by lemma 104 }
286.24/286.51	  antidomain(forward_diamond(X, Y))
286.24/286.51	
286.24/286.51	Lemma 117: antidomain(multiplication(X, antidomain(Y))) = forward_box(X, Y).
286.24/286.51	Proof:
286.24/286.51	  antidomain(multiplication(X, antidomain(Y)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(multiplication(X, c(Y)))
286.24/286.51	= { by lemma 39 }
286.24/286.51	  antidomain(multiplication(X, domain(antidomain(Y))))
286.24/286.51	= { by lemma 116 }
286.24/286.51	  antidomain(forward_diamond(X, antidomain(Y)))
286.24/286.51	= { by lemma 109 }
286.24/286.51	  forward_box(X, Y)
286.24/286.51	
286.24/286.51	Lemma 118: domain(multiplication(X, multiplication(Y, domain(Z)))) = forward_diamond(multiplication(X, Y), Z).
286.24/286.51	Proof:
286.24/286.51	  domain(multiplication(X, multiplication(Y, domain(Z))))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  domain(multiplication(multiplication(X, Y), domain(Z)))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  forward_diamond(multiplication(X, Y), Z)
286.24/286.51	
286.24/286.51	Lemma 119: multiplication(domain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(X, Y), Z).
286.24/286.51	Proof:
286.24/286.51	  multiplication(domain(X), multiplication(antidomain(Y), Z))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  multiplication(multiplication(domain(X), antidomain(Y)), Z)
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  multiplication(domain_difference(X, Y), Z)
286.24/286.51	
286.24/286.51	Lemma 120: multiplication(domain_difference(X, Y), antidomain(Y)) = domain_difference(X, Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(domain_difference(X, Y), antidomain(Y))
286.24/286.51	= { by lemma 119 }
286.24/286.51	  multiplication(domain(X), multiplication(antidomain(Y), antidomain(Y)))
286.24/286.51	= { by lemma 72 }
286.24/286.51	  multiplication(domain(X), antidomain(Y))
286.24/286.51	= { by axiom 28 (domain_difference) }
286.24/286.51	  domain_difference(X, Y)
286.24/286.51	
286.24/286.51	Lemma 121: forward_box(domain_difference(X, Y), Y) = antidomain(domain_difference(X, Y)).
286.24/286.51	Proof:
286.24/286.51	  forward_box(domain_difference(X, Y), Y)
286.24/286.51	= { by lemma 117 }
286.24/286.51	  antidomain(multiplication(domain_difference(X, Y), antidomain(Y)))
286.24/286.51	= { by lemma 120 }
286.24/286.51	  antidomain(domain_difference(X, Y))
286.24/286.51	
286.24/286.51	Lemma 122: multiplication(antidomain(X), coantidomain(domain(X))) = coantidomain(domain(X)).
286.24/286.51	Proof:
286.24/286.51	  multiplication(antidomain(X), coantidomain(domain(X)))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  multiplication(c(X), coantidomain(domain(X)))
286.24/286.51	= { by lemma 78 }
286.24/286.51	  multiplication(addition(domain(X), c(X)), coantidomain(domain(X)))
286.24/286.51	= { by lemma 56 }
286.24/286.51	  multiplication(one, coantidomain(domain(X)))
286.24/286.51	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.51	  coantidomain(domain(X))
286.24/286.51	
286.24/286.51	Lemma 123: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
286.24/286.51	Proof:
286.24/286.51	  coantidomain(multiplication(codomain(X), Y))
286.24/286.51	= { by lemma 93 }
286.24/286.51	  coantidomain(codomain(multiplication(codomain(X), Y)))
286.24/286.51	= { by axiom 24 (backward_diamond) }
286.24/286.51	  coantidomain(backward_diamond(Y, X))
286.24/286.51	
286.24/286.51	Lemma 124: multiplication(antidomain(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(antidomain(X), antidomain(Y))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  multiplication(c(X), antidomain(Y))
286.24/286.51	= { by lemma 75 }
286.24/286.51	  domain_difference(antidomain(X), Y)
286.24/286.51	
286.24/286.51	Lemma 125: multiplication(coantidomain(X), multiplication(coantidomain(X), Y)) = multiplication(coantidomain(X), Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(coantidomain(X), multiplication(coantidomain(X), Y))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  multiplication(multiplication(coantidomain(X), coantidomain(X)), Y)
286.24/286.51	= { by lemma 69 }
286.24/286.51	  multiplication(coantidomain(X), Y)
286.24/286.51	
286.24/286.51	Lemma 126: domain(domain_difference(antidomain(X), Y)) = antidomain(forward_box(antidomain(X), Y)).
286.24/286.51	Proof:
286.24/286.51	  domain(domain_difference(antidomain(X), Y))
286.24/286.51	= { by lemma 107 }
286.24/286.51	  forward_diamond(domain(antidomain(X)), antidomain(Y))
286.24/286.51	= { by lemma 39 }
286.24/286.51	  forward_diamond(c(X), antidomain(Y))
286.24/286.51	= { by lemma 113 }
286.24/286.51	  antidomain(forward_box(c(X), Y))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  antidomain(forward_box(antidomain(X), Y))
286.24/286.51	
286.24/286.51	Lemma 127: domain(multiplication(X, forward_diamond(Y, Z))) = forward_diamond(X, forward_diamond(Y, Z)).
286.24/286.51	Proof:
286.24/286.51	  domain(multiplication(X, forward_diamond(Y, Z)))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  domain(multiplication(X, domain(multiplication(Y, domain(Z)))))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  forward_diamond(X, multiplication(Y, domain(Z)))
286.24/286.51	= { by lemma 104 }
286.24/286.51	  forward_diamond(X, domain(multiplication(Y, domain(Z))))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  forward_diamond(X, forward_diamond(Y, Z))
286.24/286.51	
286.24/286.51	Lemma 128: multiplication(forward_diamond(X, Y), antidomain(Z)) = domain_difference(forward_diamond(X, Y), Z).
286.24/286.51	Proof:
286.24/286.51	  multiplication(forward_diamond(X, Y), antidomain(Z))
286.24/286.51	= { by lemma 87 }
286.24/286.51	  domain_difference(multiplication(X, domain(Y)), Z)
286.24/286.51	= { by lemma 105 }
286.24/286.51	  domain_difference(domain(multiplication(X, domain(Y))), Z)
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  domain_difference(forward_diamond(X, Y), Z)
286.24/286.51	
286.24/286.51	Lemma 129: domain_difference(domain_difference(X, antidomain(Y)), Z) = domain_difference(forward_diamond(domain(X), Y), Z).
286.24/286.51	Proof:
286.24/286.51	  domain_difference(domain_difference(X, antidomain(Y)), Z)
286.24/286.51	= { by lemma 105 }
286.24/286.51	  domain_difference(domain(domain_difference(X, antidomain(Y))), Z)
286.24/286.51	= { by lemma 77 }
286.24/286.51	  domain_difference(forward_diamond(domain(X), Y), Z)
286.24/286.51	
286.24/286.51	Lemma 130: multiplication(coantidomain(X), addition(Y, coantidomain(X))) = multiplication(coantidomain(X), addition(Y, one)).
286.24/286.51	Proof:
286.24/286.51	  multiplication(coantidomain(X), addition(Y, coantidomain(X)))
286.24/286.51	= { by axiom 5 (additive_commutativity) }
286.24/286.51	  multiplication(coantidomain(X), addition(coantidomain(X), Y))
286.24/286.51	= { by axiom 3 (right_distributivity) }
286.24/286.51	  addition(multiplication(coantidomain(X), coantidomain(X)), multiplication(coantidomain(X), Y))
286.24/286.51	= { by lemma 69 }
286.24/286.51	  addition(coantidomain(X), multiplication(coantidomain(X), Y))
286.24/286.51	= { by lemma 67 }
286.24/286.51	  multiplication(coantidomain(X), addition(Y, one))
286.24/286.51	
286.24/286.51	Lemma 131: multiplication(domain(X), multiplication(domain(Y), Z)) = multiplication(domain_difference(X, antidomain(Y)), Z).
286.24/286.51	Proof:
286.24/286.51	  multiplication(domain(X), multiplication(domain(Y), Z))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  multiplication(multiplication(domain(X), domain(Y)), Z)
286.24/286.51	= { by lemma 58 }
286.24/286.51	  multiplication(domain_difference(X, antidomain(Y)), Z)
286.24/286.51	
286.24/286.51	Lemma 132: multiplication(domain_difference(X, antidomain(Y)), Y) = multiplication(domain(X), Y).
286.24/286.51	Proof:
286.24/286.51	  multiplication(domain_difference(X, antidomain(Y)), Y)
286.24/286.51	= { by lemma 131 }
286.24/286.51	  multiplication(domain(X), multiplication(domain(Y), Y))
286.24/286.51	= { by lemma 83 }
286.24/286.51	  multiplication(domain(X), Y)
286.24/286.51	
286.24/286.51	Lemma 133: domain_difference(X, antidomain(forward_diamond(Y, Z))) = multiplication(domain(X), forward_diamond(Y, Z)).
286.24/286.51	Proof:
286.24/286.51	  domain_difference(X, antidomain(forward_diamond(Y, Z)))
286.24/286.51	= { by lemma 116 }
286.24/286.51	  domain_difference(X, antidomain(multiplication(Y, domain(Z))))
286.24/286.51	= { by lemma 58 }
286.24/286.51	  multiplication(domain(X), domain(multiplication(Y, domain(Z))))
286.24/286.51	= { by axiom 27 (forward_diamond) }
286.24/286.51	  multiplication(domain(X), forward_diamond(Y, Z))
286.24/286.51	
286.24/286.51	Lemma 134: antidomain(multiplication(X, multiplication(Y, domain(Z)))) = antidomain(forward_diamond(multiplication(X, Y), Z)).
286.24/286.51	Proof:
286.24/286.51	  antidomain(multiplication(X, multiplication(Y, domain(Z))))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  c(multiplication(X, multiplication(Y, domain(Z))))
286.24/286.51	= { by axiom 11 (multiplicative_associativity) }
286.24/286.51	  c(multiplication(multiplication(X, Y), domain(Z)))
286.24/286.51	= { by lemma 59 }
286.24/286.51	  antidomain(forward_diamond(multiplication(X, Y), Z))
286.24/286.51	
286.24/286.51	Lemma 135: multiplication(antidomain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(antidomain(X), Y), Z).
286.24/286.51	Proof:
286.24/286.51	  multiplication(antidomain(X), multiplication(antidomain(Y), Z))
286.24/286.51	= { by lemma 97 }
286.24/286.51	  multiplication(c(X), multiplication(antidomain(Y), Z))
286.24/286.51	= { by lemma 39 }
286.24/286.51	  multiplication(domain(antidomain(X)), multiplication(antidomain(Y), Z))
286.24/286.51	= { by lemma 119 }
286.24/286.51	  multiplication(domain_difference(antidomain(X), Y), Z)
286.24/286.51	
286.24/286.51	Lemma 136: multiplication(domain(X), addition(X, one)) = addition(X, domain(X)).
286.24/286.51	Proof:
286.24/286.51	  multiplication(domain(X), addition(X, one))
286.24/286.51	= { by lemma 67 }
286.24/286.51	  addition(domain(X), multiplication(domain(X), X))
286.24/286.51	= { by lemma 83 }
286.24/286.51	  addition(domain(X), X)
286.24/286.51	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  addition(X, domain(X))
286.24/286.52	
286.24/286.52	Lemma 137: multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, X)) = zero.
286.24/286.52	Proof:
286.24/286.52	  multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, X))
286.24/286.52	= { by lemma 95 }
286.24/286.52	  multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, codomain(X)))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, coantidomain(coantidomain(X))))
286.24/286.52	= { by lemma 102 }
286.24/286.52	  multiplication(coantidomain(multiplication(X, Y)), codomain(multiplication(coantidomain(coantidomain(X)), Y)))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  multiplication(coantidomain(multiplication(X, Y)), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(X)), Y))))
286.24/286.52	= { by lemma 92 }
286.24/286.52	  multiplication(addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(X)), Y))))
286.24/286.52	= { by axiom 16 (codomain2) }
286.24/286.52	  multiplication(coantidomain(multiplication(coantidomain(coantidomain(X)), Y)), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(X)), Y))))
286.24/286.52	= { by axiom 19 (codomain1) }
286.24/286.52	  zero
286.24/286.52	
286.24/286.52	Lemma 138: addition(antidomain(X), domain_difference(Y, X)) = antidomain(X).
286.24/286.52	Proof:
286.24/286.52	  addition(antidomain(X), domain_difference(Y, X))
286.24/286.52	= { by axiom 28 (domain_difference) }
286.24/286.52	  addition(antidomain(X), multiplication(domain(Y), antidomain(X)))
286.24/286.52	= { by lemma 66 }
286.24/286.52	  multiplication(addition(domain(Y), one), antidomain(X))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(one, domain(Y)), antidomain(X))
286.24/286.52	= { by lemma 55 }
286.24/286.52	  multiplication(one, antidomain(X))
286.24/286.52	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.52	  antidomain(X)
286.24/286.52	
286.24/286.52	Lemma 139: multiplication(domain(X), domain_difference(Y, X)) = zero.
286.24/286.52	Proof:
286.24/286.52	  multiplication(domain(X), domain_difference(Y, X))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  multiplication(antidomain(antidomain(X)), domain_difference(Y, X))
286.24/286.52	= { by lemma 79 }
286.24/286.52	  multiplication(antidomain(antidomain(X)), addition(antidomain(X), domain_difference(Y, X)))
286.24/286.52	= { by lemma 138 }
286.24/286.52	  multiplication(antidomain(antidomain(X)), antidomain(X))
286.24/286.52	= { by lemma 124 }
286.24/286.52	  domain_difference(antidomain(antidomain(X)), X)
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  domain_difference(domain(X), X)
286.24/286.52	= { by lemma 105 }
286.24/286.52	  domain_difference(X, X)
286.24/286.52	= { by lemma 46 }
286.24/286.52	  zero
286.24/286.52	
286.24/286.52	Lemma 140: addition(domain(X), domain_difference(X, Y)) = domain(X).
286.24/286.52	Proof:
286.24/286.52	  addition(domain(X), domain_difference(X, Y))
286.24/286.52	= { by axiom 28 (domain_difference) }
286.24/286.52	  addition(domain(X), multiplication(domain(X), antidomain(Y)))
286.24/286.52	= { by lemma 67 }
286.24/286.52	  multiplication(domain(X), addition(antidomain(Y), one))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(domain(X), addition(one, antidomain(Y)))
286.24/286.52	= { by lemma 53 }
286.24/286.52	  multiplication(domain(X), one)
286.24/286.52	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.52	  domain(X)
286.24/286.52	
286.24/286.52	Lemma 141: multiplication(addition(antidomain(Y), X), domain(Y)) = multiplication(X, domain(Y)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(addition(antidomain(Y), X), domain(Y))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(X, antidomain(Y)), domain(Y))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(antidomain(Y), X), domain(Y))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  multiplication(addition(c(Y), X), domain(Y))
286.24/286.52	= { by axiom 4 (left_distributivity) }
286.24/286.52	  addition(multiplication(c(Y), domain(Y)), multiplication(X, domain(Y)))
286.24/286.52	= { by axiom 29 (complement) }
286.24/286.52	  addition(multiplication(antidomain(domain(Y)), domain(Y)), multiplication(X, domain(Y)))
286.24/286.52	= { by axiom 21 (domain1) }
286.24/286.52	  addition(zero, multiplication(X, domain(Y)))
286.24/286.52	= { by lemma 35 }
286.24/286.52	  multiplication(X, domain(Y))
286.24/286.52	
286.24/286.52	Lemma 142: antidomain(forward_box(multiplication(X, antidomain(Y)), Y)) = antidomain(forward_box(X, Y)).
286.24/286.52	Proof:
286.24/286.52	  antidomain(forward_box(multiplication(X, antidomain(Y)), Y))
286.24/286.52	= { by lemma 113 }
286.24/286.52	  forward_diamond(multiplication(X, antidomain(Y)), antidomain(Y))
286.24/286.52	= { by lemma 118 }
286.24/286.52	  domain(multiplication(X, multiplication(antidomain(Y), domain(antidomain(Y)))))
286.24/286.52	= { by lemma 39 }
286.24/286.52	  domain(multiplication(X, multiplication(antidomain(Y), c(Y))))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  domain(multiplication(X, multiplication(antidomain(Y), antidomain(Y))))
286.24/286.52	= { by lemma 72 }
286.24/286.52	  domain(multiplication(X, antidomain(Y)))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  domain(multiplication(X, c(Y)))
286.24/286.52	= { by lemma 73 }
286.24/286.52	  forward_diamond(X, antidomain(Y))
286.24/286.52	= { by lemma 113 }
286.24/286.52	  antidomain(forward_box(X, Y))
286.24/286.52	
286.24/286.52	Lemma 143: multiplication(antidomain(X), addition(Z, multiplication(X, Y))) = multiplication(antidomain(X), Z).
286.24/286.52	Proof:
286.24/286.52	  multiplication(antidomain(X), addition(Z, multiplication(X, Y)))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(antidomain(X), addition(multiplication(X, Y), Z))
286.24/286.52	= { by axiom 3 (right_distributivity) }
286.24/286.52	  addition(multiplication(antidomain(X), multiplication(X, Y)), multiplication(antidomain(X), Z))
286.24/286.52	= { by lemma 57 }
286.24/286.52	  addition(zero, multiplication(antidomain(X), Z))
286.24/286.52	= { by lemma 35 }
286.24/286.52	  multiplication(antidomain(X), Z)
286.24/286.52	
286.24/286.52	Lemma 144: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
286.24/286.52	Proof:
286.24/286.52	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, antidomain(antidomain(Y))))
286.24/286.52	= { by lemma 71 }
286.24/286.52	  multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, antidomain(antidomain(Y))))
286.24/286.52	= { by axiom 18 (domain2) }
286.24/286.52	  multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, antidomain(antidomain(Y))))
286.24/286.52	= { by axiom 21 (domain1) }
286.24/286.52	  zero
286.24/286.52	
286.24/286.52	Lemma 145: addition(domain_difference(X, Y), multiplication(domain(X), Z)) = multiplication(domain(X), addition(Z, antidomain(Y))).
286.24/286.52	Proof:
286.24/286.52	  addition(domain_difference(X, Y), multiplication(domain(X), Z))
286.24/286.52	= { by axiom 28 (domain_difference) }
286.24/286.52	  addition(multiplication(domain(X), antidomain(Y)), multiplication(domain(X), Z))
286.24/286.52	= { by axiom 3 (right_distributivity) }
286.24/286.52	  multiplication(domain(X), addition(antidomain(Y), Z))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(domain(X), addition(Z, antidomain(Y)))
286.24/286.52	
286.24/286.52	Lemma 146: antidomain(multiplication(X, multiplication(Y, antidomain(Z)))) = forward_box(multiplication(X, Y), Z).
286.24/286.52	Proof:
286.24/286.52	  antidomain(multiplication(X, multiplication(Y, antidomain(Z))))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(multiplication(X, multiplication(Y, c(Z))))
286.24/286.52	= { by lemma 39 }
286.24/286.52	  antidomain(multiplication(X, multiplication(Y, domain(antidomain(Z)))))
286.24/286.52	= { by lemma 134 }
286.24/286.52	  antidomain(forward_diamond(multiplication(X, Y), antidomain(Z)))
286.24/286.52	= { by lemma 109 }
286.24/286.52	  forward_box(multiplication(X, Y), Z)
286.24/286.52	
286.24/286.52	Lemma 147: antidomain(forward_diamond(X, forward_diamond(Y, Z))) = antidomain(multiplication(X, forward_diamond(Y, Z))).
286.24/286.52	Proof:
286.24/286.52	  antidomain(forward_diamond(X, forward_diamond(Y, Z)))
286.24/286.52	= { by lemma 127 }
286.24/286.52	  antidomain(domain(multiplication(X, forward_diamond(Y, Z))))
286.24/286.52	= { by axiom 29 (complement) }
286.24/286.52	  c(multiplication(X, forward_diamond(Y, Z)))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(multiplication(X, forward_diamond(Y, Z)))
286.24/286.52	
286.24/286.52	Lemma 148: domain(multiplication(X, backward_box(Y, Z))) = forward_diamond(X, backward_box(Y, Z)).
286.24/286.52	Proof:
286.24/286.52	  domain(multiplication(X, backward_box(Y, Z)))
286.24/286.52	= { by axiom 26 (backward_box) }
286.24/286.52	  domain(multiplication(X, c(backward_diamond(Y, c(Z)))))
286.24/286.52	= { by lemma 73 }
286.24/286.52	  forward_diamond(X, antidomain(backward_diamond(Y, c(Z))))
286.24/286.52	= { by lemma 113 }
286.24/286.52	  antidomain(forward_box(X, backward_diamond(Y, c(Z))))
286.24/286.52	= { by lemma 86 }
286.24/286.52	  antidomain(c(forward_diamond(X, backward_box(Y, Z))))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(antidomain(forward_diamond(X, backward_box(Y, Z))))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  domain(forward_diamond(X, backward_box(Y, Z)))
286.24/286.52	= { by lemma 98 }
286.24/286.52	  forward_diamond(X, backward_box(Y, Z))
286.24/286.52	
286.24/286.52	Lemma 149: antidomain(forward_diamond(X, backward_box(Y, Z))) = antidomain(multiplication(X, backward_box(Y, Z))).
286.24/286.52	Proof:
286.24/286.52	  antidomain(forward_diamond(X, backward_box(Y, Z)))
286.24/286.52	= { by lemma 148 }
286.24/286.52	  antidomain(domain(multiplication(X, backward_box(Y, Z))))
286.24/286.52	= { by axiom 29 (complement) }
286.24/286.52	  c(multiplication(X, backward_box(Y, Z)))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(multiplication(X, backward_box(Y, Z)))
286.24/286.52	
286.24/286.52	Lemma 150: multiplication(forward_box(X, Y), antidomain(Z)) = domain_difference(forward_box(X, Y), Z).
286.24/286.52	Proof:
286.24/286.52	  multiplication(forward_box(X, Y), antidomain(Z))
286.24/286.52	= { by axiom 25 (forward_box) }
286.24/286.52	  multiplication(c(forward_diamond(X, c(Y))), antidomain(Z))
286.24/286.52	= { by lemma 75 }
286.24/286.52	  domain_difference(antidomain(forward_diamond(X, c(Y))), Z)
286.24/286.52	= { by lemma 97 }
286.24/286.52	  domain_difference(antidomain(forward_diamond(X, antidomain(Y))), Z)
286.24/286.52	= { by lemma 109 }
286.24/286.52	  domain_difference(forward_box(X, Y), Z)
286.24/286.52	
286.24/286.52	Lemma 151: multiplication(backward_box(X, Y), antidomain(Z)) = domain_difference(backward_box(X, Y), Z).
286.24/286.52	Proof:
286.24/286.52	  multiplication(backward_box(X, Y), antidomain(Z))
286.24/286.52	= { by axiom 26 (backward_box) }
286.24/286.52	  multiplication(c(backward_diamond(X, c(Y))), antidomain(Z))
286.24/286.52	= { by lemma 75 }
286.24/286.52	  domain_difference(antidomain(backward_diamond(X, c(Y))), Z)
286.24/286.52	= { by lemma 97 }
286.24/286.52	  domain_difference(antidomain(backward_diamond(X, antidomain(Y))), Z)
286.24/286.52	= { by lemma 110 }
286.24/286.52	  domain_difference(backward_box(X, Y), Z)
286.24/286.52	
286.24/286.52	Lemma 152: multiplication(domain_difference(X, antidomain(Y)), antidomain(Z)) = multiplication(domain(X), domain_difference(Y, Z)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(domain_difference(X, antidomain(Y)), antidomain(Z))
286.24/286.52	= { by lemma 131 }
286.24/286.52	  multiplication(domain(X), multiplication(domain(Y), antidomain(Z)))
286.24/286.52	= { by axiom 28 (domain_difference) }
286.24/286.52	  multiplication(domain(X), domain_difference(Y, Z))
286.24/286.52	
286.24/286.52	Lemma 153: multiplication(addition(one, X), codomain(X)) = addition(X, codomain(X)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(addition(one, X), codomain(X))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(X, one), codomain(X))
286.24/286.52	= { by lemma 66 }
286.24/286.52	  addition(codomain(X), multiplication(X, codomain(X)))
286.24/286.52	= { by lemma 81 }
286.24/286.52	  addition(codomain(X), X)
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  addition(X, codomain(X))
286.24/286.52	
286.24/286.52	Lemma 154: multiplication(antidomain(X), multiplication(addition(X, Y), Z)) = multiplication(antidomain(X), multiplication(Y, Z)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(antidomain(X), multiplication(addition(X, Y), Z))
286.24/286.52	= { by axiom 11 (multiplicative_associativity) }
286.24/286.52	  multiplication(multiplication(antidomain(X), addition(X, Y)), Z)
286.24/286.52	= { by lemma 79 }
286.24/286.52	  multiplication(multiplication(antidomain(X), Y), Z)
286.24/286.52	= { by axiom 11 (multiplicative_associativity) }
286.24/286.52	  multiplication(antidomain(X), multiplication(Y, Z))
286.24/286.52	
286.24/286.52	Lemma 155: multiplication(antidomain(X), addition(Z, addition(X, Y))) = multiplication(antidomain(X), addition(Y, Z)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(antidomain(X), addition(Z, addition(X, Y)))
286.24/286.52	= { by lemma 64 }
286.24/286.52	  multiplication(antidomain(X), addition(X, addition(Y, Z)))
286.24/286.52	= { by lemma 79 }
286.24/286.52	  multiplication(antidomain(X), addition(Y, Z))
286.24/286.52	
286.24/286.52	Lemma 156: leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) = $$true.
286.24/286.52	Proof:
286.24/286.52	  leq(multiplication(X, Y), multiplication(addition(X, Z), Y))
286.24/286.52	= { by axiom 4 (left_distributivity) }
286.24/286.52	  leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
286.24/286.52	= { by axiom 14 (order) }
286.24/286.52	  $$fresh2(addition(multiplication(X, Y), multiplication(Z, Y)), addition(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y))), multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
286.24/286.52	= { by lemma 51 }
286.24/286.52	  $$fresh2(addition(multiplication(X, Y), multiplication(Z, Y)), addition(multiplication(X, Y), multiplication(Z, Y)), multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
286.24/286.52	= { by axiom 1 (order) }
286.24/286.52	  $$true
286.24/286.52	
286.24/286.52	Lemma 157: addition(domain(X), addition(Y, domain_difference(X, Z))) = addition(Y, domain(X)).
286.24/286.52	Proof:
286.24/286.52	  addition(domain(X), addition(Y, domain_difference(X, Z)))
286.24/286.52	= { by lemma 65 }
286.24/286.52	  addition(Y, addition(domain(X), domain_difference(X, Z)))
286.24/286.52	= { by lemma 140 }
286.24/286.52	  addition(Y, domain(X))
286.24/286.52	
286.24/286.52	Lemma 158: addition(antidomain(multiplication(X, Y)), antidomain(forward_diamond(X, Y))) = antidomain(forward_diamond(X, Y)).
286.24/286.52	Proof:
286.24/286.52	  addition(antidomain(multiplication(X, Y)), antidomain(forward_diamond(X, Y)))
286.24/286.52	= { by lemma 114 }
286.24/286.52	  addition(antidomain(multiplication(X, Y)), forward_box(X, antidomain(Y)))
286.24/286.52	= { by lemma 117 }
286.24/286.52	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
286.24/286.52	= { by axiom 18 (domain2) }
286.24/286.52	  antidomain(multiplication(X, antidomain(antidomain(Y))))
286.24/286.52	= { by lemma 117 }
286.24/286.52	  forward_box(X, antidomain(Y))
286.24/286.52	= { by lemma 114 }
286.24/286.52	  antidomain(forward_diamond(X, Y))
286.24/286.52	
286.24/286.52	Lemma 159: antidomain(domain(X)) = antidomain(X).
286.24/286.52	Proof:
286.24/286.52	  antidomain(domain(X))
286.24/286.52	= { by axiom 29 (complement) }
286.24/286.52	  c(X)
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(X)
286.24/286.52	
286.24/286.52	Lemma 160: codomain(coantidomain(X)) = coantidomain(X).
286.24/286.52	Proof:
286.24/286.52	  codomain(coantidomain(X))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  coantidomain(coantidomain(coantidomain(X)))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  coantidomain(codomain(X))
286.24/286.52	= { by lemma 93 }
286.24/286.52	  coantidomain(X)
286.24/286.52	
286.24/286.52	Lemma 161: domain(antidomain(X)) = antidomain(X).
286.24/286.52	Proof:
286.24/286.52	  domain(antidomain(X))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  antidomain(antidomain(antidomain(X)))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  antidomain(domain(X))
286.24/286.52	= { by axiom 29 (complement) }
286.24/286.52	  c(X)
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(X)
286.24/286.52	
286.24/286.52	Lemma 162: backward_diamond(one, X) = codomain(X).
286.24/286.52	Proof:
286.24/286.52	  backward_diamond(one, X)
286.24/286.52	= { by axiom 24 (backward_diamond) }
286.24/286.52	  codomain(multiplication(codomain(X), one))
286.24/286.52	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.52	  codomain(codomain(X))
286.24/286.52	= { by lemma 94 }
286.24/286.52	  codomain(X)
286.24/286.52	
286.24/286.52	Lemma 163: forward_diamond(one, X) = domain(X).
286.24/286.52	Proof:
286.24/286.52	  forward_diamond(one, X)
286.24/286.52	= { by axiom 27 (forward_diamond) }
286.24/286.52	  domain(multiplication(one, domain(X)))
286.24/286.52	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.52	  domain(domain(X))
286.24/286.52	= { by lemma 42 }
286.24/286.52	  antidomain(c(X))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  antidomain(antidomain(X))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  domain(X)
286.24/286.52	
286.24/286.52	Lemma 164: domain_difference(antidomain(X), antidomain(Y)) = multiplication(antidomain(X), domain(Y)).
286.24/286.52	Proof:
286.24/286.52	  domain_difference(antidomain(X), antidomain(Y))
286.24/286.52	= { by lemma 58 }
286.24/286.52	  multiplication(domain(antidomain(X)), domain(Y))
286.24/286.52	= { by lemma 39 }
286.24/286.52	  multiplication(c(X), domain(Y))
286.24/286.52	= { by lemma 97 }
286.24/286.52	  multiplication(antidomain(X), domain(Y))
286.24/286.52	
286.24/286.52	Lemma 165: backward_diamond(X, multiplication(codomain(Y), Z)) = backward_diamond(X, backward_diamond(Z, Y)).
286.24/286.52	Proof:
286.24/286.52	  backward_diamond(X, multiplication(codomain(Y), Z))
286.24/286.52	= { by axiom 24 (backward_diamond) }
286.24/286.52	  codomain(multiplication(codomain(multiplication(codomain(Y), Z)), X))
286.24/286.52	= { by axiom 24 (backward_diamond) }
286.24/286.52	  codomain(multiplication(backward_diamond(Z, Y), X))
286.24/286.52	= { by lemma 81 }
286.24/286.52	  codomain(multiplication(multiplication(backward_diamond(Z, Y), codomain(backward_diamond(Z, Y))), X))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  codomain(multiplication(multiplication(backward_diamond(Z, Y), coantidomain(coantidomain(backward_diamond(Z, Y)))), X))
286.24/286.52	= { by lemma 92 }
286.24/286.52	  codomain(multiplication(multiplication(addition(backward_diamond(Z, Y), coantidomain(backward_diamond(Z, Y))), coantidomain(coantidomain(backward_diamond(Z, Y)))), X))
286.24/286.52	= { by lemma 90 }
286.24/286.52	  codomain(multiplication(multiplication(one, coantidomain(coantidomain(backward_diamond(Z, Y)))), X))
286.24/286.52	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.52	  codomain(multiplication(coantidomain(coantidomain(backward_diamond(Z, Y))), X))
286.24/286.52	= { by axiom 20 (codomain4) }
286.24/286.52	  codomain(multiplication(codomain(backward_diamond(Z, Y)), X))
286.24/286.52	= { by axiom 24 (backward_diamond) }
286.24/286.52	  backward_diamond(X, backward_diamond(Z, Y))
286.24/286.52	
286.24/286.52	Lemma 166: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z).
286.24/286.52	Proof:
286.24/286.52	  domain_difference(multiplication(X, domain(Y)), Z)
286.24/286.52	= { by axiom 28 (domain_difference) }
286.24/286.52	  multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
286.24/286.52	= { by axiom 27 (forward_diamond) }
286.24/286.52	  multiplication(forward_diamond(X, Y), antidomain(Z))
286.24/286.52	= { by lemma 128 }
286.24/286.52	  domain_difference(forward_diamond(X, Y), Z)
286.24/286.52	
286.24/286.52	Lemma 167: multiplication(addition(Y, addition(X, Z)), coantidomain(X)) = multiplication(addition(Y, Z), coantidomain(X)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(addition(Y, addition(X, Z)), coantidomain(X))
286.24/286.52	= { by lemma 64 }
286.24/286.52	  multiplication(addition(X, addition(Z, Y)), coantidomain(X))
286.24/286.52	= { by lemma 78 }
286.24/286.52	  multiplication(addition(Z, Y), coantidomain(X))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(Y, Z), coantidomain(X))
286.24/286.52	
286.24/286.52	Lemma 168: multiplication(domain(X), addition(Z, domain_difference(Y, X))) = multiplication(domain(X), Z).
286.24/286.52	Proof:
286.24/286.52	  multiplication(domain(X), addition(Z, domain_difference(Y, X)))
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  multiplication(antidomain(antidomain(X)), addition(Z, domain_difference(Y, X)))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(antidomain(antidomain(X)), addition(domain_difference(Y, X), Z))
286.24/286.52	= { by lemma 155 }
286.24/286.52	  multiplication(antidomain(antidomain(X)), addition(Z, addition(antidomain(X), domain_difference(Y, X))))
286.24/286.52	= { by lemma 138 }
286.24/286.52	  multiplication(antidomain(antidomain(X)), addition(Z, antidomain(X)))
286.24/286.52	= { by lemma 96 }
286.24/286.52	  multiplication(antidomain(antidomain(X)), Z)
286.24/286.52	= { by axiom 23 (domain4) }
286.24/286.52	  multiplication(domain(X), Z)
286.24/286.52	
286.24/286.52	Lemma 169: multiplication(domain(X), addition(X, antidomain(Y))) = addition(X, domain_difference(X, Y)).
286.24/286.52	Proof:
286.24/286.52	  multiplication(domain(X), addition(X, antidomain(Y)))
286.24/286.52	= { by lemma 145 }
286.24/286.52	  addition(domain_difference(X, Y), multiplication(domain(X), X))
286.24/286.52	= { by lemma 83 }
286.24/286.52	  addition(domain_difference(X, Y), X)
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  addition(X, domain_difference(X, Y))
286.24/286.52	
286.24/286.52	Lemma 170: multiplication(antidomain(X), addition(X, one)) = antidomain(X).
286.24/286.52	Proof:
286.24/286.52	  multiplication(antidomain(X), addition(X, one))
286.24/286.52	= { by lemma 54 }
286.24/286.52	  multiplication(antidomain(X), addition(X, addition(one, codomain(?))))
286.24/286.52	= { by lemma 65 }
286.24/286.52	  multiplication(antidomain(X), addition(one, addition(X, codomain(?))))
286.24/286.52	= { by lemma 155 }
286.24/286.52	  multiplication(antidomain(X), addition(codomain(?), one))
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(antidomain(X), addition(one, codomain(?)))
286.24/286.52	= { by lemma 54 }
286.24/286.52	  multiplication(antidomain(X), one)
286.24/286.52	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.52	  antidomain(X)
286.24/286.52	
286.24/286.52	Lemma 171: addition(multiplication(W, multiplication(Z, X)), multiplication(Y, X)) = multiplication(addition(Y, multiplication(W, Z)), X).
286.24/286.52	Proof:
286.24/286.52	  addition(multiplication(W, multiplication(Z, X)), multiplication(Y, X))
286.24/286.52	= { by axiom 11 (multiplicative_associativity) }
286.24/286.52	  addition(multiplication(multiplication(W, Z), X), multiplication(Y, X))
286.24/286.52	= { by axiom 4 (left_distributivity) }
286.24/286.52	  multiplication(addition(multiplication(W, Z), Y), X)
286.24/286.52	= { by axiom 5 (additive_commutativity) }
286.24/286.52	  multiplication(addition(Y, multiplication(W, Z)), X)
286.24/286.52	
286.24/286.52	Lemma 172: multiplication(addition(X, multiplication(Y, Z)), coantidomain(X)) = multiplication(Y, multiplication(Z, coantidomain(X))).
286.24/286.52	Proof:
286.24/286.52	  multiplication(addition(X, multiplication(Y, Z)), coantidomain(X))
286.24/286.52	= { by lemma 171 }
286.24/286.52	  addition(multiplication(Y, multiplication(Z, coantidomain(X))), multiplication(X, coantidomain(X)))
286.24/286.52	= { by axiom 19 (codomain1) }
286.24/286.52	  addition(multiplication(Y, multiplication(Z, coantidomain(X))), zero)
286.24/286.52	= { by axiom 9 (additive_identity) }
286.24/286.53	  multiplication(Y, multiplication(Z, coantidomain(X)))
286.24/286.53	
286.24/286.53	Lemma 173: addition(multiplication(X, Y), addition(W, multiplication(X, Z))) = addition(W, multiplication(X, addition(Y, Z))).
286.24/286.53	Proof:
286.24/286.53	  addition(multiplication(X, Y), addition(W, multiplication(X, Z)))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  addition(multiplication(X, Y), addition(multiplication(X, Z), W))
286.24/286.53	= { by axiom 12 (additive_associativity) }
286.24/286.53	  addition(addition(multiplication(X, Y), multiplication(X, Z)), W)
286.24/286.53	= { by axiom 3 (right_distributivity) }
286.24/286.53	  addition(multiplication(X, addition(Y, Z)), W)
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  addition(W, multiplication(X, addition(Y, Z)))
286.24/286.53	
286.24/286.53	Lemma 175: addition(domain_difference(X, Y), domain_difference(X, Z)) = multiplication(domain(X), addition(antidomain(Y), antidomain(Z))).
286.24/286.53	Proof:
286.24/286.53	  addition(domain_difference(X, Y), domain_difference(X, Z))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  addition(domain_difference(X, Z), domain_difference(X, Y))
286.24/286.53	= { by axiom 28 (domain_difference) }
286.24/286.53	  addition(domain_difference(X, Z), multiplication(domain(X), antidomain(Y)))
286.24/286.53	= { by lemma 145 }
286.24/286.53	  multiplication(domain(X), addition(antidomain(Y), antidomain(Z)))
286.24/286.53	
286.24/286.53	Lemma 175: multiplication(domain(X), addition(antidomain(Y), antidomain(Z))) = addition(domain_difference(X, Y), domain_difference(X, Z)).
286.24/286.53	Proof:
286.24/286.53	  multiplication(domain(X), addition(antidomain(Y), antidomain(Z)))
286.24/286.53	= { by lemma 145 }
286.24/286.53	  addition(domain_difference(X, Z), multiplication(domain(X), antidomain(Y)))
286.24/286.53	= { by axiom 28 (domain_difference) }
286.24/286.53	  addition(domain_difference(X, Z), domain_difference(X, Y))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  addition(domain_difference(X, Y), domain_difference(X, Z))
286.24/286.53	
286.24/286.53	Lemma 176: multiplication(addition(X, antidomain(Y)), domain_difference(Y, Z)) = multiplication(X, domain_difference(Y, Z)).
286.24/286.53	Proof:
286.24/286.53	  multiplication(addition(X, antidomain(Y)), domain_difference(Y, Z))
286.24/286.53	= { by lemma 159 }
286.24/286.53	  multiplication(addition(X, antidomain(domain(Y))), domain_difference(Y, Z))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  multiplication(addition(antidomain(domain(Y)), X), domain_difference(Y, Z))
286.24/286.53	= { by axiom 28 (domain_difference) }
286.24/286.53	  multiplication(addition(antidomain(domain(Y)), X), multiplication(domain(Y), antidomain(Z)))
286.24/286.53	= { by axiom 4 (left_distributivity) }
286.24/286.53	  addition(multiplication(antidomain(domain(Y)), multiplication(domain(Y), antidomain(Z))), multiplication(X, multiplication(domain(Y), antidomain(Z))))
286.24/286.53	= { by lemma 57 }
286.24/286.53	  addition(zero, multiplication(X, multiplication(domain(Y), antidomain(Z))))
286.24/286.53	= { by lemma 35 }
286.24/286.53	  multiplication(X, multiplication(domain(Y), antidomain(Z)))
286.24/286.53	= { by axiom 28 (domain_difference) }
286.24/286.53	  multiplication(X, domain_difference(Y, Z))
286.24/286.53	
286.24/286.53	Lemma 177: antidomain(multiplication(domain(X), domain_difference(Y, Z))) = forward_box(domain_difference(X, antidomain(Y)), Z).
286.24/286.53	Proof:
286.24/286.53	  antidomain(multiplication(domain(X), domain_difference(Y, Z)))
286.24/286.53	= { by lemma 152 }
286.24/286.53	  antidomain(multiplication(domain_difference(X, antidomain(Y)), antidomain(Z)))
286.24/286.53	= { by lemma 117 }
286.24/286.53	  forward_box(domain_difference(X, antidomain(Y)), Z)
286.24/286.53	
286.24/286.53	Lemma 178: multiplication(antidomain(addition(X, Y)), addition(X, Z)) = multiplication(antidomain(addition(X, Y)), Z).
286.24/286.53	Proof:
286.24/286.53	  multiplication(antidomain(addition(X, Y)), addition(X, Z))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  multiplication(antidomain(addition(X, Y)), addition(Z, X))
286.24/286.53	= { by lemma 79 }
286.24/286.53	  multiplication(antidomain(addition(X, Y)), addition(addition(X, Y), addition(Z, X)))
286.24/286.53	= { by lemma 64 }
286.24/286.53	  multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, addition(X, Y))))
286.24/286.53	= { by lemma 51 }
286.24/286.53	  multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, Y)))
286.24/286.53	= { by lemma 96 }
286.24/286.53	  multiplication(antidomain(addition(X, Y)), Z)
286.24/286.53	
286.24/286.53	Lemma 179: multiplication(addition(X, multiplication(antidomain(multiplication(Y, Z)), Y)), Z) = multiplication(X, Z).
286.24/286.53	Proof:
286.24/286.53	  multiplication(addition(X, multiplication(antidomain(multiplication(Y, Z)), Y)), Z)
286.24/286.53	= { by lemma 171 }
286.24/286.53	  addition(multiplication(antidomain(multiplication(Y, Z)), multiplication(Y, Z)), multiplication(X, Z))
286.24/286.53	= { by axiom 21 (domain1) }
286.24/286.53	  addition(zero, multiplication(X, Z))
286.24/286.53	= { by lemma 35 }
286.24/286.53	  multiplication(X, Z)
286.24/286.53	
286.24/286.53	Lemma 180: domain_difference(antidomain(addition(X, domain(Y))), Y) = antidomain(addition(X, domain(Y))).
286.24/286.53	Proof:
286.24/286.53	  domain_difference(antidomain(addition(X, domain(Y))), Y)
286.24/286.53	= { by lemma 124 }
286.24/286.53	  multiplication(antidomain(addition(X, domain(Y))), antidomain(Y))
286.24/286.53	= { by lemma 96 }
286.24/286.53	  multiplication(antidomain(addition(X, domain(Y))), addition(antidomain(Y), addition(X, domain(Y))))
286.24/286.53	= { by lemma 65 }
286.24/286.53	  multiplication(antidomain(addition(X, domain(Y))), addition(X, addition(antidomain(Y), domain(Y))))
286.24/286.53	= { by lemma 47 }
286.24/286.53	  multiplication(antidomain(addition(X, domain(Y))), addition(X, one))
286.24/286.53	= { by lemma 178 }
286.24/286.53	  multiplication(antidomain(addition(X, domain(Y))), one)
286.24/286.53	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.53	  antidomain(addition(X, domain(Y)))
286.24/286.53	
286.24/286.53	Lemma 181: addition(antidomain(X), antidomain(addition(Y, domain(X)))) = antidomain(X).
286.24/286.53	Proof:
286.24/286.53	  addition(antidomain(X), antidomain(addition(Y, domain(X))))
286.24/286.53	= { by lemma 180 }
286.24/286.53	  addition(antidomain(X), domain_difference(antidomain(addition(Y, domain(X))), X))
286.24/286.53	= { by lemma 138 }
286.24/286.53	  antidomain(X)
286.24/286.53	
286.24/286.53	Lemma 182: forward_diamond(domain(X), addition(Y, domain(X))) = domain(X).
286.24/286.53	Proof:
286.24/286.53	  forward_diamond(domain(X), addition(Y, domain(X)))
286.24/286.53	= { by axiom 23 (domain4) }
286.24/286.53	  forward_diamond(antidomain(antidomain(X)), addition(Y, domain(X)))
286.24/286.53	= { by lemma 181 }
286.24/286.53	  forward_diamond(antidomain(addition(antidomain(X), antidomain(addition(Y, domain(X))))), addition(Y, domain(X)))
286.24/286.53	= { by lemma 161 }
286.24/286.53	  forward_diamond(antidomain(addition(antidomain(X), domain(antidomain(addition(Y, domain(X)))))), addition(Y, domain(X)))
286.24/286.53	= { by lemma 161 }
286.24/286.53	  forward_diamond(domain(antidomain(addition(antidomain(X), domain(antidomain(addition(Y, domain(X))))))), addition(Y, domain(X)))
286.24/286.53	= { by lemma 77 }
286.24/286.53	  domain(domain_difference(antidomain(addition(antidomain(X), domain(antidomain(addition(Y, domain(X)))))), antidomain(addition(Y, domain(X)))))
286.24/286.53	= { by lemma 180 }
286.24/286.53	  domain(antidomain(addition(antidomain(X), domain(antidomain(addition(Y, domain(X)))))))
286.24/286.53	= { by lemma 161 }
286.24/286.53	  antidomain(addition(antidomain(X), domain(antidomain(addition(Y, domain(X))))))
286.24/286.53	= { by lemma 161 }
286.24/286.53	  antidomain(addition(antidomain(X), antidomain(addition(Y, domain(X)))))
286.24/286.53	= { by lemma 181 }
286.24/286.53	  antidomain(antidomain(X))
286.24/286.53	= { by axiom 23 (domain4) }
286.24/286.53	  domain(X)
286.24/286.53	
286.24/286.53	Lemma 183: multiplication(coantidomain(X), antidomain(codomain(X))) = coantidomain(X).
286.24/286.53	Proof:
286.24/286.53	  multiplication(coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by lemma 35 }
286.24/286.53	  addition(zero, multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by lemma 144 }
286.24/286.53	  addition(multiplication(antidomain(multiplication(coantidomain(X), codomain(X))), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by axiom 20 (codomain4) }
286.24/286.53	  addition(multiplication(antidomain(multiplication(coantidomain(X), coantidomain(coantidomain(X)))), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by axiom 19 (codomain1) }
286.24/286.53	  addition(multiplication(antidomain(zero), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by lemma 48 }
286.24/286.53	  addition(multiplication(one, multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.53	  addition(multiplication(coantidomain(X), domain(codomain(X))), multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.53	= { by axiom 3 (right_distributivity) }
286.24/286.53	  multiplication(coantidomain(X), addition(domain(codomain(X)), antidomain(codomain(X))))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  multiplication(coantidomain(X), addition(antidomain(codomain(X)), domain(codomain(X))))
286.24/286.53	= { by lemma 47 }
286.24/286.53	  multiplication(coantidomain(X), one)
286.24/286.53	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.53	  coantidomain(X)
286.24/286.53	
286.24/286.53	Lemma 184: antidomain(codomain(X)) = coantidomain(X).
286.24/286.53	Proof:
286.24/286.53	  antidomain(codomain(X))
286.24/286.53	= { by axiom 2 (order_1) }
286.24/286.53	  $$fresh($$true, $$true, coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by lemma 156 }
286.24/286.53	  $$fresh(leq(multiplication(coantidomain(X), antidomain(codomain(X))), multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by lemma 183 }
286.24/286.53	  $$fresh(leq(coantidomain(X), multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by lemma 31 }
286.24/286.53	  $$fresh(leq(coantidomain(X), multiplication(one, antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.53	  $$fresh(leq(coantidomain(X), antidomain(codomain(X))), $$true, coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by axiom 13 (order_1) }
286.24/286.53	  addition(coantidomain(X), antidomain(codomain(X)))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  addition(antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 13 (order_1) }
286.24/286.53	  $$fresh(leq(antidomain(codomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 20 (codomain4) }
286.24/286.53	  $$fresh(leq(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), one), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by lemma 31 }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), addition(coantidomain(X), coantidomain(coantidomain(X)))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by lemma 96 }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 20 (codomain4) }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), multiplication(one, coantidomain(X))), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by lemma 32 }
286.24/286.53	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), multiplication(addition(antidomain(codomain(X)), antidomain(antidomain(codomain(X)))), coantidomain(X))), $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by lemma 156 }
286.24/286.53	  $$fresh($$true, $$true, antidomain(codomain(X)), coantidomain(X))
286.24/286.53	= { by axiom 2 (order_1) }
286.24/286.53	  coantidomain(X)
286.24/286.53	
286.24/286.53	Lemma 185: antidomain(coantidomain(X)) = codomain(X).
286.24/286.53	Proof:
286.24/286.53	  antidomain(coantidomain(X))
286.24/286.53	= { by lemma 160 }
286.24/286.53	  antidomain(codomain(coantidomain(X)))
286.24/286.53	= { by lemma 184 }
286.24/286.53	  coantidomain(coantidomain(X))
286.24/286.53	= { by axiom 20 (codomain4) }
286.24/286.53	  codomain(X)
286.24/286.53	
286.24/286.53	Lemma 186: domain(codomain(X)) = codomain(X).
286.24/286.53	Proof:
286.24/286.53	  domain(codomain(X))
286.24/286.53	= { by axiom 23 (domain4) }
286.24/286.53	  antidomain(antidomain(codomain(X)))
286.24/286.53	= { by lemma 184 }
286.24/286.53	  antidomain(coantidomain(X))
286.24/286.53	= { by lemma 185 }
286.24/286.53	  codomain(X)
286.24/286.53	
286.24/286.53	Lemma 187: backward_box(one, X) = coantidomain(antidomain(X)).
286.24/286.53	Proof:
286.24/286.53	  backward_box(one, X)
286.24/286.53	= { by axiom 26 (backward_box) }
286.24/286.53	  c(backward_diamond(one, c(X)))
286.24/286.53	= { by lemma 40 }
286.24/286.53	  c(codomain(codomain(c(X))))
286.24/286.53	= { by lemma 97 }
286.24/286.53	  antidomain(codomain(codomain(c(X))))
286.24/286.53	= { by lemma 94 }
286.24/286.53	  antidomain(codomain(c(X)))
286.24/286.53	= { by lemma 97 }
286.24/286.53	  antidomain(codomain(antidomain(X)))
286.24/286.53	= { by lemma 184 }
286.24/286.53	  coantidomain(antidomain(X))
286.24/286.53	
286.24/286.53	Lemma 188: coantidomain(backward_diamond(X, Y)) = antidomain(backward_diamond(X, Y)).
286.24/286.53	Proof:
286.24/286.53	  coantidomain(backward_diamond(X, Y))
286.24/286.53	= { by lemma 123 }
286.24/286.53	  coantidomain(multiplication(codomain(Y), X))
286.24/286.53	= { by lemma 184 }
286.24/286.53	  antidomain(codomain(multiplication(codomain(Y), X)))
286.24/286.53	= { by axiom 24 (backward_diamond) }
286.24/286.53	  antidomain(backward_diamond(X, Y))
286.24/286.53	
286.24/286.53	Lemma 189: domain(backward_diamond(X, Y)) = backward_diamond(X, Y).
286.24/286.53	Proof:
286.24/286.53	  domain(backward_diamond(X, Y))
286.24/286.53	= { by axiom 24 (backward_diamond) }
286.24/286.53	  domain(codomain(multiplication(codomain(Y), X)))
286.24/286.53	= { by lemma 186 }
286.24/286.53	  codomain(multiplication(codomain(Y), X))
286.24/286.53	= { by axiom 24 (backward_diamond) }
286.24/286.53	  backward_diamond(X, Y)
286.24/286.53	
286.24/286.53	Lemma 190: backward_diamond(backward_diamond(X, coantidomain(Y)), multiplication(coantidomain(Y), X)) = backward_diamond(X, coantidomain(Y)).
286.24/286.53	Proof:
286.24/286.53	  backward_diamond(backward_diamond(X, coantidomain(Y)), multiplication(coantidomain(Y), X))
286.24/286.53	= { by axiom 24 (backward_diamond) }
286.24/286.53	  codomain(multiplication(codomain(multiplication(coantidomain(Y), X)), backward_diamond(X, coantidomain(Y))))
286.24/286.53	= { by lemma 68 }
286.24/286.53	  codomain(multiplication(codomain(multiplication(coantidomain(Y), X)), addition(backward_diamond(X, coantidomain(Y)), coantidomain(codomain(multiplication(coantidomain(Y), X))))))
286.24/286.53	= { by axiom 24 (backward_diamond) }
286.24/286.53	  backward_diamond(addition(backward_diamond(X, coantidomain(Y)), coantidomain(codomain(multiplication(coantidomain(Y), X)))), multiplication(coantidomain(Y), X))
286.24/286.53	= { by lemma 93 }
286.24/286.53	  backward_diamond(addition(backward_diamond(X, coantidomain(Y)), coantidomain(multiplication(coantidomain(Y), X))), multiplication(coantidomain(Y), X))
286.24/286.53	= { by axiom 5 (additive_commutativity) }
286.24/286.53	  backward_diamond(addition(coantidomain(multiplication(coantidomain(Y), X)), backward_diamond(X, coantidomain(Y))), multiplication(coantidomain(Y), X))
286.24/286.53	= { by lemma 102 }
286.24/286.53	  backward_diamond(addition(coantidomain(multiplication(coantidomain(Y), X)), codomain(multiplication(coantidomain(Y), X))), multiplication(coantidomain(Y), X))
286.24/286.53	= { by lemma 44 }
286.24/286.53	  backward_diamond(one, multiplication(coantidomain(Y), X))
286.24/286.53	= { by lemma 162 }
286.24/286.53	  codomain(multiplication(coantidomain(Y), X))
286.24/286.53	= { by lemma 102 }
286.24/286.54	  backward_diamond(X, coantidomain(Y))
286.24/286.54	
286.24/286.54	Lemma 191: forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)) = backward_diamond(X, Y).
286.24/286.54	Proof:
286.24/286.54	  forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y))
286.24/286.54	= { by lemma 112 }
286.24/286.54	  forward_diamond(addition(backward_diamond(X, Y), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 95 }
286.24/286.54	  forward_diamond(addition(backward_diamond(X, codomain(Y)), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y))
286.24/286.54	= { by axiom 20 (codomain4) }
286.24/286.54	  forward_diamond(addition(backward_diamond(X, coantidomain(coantidomain(Y))), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 190 }
286.24/286.54	  forward_diamond(addition(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 95 }
286.24/286.54	  forward_diamond(addition(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)), antidomain(backward_diamond(X, codomain(Y)))), backward_diamond(X, Y))
286.24/286.54	= { by axiom 20 (codomain4) }
286.24/286.54	  forward_diamond(addition(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)), antidomain(backward_diamond(X, coantidomain(coantidomain(Y))))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 190 }
286.24/286.54	  forward_diamond(addition(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)), antidomain(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 188 }
286.24/286.54	  forward_diamond(addition(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)), coantidomain(backward_diamond(backward_diamond(X, coantidomain(coantidomain(Y))), multiplication(coantidomain(coantidomain(Y)), X)))), backward_diamond(X, Y))
286.24/286.54	= { by lemma 90 }
286.24/286.54	  forward_diamond(one, backward_diamond(X, Y))
286.24/286.54	= { by lemma 163 }
286.24/286.54	  domain(backward_diamond(X, Y))
286.24/286.54	= { by lemma 189 }
286.24/286.54	  backward_diamond(X, Y)
286.24/286.54	
286.24/286.54	Lemma 192: domain_difference(X, antidomain(backward_diamond(Y, Z))) = multiplication(domain(X), backward_diamond(Y, Z)).
286.24/286.54	Proof:
286.24/286.54	  domain_difference(X, antidomain(backward_diamond(Y, Z)))
286.24/286.54	= { by lemma 188 }
286.24/286.54	  domain_difference(X, coantidomain(backward_diamond(Y, Z)))
286.24/286.54	= { by lemma 120 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), antidomain(coantidomain(backward_diamond(Y, Z))))
286.24/286.54	= { by lemma 123 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), antidomain(coantidomain(multiplication(codomain(Z), Y))))
286.24/286.54	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(multiplication(codomain(Z), Y))), one))
286.24/286.54	= { by lemma 31 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(multiplication(codomain(Z), Y))), addition(coantidomain(multiplication(codomain(Z), Y)), coantidomain(coantidomain(multiplication(codomain(Z), Y))))))
286.24/286.54	= { by lemma 79 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(multiplication(codomain(Z), Y))), coantidomain(coantidomain(multiplication(codomain(Z), Y)))))
286.24/286.54	= { by lemma 123 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(backward_diamond(Y, Z))), coantidomain(coantidomain(multiplication(codomain(Z), Y)))))
286.24/286.54	= { by axiom 20 (codomain4) }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(backward_diamond(Y, Z))), codomain(multiplication(codomain(Z), Y))))
286.24/286.54	= { by axiom 24 (backward_diamond) }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), multiplication(antidomain(coantidomain(backward_diamond(Y, Z))), backward_diamond(Y, Z)))
286.24/286.54	= { by axiom 11 (multiplicative_associativity) }
286.24/286.54	  multiplication(multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), antidomain(coantidomain(backward_diamond(Y, Z)))), backward_diamond(Y, Z))
286.24/286.54	= { by lemma 120 }
286.24/286.54	  multiplication(domain_difference(X, coantidomain(backward_diamond(Y, Z))), backward_diamond(Y, Z))
286.24/286.54	= { by lemma 188 }
286.24/286.54	  multiplication(domain_difference(X, antidomain(backward_diamond(Y, Z))), backward_diamond(Y, Z))
286.24/286.54	= { by lemma 132 }
286.24/286.54	  multiplication(domain(X), backward_diamond(Y, Z))
286.24/286.54	
286.24/286.54	Lemma 193: multiplication(domain(X), codomain(Y)) = domain_difference(X, coantidomain(Y)).
286.24/286.54	Proof:
286.24/286.54	  multiplication(domain(X), codomain(Y))
286.24/286.54	= { by lemma 85 }
286.24/286.54	  multiplication(domain(X), backward_diamond(Y, one))
286.24/286.54	= { by lemma 192 }
286.24/286.54	  domain_difference(X, antidomain(backward_diamond(Y, one)))
286.24/286.54	= { by lemma 85 }
286.24/286.54	  domain_difference(X, antidomain(codomain(Y)))
286.24/286.54	= { by lemma 184 }
286.24/286.54	  domain_difference(X, coantidomain(Y))
286.24/286.54	
286.24/286.54	Lemma 194: multiplication(domain(X), coantidomain(Y)) = domain_difference(X, codomain(Y)).
286.24/286.54	Proof:
286.24/286.54	  multiplication(domain(X), coantidomain(Y))
286.24/286.54	= { by lemma 103 }
286.24/286.54	  multiplication(domain(X), backward_diamond(coantidomain(Y), coantidomain(Y)))
286.24/286.54	= { by lemma 192 }
286.24/286.54	  domain_difference(X, antidomain(backward_diamond(coantidomain(Y), coantidomain(Y))))
286.24/286.54	= { by lemma 103 }
286.24/286.54	  domain_difference(X, antidomain(coantidomain(Y)))
286.24/286.54	= { by lemma 185 }
286.24/286.54	  domain_difference(X, codomain(Y))
286.24/286.54	
286.24/286.54	Lemma 195: domain_difference(antidomain(X), coantidomain(Y)) = multiplication(antidomain(X), codomain(Y)).
286.24/286.54	Proof:
286.24/286.54	  domain_difference(antidomain(X), coantidomain(Y))
286.24/286.54	= { by lemma 193 }
286.24/286.54	  multiplication(domain(antidomain(X)), codomain(Y))
286.24/286.54	= { by lemma 161 }
286.24/286.54	  multiplication(antidomain(X), codomain(Y))
286.24/286.54	
286.24/286.54	Lemma 196: domain_difference(antidomain(X), codomain(Y)) = multiplication(antidomain(X), coantidomain(Y)).
286.24/286.54	Proof:
286.24/286.54	  domain_difference(antidomain(X), codomain(Y))
286.24/286.54	= { by lemma 194 }
286.24/286.54	  multiplication(domain(antidomain(X)), coantidomain(Y))
286.24/286.54	= { by lemma 161 }
286.24/286.54	  multiplication(antidomain(X), coantidomain(Y))
286.24/286.54	
286.24/286.54	Lemma 197: coantidomain(multiplication(codomain(X), Y)) = antidomain(backward_diamond(Y, X)).
286.24/286.54	Proof:
286.24/286.54	  coantidomain(multiplication(codomain(X), Y))
286.24/286.54	= { by lemma 93 }
286.24/286.54	  coantidomain(codomain(multiplication(codomain(X), Y)))
286.24/286.54	= { by axiom 24 (backward_diamond) }
286.24/286.54	  coantidomain(backward_diamond(Y, X))
286.24/286.54	= { by lemma 188 }
286.24/286.54	  antidomain(backward_diamond(Y, X))
286.24/286.54	
286.24/286.54	Lemma 198: multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(Y))) = domain_difference(X, codomain(antidomain(Y))).
286.24/286.54	Proof:
286.24/286.54	  multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(Y)))
286.24/286.54	= { by lemma 131 }
286.24/286.54	  multiplication(domain(X), multiplication(domain(Y), coantidomain(antidomain(Y))))
286.24/286.54	= { by lemma 88 }
286.24/286.54	  multiplication(domain(X), coantidomain(antidomain(Y)))
286.24/286.54	= { by lemma 194 }
286.24/286.54	  domain_difference(X, codomain(antidomain(Y)))
286.24/286.54	
286.24/286.54	Lemma 199: backward_box(X, zero) = coantidomain(X).
286.24/286.54	Proof:
286.24/286.54	  backward_box(X, zero)
286.24/286.54	= { by axiom 26 (backward_box) }
286.24/286.54	  c(backward_diamond(X, c(zero)))
286.24/286.54	= { by lemma 49 }
286.24/286.54	  c(backward_diamond(X, one))
286.24/286.54	= { by lemma 85 }
286.24/286.54	  c(codomain(X))
286.24/286.54	= { by lemma 97 }
286.24/286.54	  antidomain(codomain(X))
286.24/286.54	= { by lemma 184 }
286.24/286.54	  coantidomain(X)
286.24/286.54	
286.24/286.54	Lemma 200: antidomain(multiplication(X, coantidomain(antidomain(Y)))) = forward_box(X, codomain(antidomain(Y))).
286.24/286.54	Proof:
286.24/286.54	  antidomain(multiplication(X, coantidomain(antidomain(Y))))
286.24/286.54	= { by lemma 187 }
286.24/286.54	  antidomain(multiplication(X, backward_box(one, Y)))
286.24/286.54	= { by lemma 149 }
286.24/286.54	  antidomain(forward_diamond(X, backward_box(one, Y)))
286.24/286.54	= { by lemma 97 }
286.24/286.54	  c(forward_diamond(X, backward_box(one, Y)))
286.24/286.54	= { by lemma 86 }
286.24/286.54	  forward_box(X, backward_diamond(one, c(Y)))
286.24/286.54	= { by lemma 40 }
286.24/286.54	  forward_box(X, codomain(codomain(c(Y))))
286.24/286.54	= { by lemma 94 }
286.24/286.54	  forward_box(X, codomain(c(Y)))
286.24/286.54	= { by lemma 97 }
286.24/286.54	  forward_box(X, codomain(antidomain(Y)))
286.24/286.54	
286.24/286.54	Lemma 201: forward_box(coantidomain(X), codomain(X)) = codomain(X).
286.24/286.54	Proof:
286.24/286.54	  forward_box(coantidomain(X), codomain(X))
286.24/286.54	= { by lemma 117 }
286.24/286.54	  antidomain(multiplication(coantidomain(X), antidomain(codomain(X))))
286.24/286.54	= { by lemma 183 }
286.24/286.54	  antidomain(coantidomain(X))
286.24/286.54	= { by lemma 185 }
286.24/286.54	  codomain(X)
286.24/286.54	
286.24/286.54	Lemma 202: domain(multiplication(X, coantidomain(antidomain(Y)))) = forward_diamond(X, coantidomain(antidomain(Y))).
286.24/286.54	Proof:
286.24/286.54	  domain(multiplication(X, coantidomain(antidomain(Y))))
286.24/286.54	= { by axiom 23 (domain4) }
286.24/286.54	  antidomain(antidomain(multiplication(X, coantidomain(antidomain(Y)))))
286.24/286.54	= { by lemma 200 }
286.24/286.54	  antidomain(forward_box(X, codomain(antidomain(Y))))
286.24/286.54	= { by lemma 97 }
286.24/286.54	  antidomain(forward_box(X, codomain(c(Y))))
286.24/286.54	= { by lemma 94 }
286.24/286.54	  antidomain(forward_box(X, codomain(codomain(c(Y)))))
286.24/286.54	= { by lemma 40 }
286.24/286.54	  antidomain(forward_box(X, backward_diamond(one, c(Y))))
286.24/286.54	= { by lemma 86 }
286.24/286.54	  antidomain(c(forward_diamond(X, backward_box(one, Y))))
286.24/286.54	= { by lemma 97 }
286.24/286.54	  antidomain(antidomain(forward_diamond(X, backward_box(one, Y))))
286.24/286.54	= { by axiom 23 (domain4) }
286.24/286.54	  domain(forward_diamond(X, backward_box(one, Y)))
286.24/286.54	= { by lemma 98 }
286.24/286.54	  forward_diamond(X, backward_box(one, Y))
286.24/286.54	= { by lemma 187 }
286.24/286.54	  forward_diamond(X, coantidomain(antidomain(Y)))
286.24/286.54	
286.24/286.54	Lemma 203: forward_box(antidomain(X), codomain(domain(X))) = codomain(domain(X)).
286.24/286.54	Proof:
286.24/286.54	  forward_box(antidomain(X), codomain(domain(X)))
286.24/286.54	= { by axiom 23 (domain4) }
286.24/286.54	  forward_box(antidomain(X), codomain(antidomain(antidomain(X))))
286.24/286.54	= { by lemma 200 }
286.24/286.54	  antidomain(multiplication(antidomain(X), coantidomain(antidomain(antidomain(X)))))
286.24/286.54	= { by axiom 23 (domain4) }
286.24/286.54	  antidomain(multiplication(antidomain(X), coantidomain(domain(X))))
286.24/286.54	= { by lemma 122 }
286.24/286.54	  antidomain(coantidomain(domain(X)))
286.24/286.54	= { by lemma 185 }
286.24/286.54	  codomain(domain(X))
286.24/286.54	
286.24/286.54	Lemma 204: domain(multiplication(X, coantidomain(Y))) = forward_diamond(X, coantidomain(Y)).
286.24/286.54	Proof:
286.24/286.54	  domain(multiplication(X, coantidomain(Y)))
286.24/286.54	= { by lemma 199 }
286.24/286.54	  domain(multiplication(X, backward_box(Y, zero)))
286.24/286.54	= { by lemma 148 }
286.24/286.54	  forward_diamond(X, backward_box(Y, zero))
286.24/286.54	= { by lemma 199 }
286.24/286.54	  forward_diamond(X, coantidomain(Y))
286.24/286.54	
286.24/286.54	Lemma 205: domain_difference(coantidomain(X), antidomain(Y)) = multiplication(coantidomain(X), domain(Y)).
286.24/286.54	Proof:
286.24/286.54	  domain_difference(coantidomain(X), antidomain(Y))
286.24/286.54	= { by lemma 199 }
286.24/286.54	  domain_difference(backward_box(X, zero), antidomain(Y))
286.24/286.54	= { by lemma 151 }
286.24/286.54	  multiplication(backward_box(X, zero), antidomain(antidomain(Y)))
286.24/286.54	= { by lemma 199 }
286.24/286.54	  multiplication(coantidomain(X), antidomain(antidomain(Y)))
286.24/286.54	= { by axiom 23 (domain4) }
286.24/286.54	  multiplication(coantidomain(X), domain(Y))
286.24/286.54	
286.24/286.54	Lemma 206: multiplication(codomain(X), antidomain(Y)) = domain_difference(codomain(X), Y).
286.24/286.54	Proof:
286.24/286.54	  multiplication(codomain(X), antidomain(Y))
286.24/286.54	= { by lemma 201 }
286.24/286.54	  multiplication(forward_box(coantidomain(X), codomain(X)), antidomain(Y))
286.24/286.54	= { by lemma 150 }
286.24/286.54	  domain_difference(forward_box(coantidomain(X), codomain(X)), Y)
286.24/286.54	= { by lemma 201 }
286.24/286.57	  domain_difference(codomain(X), Y)
286.24/286.57	
286.24/286.57	Lemma 207: antidomain(multiplication(X, codomain(Y))) = forward_box(X, coantidomain(Y)).
286.24/286.57	Proof:
286.24/286.57	  antidomain(multiplication(X, codomain(Y)))
286.24/286.57	= { by lemma 85 }
286.24/286.57	  antidomain(multiplication(X, backward_diamond(Y, one)))
286.24/286.57	= { by lemma 48 }
286.24/286.57	  antidomain(multiplication(X, backward_diamond(Y, antidomain(zero))))
286.24/286.57	= { by lemma 34 }
286.24/286.57	  antidomain(multiplication(X, backward_diamond(Y, antidomain(antidomain(one)))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(multiplication(X, backward_diamond(Y, domain(one))))
286.24/286.57	= { by lemma 189 }
286.24/286.57	  antidomain(multiplication(X, domain(backward_diamond(Y, domain(one)))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, domain(one))))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, antidomain(antidomain(one)))))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, antidomain(c(one)))))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, c(c(one)))))))
286.24/286.57	= { by lemma 39 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, domain(antidomain(c(one))))))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, domain(c(c(one))))))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, antidomain(antidomain(c(c(one)))))))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, antidomain(c(c(c(one)))))))))
286.24/286.57	= { by lemma 42 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, domain(domain(c(c(one)))))))))
286.24/286.57	= { by lemma 43 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, domain(c(domain(c(one)))))))))
286.24/286.57	= { by lemma 43 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, c(domain(domain(c(one)))))))))
286.24/286.57	= { by lemma 41 }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, c(forward_diamond(one, c(one))))))))
286.24/286.57	= { by axiom 25 (forward_box) }
286.24/286.57	  antidomain(multiplication(X, antidomain(antidomain(backward_diamond(Y, forward_box(one, one))))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(c(backward_diamond(Y, forward_box(one, one))))))
286.24/286.57	= { by axiom 25 (forward_box) }
286.24/286.57	  antidomain(multiplication(X, antidomain(c(backward_diamond(Y, c(forward_diamond(one, c(one))))))))
286.24/286.57	= { by axiom 26 (backward_box) }
286.24/286.57	  antidomain(multiplication(X, antidomain(backward_box(Y, forward_diamond(one, c(one))))))
286.24/286.57	= { by lemma 41 }
286.24/286.57	  antidomain(multiplication(X, antidomain(backward_box(Y, domain(domain(c(one)))))))
286.24/286.57	= { by lemma 111 }
286.24/286.57	  antidomain(multiplication(X, antidomain(backward_box(Y, domain(c(one))))))
286.24/286.57	= { by lemma 111 }
286.24/286.57	  antidomain(multiplication(X, antidomain(backward_box(Y, c(one)))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(multiplication(X, antidomain(backward_box(Y, antidomain(one)))))
286.24/286.57	= { by lemma 117 }
286.24/286.57	  forward_box(X, backward_box(Y, antidomain(one)))
286.24/286.57	= { by lemma 34 }
286.24/286.57	  forward_box(X, backward_box(Y, zero))
286.24/286.57	= { by lemma 199 }
286.24/286.57	  forward_box(X, coantidomain(Y))
286.24/286.57	
286.24/286.57	Lemma 208: antidomain(forward_box(X, coantidomain(Y))) = forward_diamond(X, codomain(Y)).
286.24/286.57	Proof:
286.24/286.57	  antidomain(forward_box(X, coantidomain(Y)))
286.24/286.57	= { by lemma 207 }
286.24/286.57	  antidomain(antidomain(multiplication(X, codomain(Y))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  domain(multiplication(X, codomain(Y)))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  domain(multiplication(X, forward_box(coantidomain(Y), codomain(Y))))
286.24/286.57	= { by axiom 25 (forward_box) }
286.24/286.57	  domain(multiplication(X, c(forward_diamond(coantidomain(Y), c(codomain(Y))))))
286.24/286.57	= { by lemma 73 }
286.24/286.57	  forward_diamond(X, antidomain(forward_diamond(coantidomain(Y), c(codomain(Y)))))
286.24/286.57	= { by lemma 113 }
286.24/286.57	  antidomain(forward_box(X, forward_diamond(coantidomain(Y), c(codomain(Y)))))
286.24/286.57	= { by axiom 25 (forward_box) }
286.24/286.57	  antidomain(c(forward_diamond(X, c(forward_diamond(coantidomain(Y), c(codomain(Y)))))))
286.24/286.57	= { by axiom 25 (forward_box) }
286.24/286.57	  antidomain(c(forward_diamond(X, forward_box(coantidomain(Y), codomain(Y)))))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  antidomain(antidomain(forward_diamond(X, forward_box(coantidomain(Y), codomain(Y)))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  domain(forward_diamond(X, forward_box(coantidomain(Y), codomain(Y))))
286.24/286.57	= { by lemma 98 }
286.24/286.57	  forward_diamond(X, forward_box(coantidomain(Y), codomain(Y)))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  forward_diamond(X, codomain(Y))
286.24/286.57	
286.24/286.57	Lemma 209: antidomain(forward_diamond(X, codomain(Y))) = forward_box(X, coantidomain(Y)).
286.24/286.57	Proof:
286.24/286.57	  antidomain(forward_diamond(X, codomain(Y)))
286.24/286.57	= { by lemma 208 }
286.24/286.57	  antidomain(antidomain(forward_box(X, coantidomain(Y))))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  domain(forward_box(X, coantidomain(Y)))
286.24/286.57	= { by lemma 99 }
286.24/286.57	  forward_box(X, coantidomain(Y))
286.24/286.57	
286.24/286.57	Lemma 210: domain(domain_difference(codomain(X), Y)) = antidomain(forward_box(codomain(X), Y)).
286.24/286.57	Proof:
286.24/286.57	  domain(domain_difference(codomain(X), Y))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  domain(domain_difference(forward_box(coantidomain(X), codomain(X)), Y))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(antidomain(domain_difference(forward_box(coantidomain(X), codomain(X)), Y)))
286.24/286.57	= { by lemma 121 }
286.24/286.57	  antidomain(forward_box(domain_difference(forward_box(coantidomain(X), codomain(X)), Y), Y))
286.24/286.57	= { by lemma 150 }
286.24/286.57	  antidomain(forward_box(multiplication(forward_box(coantidomain(X), codomain(X)), antidomain(Y)), Y))
286.24/286.57	= { by lemma 142 }
286.24/286.57	  antidomain(forward_box(forward_box(coantidomain(X), codomain(X)), Y))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  antidomain(forward_box(codomain(X), Y))
286.24/286.57	
286.24/286.57	Lemma 211: multiplication(addition(antidomain(Y), X), domain_difference(Y, Z)) = multiplication(X, domain_difference(Y, Z)).
286.24/286.57	Proof:
286.24/286.57	  multiplication(addition(antidomain(Y), X), domain_difference(Y, Z))
286.24/286.57	= { by axiom 5 (additive_commutativity) }
286.24/286.57	  multiplication(addition(X, antidomain(Y)), domain_difference(Y, Z))
286.24/286.57	= { by lemma 176 }
286.24/286.57	  multiplication(X, domain_difference(Y, Z))
286.24/286.57	
286.24/286.57	Lemma 212: multiplication(domain_difference(X, Y), addition(Z, antidomain(X))) = multiplication(domain_difference(X, Y), Z).
286.24/286.57	Proof:
286.24/286.57	  multiplication(domain_difference(X, Y), addition(Z, antidomain(X)))
286.24/286.57	= { by lemma 119 }
286.24/286.57	  multiplication(domain(X), multiplication(antidomain(Y), addition(Z, antidomain(X))))
286.24/286.57	= { by lemma 161 }
286.24/286.57	  multiplication(domain(X), multiplication(domain(antidomain(Y)), addition(Z, antidomain(X))))
286.24/286.57	= { by lemma 145 }
286.24/286.57	  multiplication(domain(X), addition(domain_difference(antidomain(Y), X), multiplication(domain(antidomain(Y)), Z)))
286.24/286.57	= { by lemma 161 }
286.24/286.57	  multiplication(domain(X), addition(domain_difference(antidomain(Y), X), multiplication(antidomain(Y), Z)))
286.24/286.57	= { by axiom 5 (additive_commutativity) }
286.24/286.57	  multiplication(domain(X), addition(multiplication(antidomain(Y), Z), domain_difference(antidomain(Y), X)))
286.24/286.57	= { by lemma 168 }
286.24/286.57	  multiplication(domain(X), multiplication(antidomain(Y), Z))
286.24/286.57	= { by lemma 119 }
286.24/286.57	  multiplication(domain_difference(X, Y), Z)
286.24/286.57	
286.24/286.57	Lemma 213: multiplication(antidomain(Y), domain(X)) = domain_difference(X, Y).
286.24/286.57	Proof:
286.24/286.57	  multiplication(antidomain(Y), domain(X))
286.24/286.57	= { by lemma 164 }
286.24/286.57	  domain_difference(antidomain(Y), antidomain(X))
286.24/286.57	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.57	  multiplication(one, domain_difference(antidomain(Y), antidomain(X)))
286.24/286.57	= { by lemma 47 }
286.24/286.57	  multiplication(addition(antidomain(antidomain(X)), domain(antidomain(X))), domain_difference(antidomain(Y), antidomain(X)))
286.24/286.57	= { by axiom 5 (additive_commutativity) }
286.24/286.57	  multiplication(addition(domain(antidomain(X)), antidomain(antidomain(X))), domain_difference(antidomain(Y), antidomain(X)))
286.24/286.57	= { by axiom 4 (left_distributivity) }
286.24/286.57	  addition(multiplication(domain(antidomain(X)), domain_difference(antidomain(Y), antidomain(X))), multiplication(antidomain(antidomain(X)), domain_difference(antidomain(Y), antidomain(X))))
286.24/286.57	= { by lemma 139 }
286.24/286.57	  addition(zero, multiplication(antidomain(antidomain(X)), domain_difference(antidomain(Y), antidomain(X))))
286.24/286.57	= { by lemma 35 }
286.24/286.57	  multiplication(antidomain(antidomain(X)), domain_difference(antidomain(Y), antidomain(X)))
286.24/286.57	= { by lemma 164 }
286.24/286.57	  multiplication(antidomain(antidomain(X)), multiplication(antidomain(Y), domain(X)))
286.24/286.57	= { by lemma 135 }
286.24/286.57	  multiplication(domain_difference(antidomain(antidomain(X)), Y), domain(X))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  multiplication(domain_difference(domain(X), Y), domain(X))
286.24/286.57	= { by lemma 74 }
286.24/286.57	  multiplication(domain_difference(forward_diamond(X, one), Y), domain(X))
286.24/286.57	= { by lemma 74 }
286.24/286.57	  multiplication(domain_difference(forward_diamond(X, one), Y), forward_diamond(X, one))
286.24/286.57	= { by lemma 212 }
286.24/286.57	  multiplication(domain_difference(forward_diamond(X, one), Y), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one))))
286.24/286.57	= { by lemma 89 }
286.24/286.57	  multiplication(domain_difference(forward_diamond(X, one), Y), one)
286.24/286.57	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.57	  domain_difference(forward_diamond(X, one), Y)
286.24/286.57	= { by lemma 74 }
286.24/286.57	  domain_difference(domain(X), Y)
286.24/286.57	= { by lemma 105 }
286.24/286.57	  domain_difference(X, Y)
286.24/286.57	
286.24/286.57	Lemma 214: domain_difference(X, antidomain(Y)) = domain_difference(Y, antidomain(X)).
286.24/286.57	Proof:
286.24/286.57	  domain_difference(X, antidomain(Y))
286.24/286.57	= { by lemma 213 }
286.24/286.57	  multiplication(antidomain(antidomain(Y)), domain(X))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  multiplication(domain(Y), domain(X))
286.24/286.57	= { by lemma 58 }
286.24/286.57	  domain_difference(Y, antidomain(X))
286.24/286.57	
286.24/286.57	Lemma 215: forward_diamond(antidomain(X), Y) = domain(domain_difference(Y, X)).
286.24/286.57	Proof:
286.24/286.57	  forward_diamond(antidomain(X), Y)
286.24/286.57	= { by axiom 27 (forward_diamond) }
286.24/286.57	  domain(multiplication(antidomain(X), domain(Y)))
286.24/286.57	= { by lemma 213 }
286.24/286.57	  domain(domain_difference(Y, X))
286.24/286.57	
286.24/286.57	Lemma 216: domain_difference(antidomain(X), Y) = domain_difference(antidomain(Y), X).
286.24/286.57	Proof:
286.24/286.57	  domain_difference(antidomain(X), Y)
286.24/286.57	= { by lemma 213 }
286.24/286.57	  multiplication(antidomain(Y), domain(antidomain(X)))
286.24/286.57	= { by lemma 161 }
286.24/286.57	  multiplication(antidomain(Y), antidomain(X))
286.24/286.57	= { by lemma 124 }
286.24/286.57	  domain_difference(antidomain(Y), X)
286.24/286.57	
286.24/286.57	Lemma 217: codomain(domain_difference(codomain(Y), X)) = backward_diamond(antidomain(X), Y).
286.24/286.57	Proof:
286.24/286.57	  codomain(domain_difference(codomain(Y), X))
286.24/286.57	= { by lemma 206 }
286.24/286.57	  codomain(multiplication(codomain(Y), antidomain(X)))
286.24/286.57	= { by axiom 24 (backward_diamond) }
286.24/286.57	  backward_diamond(antidomain(X), Y)
286.24/286.57	
286.24/286.57	Lemma 218: multiplication(codomain(X), domain(Y)) = domain_difference(Y, coantidomain(X)).
286.24/286.57	Proof:
286.24/286.57	  multiplication(codomain(X), domain(Y))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  multiplication(forward_box(coantidomain(X), codomain(X)), domain(Y))
286.24/286.57	= { by lemma 117 }
286.24/286.57	  multiplication(antidomain(multiplication(coantidomain(X), antidomain(codomain(X)))), domain(Y))
286.24/286.57	= { by lemma 97 }
286.24/286.57	  multiplication(c(multiplication(coantidomain(X), antidomain(codomain(X)))), domain(Y))
286.24/286.57	= { by lemma 39 }
286.24/286.57	  multiplication(domain(antidomain(multiplication(coantidomain(X), antidomain(codomain(X))))), domain(Y))
286.24/286.57	= { by lemma 58 }
286.24/286.57	  domain_difference(antidomain(multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(Y))
286.24/286.57	= { by lemma 117 }
286.24/286.57	  domain_difference(forward_box(coantidomain(X), codomain(X)), antidomain(Y))
286.24/286.57	= { by lemma 201 }
286.24/286.57	  domain_difference(codomain(X), antidomain(Y))
286.24/286.57	= { by lemma 214 }
286.24/286.57	  domain_difference(Y, antidomain(codomain(X)))
286.24/286.57	= { by lemma 184 }
286.24/286.57	  domain_difference(Y, coantidomain(X))
286.24/286.57	
286.24/286.57	Lemma 219: domain(domain_difference(Y, codomain(X))) = forward_diamond(coantidomain(X), Y).
286.24/286.57	Proof:
286.24/286.57	  domain(domain_difference(Y, codomain(X)))
286.24/286.57	= { by lemma 194 }
286.24/286.57	  domain(multiplication(domain(Y), coantidomain(X)))
286.24/286.57	= { by lemma 204 }
286.24/286.57	  forward_diamond(domain(Y), coantidomain(X))
286.24/286.57	= { by lemma 77 }
286.24/286.57	  domain(domain_difference(Y, antidomain(coantidomain(X))))
286.24/286.57	= { by lemma 214 }
286.24/286.57	  domain(domain_difference(coantidomain(X), antidomain(Y)))
286.24/286.57	= { by lemma 199 }
286.24/286.57	  domain(domain_difference(backward_box(X, zero), antidomain(Y)))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  antidomain(antidomain(domain_difference(backward_box(X, zero), antidomain(Y))))
286.24/286.57	= { by lemma 121 }
286.24/286.57	  antidomain(forward_box(domain_difference(backward_box(X, zero), antidomain(Y)), antidomain(Y)))
286.24/286.57	= { by lemma 151 }
286.24/286.57	  antidomain(forward_box(multiplication(backward_box(X, zero), antidomain(antidomain(Y))), antidomain(Y)))
286.24/286.57	= { by lemma 142 }
286.24/286.57	  antidomain(forward_box(backward_box(X, zero), antidomain(Y)))
286.24/286.57	= { by lemma 199 }
286.24/286.57	  antidomain(forward_box(coantidomain(X), antidomain(Y)))
286.24/286.57	= { by lemma 114 }
286.24/286.57	  antidomain(antidomain(forward_diamond(coantidomain(X), Y)))
286.24/286.57	= { by axiom 23 (domain4) }
286.24/286.57	  domain(forward_diamond(coantidomain(X), Y))
286.24/286.57	= { by lemma 98 }
286.24/286.57	  forward_diamond(coantidomain(X), Y)
286.24/286.57	
286.24/286.57	Lemma 220: multiplication(antidomain(X), coantidomain(Y)) = domain_difference(coantidomain(Y), X).
286.24/286.57	Proof:
286.24/286.57	  multiplication(antidomain(X), coantidomain(Y))
286.24/286.57	= { by lemma 196 }
286.24/286.57	  domain_difference(antidomain(X), codomain(Y))
286.24/286.57	= { by lemma 216 }
286.24/286.57	  domain_difference(antidomain(codomain(Y)), X)
286.24/286.57	= { by lemma 184 }
286.24/286.57	  domain_difference(coantidomain(Y), X)
286.24/286.57	
286.24/286.57	Lemma 221: multiplication(antidomain(Y), codomain(X)) = domain_difference(codomain(X), Y).
286.24/286.57	Proof:
286.24/286.57	  multiplication(antidomain(Y), codomain(X))
286.24/286.57	= { by lemma 195 }
286.24/286.57	  domain_difference(antidomain(Y), coantidomain(X))
286.24/286.57	= { by lemma 216 }
286.24/286.57	  domain_difference(antidomain(coantidomain(X)), Y)
286.24/286.57	= { by lemma 185 }
286.24/286.58	  domain_difference(codomain(X), Y)
286.24/286.58	
286.24/286.58	Lemma 222: domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))) = forward_diamond(X, Y).
286.24/286.58	Proof:
286.24/286.58	  domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.58	  multiplication(one, domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))))
286.24/286.58	= { by lemma 89 }
286.24/286.58	  multiplication(addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))))
286.24/286.58	= { by lemma 176 }
286.24/286.58	  multiplication(forward_diamond(X, Y), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))))
286.24/286.58	= { by lemma 211 }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))))
286.24/286.58	= { by lemma 218 }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), domain(forward_diamond(X, Y))))
286.24/286.58	= { by axiom 11 (multiplicative_associativity) }
286.24/286.58	  multiplication(multiplication(addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)), codomain(forward_diamond(X, Y))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 141 }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), multiplication(addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)), codomain(forward_diamond(X, Y)))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), multiplication(addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))), codomain(forward_diamond(X, Y)))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 4 (left_distributivity) }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), addition(multiplication(forward_diamond(X, Y), codomain(forward_diamond(X, Y))), multiplication(antidomain(forward_diamond(X, Y)), codomain(forward_diamond(X, Y))))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 81 }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), addition(forward_diamond(X, Y), multiplication(antidomain(forward_diamond(X, Y)), codomain(forward_diamond(X, Y))))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), addition(multiplication(antidomain(forward_diamond(X, Y)), codomain(forward_diamond(X, Y))), forward_diamond(X, Y))), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 12 (additive_associativity) }
286.24/286.58	  multiplication(addition(addition(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), codomain(forward_diamond(X, Y)))), forward_diamond(X, Y)), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 67 }
286.24/286.58	  multiplication(addition(multiplication(antidomain(forward_diamond(X, Y)), addition(codomain(forward_diamond(X, Y)), one)), forward_diamond(X, Y)), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  multiplication(addition(multiplication(antidomain(forward_diamond(X, Y)), addition(one, codomain(forward_diamond(X, Y)))), forward_diamond(X, Y)), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 54 }
286.24/286.58	  multiplication(addition(multiplication(antidomain(forward_diamond(X, Y)), one), forward_diamond(X, Y)), domain(forward_diamond(X, Y)))
286.24/286.58	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.58	  multiplication(addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 141 }
286.24/286.58	  multiplication(forward_diamond(X, Y), domain(forward_diamond(X, Y)))
286.24/286.58	= { by lemma 98 }
286.24/286.58	  multiplication(forward_diamond(X, Y), forward_diamond(X, Y))
286.24/286.58	= { by axiom 27 (forward_diamond) }
286.24/286.58	  multiplication(forward_diamond(X, Y), domain(multiplication(X, domain(Y))))
286.24/286.58	= { by axiom 23 (domain4) }
286.24/286.58	  multiplication(forward_diamond(X, Y), antidomain(antidomain(multiplication(X, domain(Y)))))
286.24/286.58	= { by lemma 87 }
286.24/286.58	  domain_difference(multiplication(X, domain(Y)), antidomain(multiplication(X, domain(Y))))
286.24/286.58	= { by axiom 28 (domain_difference) }
286.24/286.58	  multiplication(domain(multiplication(X, domain(Y))), antidomain(antidomain(multiplication(X, domain(Y)))))
286.24/286.58	= { by axiom 23 (domain4) }
286.24/286.58	  multiplication(antidomain(antidomain(multiplication(X, domain(Y)))), antidomain(antidomain(multiplication(X, domain(Y)))))
286.24/286.58	= { by lemma 72 }
286.24/286.58	  antidomain(antidomain(multiplication(X, domain(Y))))
286.24/286.58	= { by axiom 23 (domain4) }
286.24/286.58	  domain(multiplication(X, domain(Y)))
286.24/286.58	= { by axiom 27 (forward_diamond) }
286.24/286.58	  forward_diamond(X, Y)
286.24/286.58	
286.24/286.58	Lemma 223: addition(codomain(X), antidomain(addition(Y, coantidomain(X)))) = codomain(X).
286.24/286.58	Proof:
286.24/286.58	  addition(codomain(X), antidomain(addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 186 }
286.24/286.58	  addition(domain(codomain(X)), antidomain(addition(Y, coantidomain(X))))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  addition(domain(codomain(X)), antidomain(addition(coantidomain(X), Y)))
286.24/286.58	= { by axiom 7 (multiplicative_right_identity) }
286.24/286.58	  addition(domain(codomain(X)), multiplication(antidomain(addition(coantidomain(X), Y)), one))
286.24/286.58	= { by lemma 44 }
286.24/286.58	  addition(domain(codomain(X)), multiplication(antidomain(addition(coantidomain(X), Y)), addition(coantidomain(X), codomain(X))))
286.24/286.58	= { by lemma 178 }
286.24/286.58	  addition(domain(codomain(X)), multiplication(antidomain(addition(coantidomain(X), Y)), codomain(X)))
286.24/286.58	= { by lemma 221 }
286.24/286.58	  addition(domain(codomain(X)), domain_difference(codomain(X), addition(coantidomain(X), Y)))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  addition(domain(codomain(X)), domain_difference(codomain(X), addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 140 }
286.24/286.58	  domain(codomain(X))
286.24/286.58	= { by lemma 186 }
286.24/286.58	  codomain(X)
286.24/286.58	
286.24/286.58	Lemma 224: domain_difference(addition(Y, coantidomain(X)), codomain(X)) = coantidomain(X).
286.24/286.58	Proof:
286.24/286.58	  domain_difference(addition(Y, coantidomain(X)), codomain(X))
286.24/286.58	= { by lemma 94 }
286.24/286.58	  domain_difference(addition(Y, coantidomain(X)), codomain(codomain(X)))
286.24/286.58	= { by lemma 223 }
286.24/286.58	  domain_difference(addition(Y, coantidomain(X)), codomain(addition(codomain(X), antidomain(addition(Y, coantidomain(X))))))
286.24/286.58	= { by lemma 213 }
286.24/286.58	  multiplication(antidomain(codomain(addition(codomain(X), antidomain(addition(Y, coantidomain(X)))))), domain(addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 184 }
286.24/286.58	  multiplication(coantidomain(addition(codomain(X), antidomain(addition(Y, coantidomain(X))))), domain(addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 205 }
286.24/286.58	  domain_difference(coantidomain(addition(codomain(X), antidomain(addition(Y, coantidomain(X))))), antidomain(addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 161 }
286.24/286.58	  domain_difference(coantidomain(addition(codomain(X), domain(antidomain(addition(Y, coantidomain(X)))))), antidomain(addition(Y, coantidomain(X))))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  domain_difference(coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))), antidomain(addition(Y, coantidomain(X))))
286.24/286.58	= { by lemma 220 }
286.24/286.58	  multiplication(antidomain(antidomain(addition(Y, coantidomain(X)))), coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by lemma 92 }
286.24/286.58	  multiplication(addition(antidomain(antidomain(addition(Y, coantidomain(X)))), addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))), coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by lemma 51 }
286.24/286.58	  multiplication(addition(antidomain(antidomain(addition(Y, coantidomain(X)))), addition(domain(antidomain(addition(Y, coantidomain(X)))), addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X)))), coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by lemma 64 }
286.24/286.58	  multiplication(addition(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X)), addition(antidomain(antidomain(addition(Y, coantidomain(X)))), domain(antidomain(addition(Y, coantidomain(X)))))), coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by lemma 78 }
286.24/286.58	  multiplication(addition(antidomain(antidomain(addition(Y, coantidomain(X)))), domain(antidomain(addition(Y, coantidomain(X))))), coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by lemma 47 }
286.24/286.58	  multiplication(one, coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X))))
286.24/286.58	= { by axiom 6 (multiplicative_left_identity) }
286.24/286.58	  coantidomain(addition(domain(antidomain(addition(Y, coantidomain(X)))), codomain(X)))
286.24/286.58	= { by axiom 5 (additive_commutativity) }
286.24/286.58	  coantidomain(addition(codomain(X), domain(antidomain(addition(Y, coantidomain(X))))))
286.24/286.58	= { by lemma 161 }
286.24/286.58	  coantidomain(addition(codomain(X), antidomain(addition(Y, coantidomain(X)))))
286.24/286.58	= { by lemma 223 }
286.24/286.58	  coantidomain(codomain(X))
286.24/286.58	= { by lemma 93 }
286.24/286.58	  coantidomain(X)
286.24/286.58	
286.24/286.58	Lemma 225: forward_box(domain_difference(X, Y), Z) = forward_box(domain_difference(X, Z), Y).
286.24/286.58	Proof:
286.24/286.58	  forward_box(domain_difference(X, Y), Z)
286.24/286.58	= { by lemma 101 }
286.24/286.58	  forward_box(domain_difference(X, domain(Y)), Z)
286.24/286.58	= { by axiom 23 (domain4) }
286.24/286.58	  forward_box(domain_difference(X, antidomain(antidomain(Y))), Z)
286.24/286.58	= { by lemma 177 }
286.24/286.58	  antidomain(multiplication(domain(X), domain_difference(antidomain(Y), Z)))
286.24/286.58	= { by lemma 216 }
286.24/286.58	  antidomain(multiplication(domain(X), domain_difference(antidomain(Z), Y)))
286.24/286.58	= { by lemma 177 }
286.24/286.58	  forward_box(domain_difference(X, antidomain(antidomain(Z))), Y)
286.24/286.58	= { by axiom 23 (domain4) }
286.24/286.58	  forward_box(domain_difference(X, domain(Z)), Y)
286.24/286.58	= { by lemma 101 }
286.24/286.58	  forward_box(domain_difference(X, Z), Y)
286.24/286.58	
286.24/286.58	Lemma 226: domain_difference(coantidomain(X), codomain(Y)) = multiplication(coantidomain(X), coantidomain(Y)).
286.24/286.58	Proof:
286.24/286.58	  domain_difference(coantidomain(X), codomain(Y))
286.24/286.58	= { by lemma 199 }
286.24/286.58	  domain_difference(backward_box(X, zero), codomain(Y))
286.24/286.58	= { by lemma 151 }
286.24/286.58	  multiplication(backward_box(X, zero), antidomain(codomain(Y)))
286.24/286.58	= { by lemma 199 }
286.24/286.58	  multiplication(coantidomain(X), antidomain(codomain(Y)))
286.24/286.58	= { by lemma 184 }
286.24/286.58	  multiplication(coantidomain(X), coantidomain(Y))
286.24/286.58	
286.24/286.58	Lemma 227: multiplication(antidomain(Z), forward_diamond(X, Y)) = domain_difference(forward_diamond(X, Y), Z).
286.24/286.58	Proof:
286.24/286.58	  multiplication(antidomain(Z), forward_diamond(X, Y))
286.24/286.58	= { by axiom 27 (forward_diamond) }
286.24/286.58	  multiplication(antidomain(Z), domain(multiplication(X, domain(Y))))
286.24/286.58	= { by lemma 213 }
286.24/286.58	  domain_difference(multiplication(X, domain(Y)), Z)
286.24/286.58	= { by lemma 166 }
286.24/286.59	  domain_difference(forward_diamond(X, Y), Z)
286.24/286.59	
286.24/286.59	Lemma 228: forward_diamond(addition(X, multiplication(antidomain(forward_diamond(Y, Z)), Y)), Z) = forward_diamond(X, Z).
286.24/286.59	Proof:
286.24/286.59	  forward_diamond(addition(X, multiplication(antidomain(forward_diamond(Y, Z)), Y)), Z)
286.24/286.59	= { by lemma 116 }
286.24/286.59	  forward_diamond(addition(X, multiplication(antidomain(multiplication(Y, domain(Z))), Y)), Z)
286.24/286.59	= { by axiom 27 (forward_diamond) }
286.24/286.59	  domain(multiplication(addition(X, multiplication(antidomain(multiplication(Y, domain(Z))), Y)), domain(Z)))
286.24/286.59	= { by lemma 179 }
286.24/286.59	  domain(multiplication(X, domain(Z)))
286.24/286.59	= { by axiom 27 (forward_diamond) }
286.34/286.61	  forward_diamond(X, Z)
286.34/286.61	
286.34/286.61	Lemma 229: antidomain(addition(X, domain(X))) = antidomain(X).
286.34/286.61	Proof:
286.34/286.61	  antidomain(addition(X, domain(X)))
286.34/286.61	= { by lemma 159 }
286.34/286.61	  antidomain(domain(addition(X, domain(X))))
286.34/286.61	= { by lemma 136 }
286.34/286.61	  antidomain(domain(multiplication(domain(X), addition(X, one))))
286.34/286.61	= { by lemma 74 }
286.34/286.61	  antidomain(domain(multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 163 }
286.34/286.61	  antidomain(forward_diamond(one, multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 89 }
286.34/286.61	  antidomain(forward_diamond(addition(forward_diamond(X, one), antidomain(forward_diamond(X, one))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 76 }
286.34/286.61	  antidomain(forward_diamond(addition(forward_diamond(X, one), domain_difference(one, forward_diamond(X, one))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 48 }
286.34/286.61	  antidomain(forward_diamond(addition(forward_diamond(X, one), domain_difference(antidomain(zero), forward_diamond(X, one))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 124 }
286.34/286.61	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(zero), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by axiom 21 (domain1) }
286.34/286.61	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(antidomain(forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one)))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.61	= { by lemma 114 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(forward_box(antidomain(forward_diamond(X, one)), antidomain(multiplication(forward_diamond(X, one), addition(X, one)))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(antidomain(multiplication(antidomain(forward_diamond(X, one)), antidomain(antidomain(multiplication(forward_diamond(X, one), addition(X, one)))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by axiom 18 (domain2) }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(antidomain(multiplication(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one)))), antidomain(multiplication(antidomain(forward_diamond(X, one)), antidomain(antidomain(multiplication(forward_diamond(X, one), addition(X, one))))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 57 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(antidomain(zero), antidomain(multiplication(antidomain(forward_diamond(X, one)), antidomain(antidomain(multiplication(forward_diamond(X, one), addition(X, one))))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 48 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(one, antidomain(multiplication(antidomain(forward_diamond(X, one)), antidomain(antidomain(multiplication(forward_diamond(X, one), addition(X, one))))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(one, forward_box(antidomain(forward_diamond(X, one)), antidomain(multiplication(forward_diamond(X, one), addition(X, one))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by axiom 25 (forward_box) }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(one, c(forward_diamond(antidomain(forward_diamond(X, one)), c(antidomain(multiplication(forward_diamond(X, one), addition(X, one))))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by axiom 29 (complement) }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(addition(one, antidomain(domain(forward_diamond(antidomain(forward_diamond(X, one)), c(antidomain(multiplication(forward_diamond(X, one), addition(X, one)))))))), forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 53 }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(multiplication(one, forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one))))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.62	  antidomain(forward_diamond(addition(forward_diamond(X, one), multiplication(antidomain(forward_diamond(antidomain(forward_diamond(X, one)), multiplication(forward_diamond(X, one), addition(X, one)))), antidomain(forward_diamond(X, one)))), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 228 }
286.34/286.62	  antidomain(forward_diamond(forward_diamond(X, one), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 74 }
286.34/286.62	  antidomain(forward_diamond(domain(X), multiplication(forward_diamond(X, one), addition(X, one))))
286.34/286.62	= { by lemma 74 }
286.34/286.62	  antidomain(forward_diamond(domain(X), multiplication(domain(X), addition(X, one))))
286.34/286.62	= { by lemma 136 }
286.34/286.62	  antidomain(forward_diamond(domain(X), addition(X, domain(X))))
286.34/286.62	= { by lemma 182 }
286.34/286.62	  antidomain(domain(X))
286.34/286.62	= { by lemma 159 }
286.34/286.62	  antidomain(X)
286.34/286.62	
286.34/286.62	Lemma 231: multiplication(domain(X), forward_diamond(Z, Y)) = multiplication(forward_diamond(Z, Y), domain(X)).
286.34/286.62	Proof:
286.34/286.62	  multiplication(domain(X), forward_diamond(Z, Y))
286.34/286.62	= { by lemma 133 }
286.34/286.62	  domain_difference(X, antidomain(forward_diamond(Z, Y)))
286.34/286.62	= { by lemma 214 }
286.34/286.62	  domain_difference(forward_diamond(Z, Y), antidomain(X))
286.34/286.62	= { by lemma 128 }
286.34/286.62	  multiplication(forward_diamond(Z, Y), antidomain(antidomain(X)))
286.34/286.62	= { by lemma 87 }
286.34/286.62	  domain_difference(multiplication(Z, domain(Y)), antidomain(X))
286.34/286.62	= { by lemma 58 }
286.34/286.62	  multiplication(domain(multiplication(Z, domain(Y))), domain(X))
286.34/286.62	= { by axiom 27 (forward_diamond) }
286.34/286.62	  multiplication(forward_diamond(Z, Y), domain(X))
286.34/286.62	
286.34/286.62	Lemma 231: multiplication(forward_diamond(Z, Y), domain(X)) = multiplication(domain(X), forward_diamond(Z, Y)).
286.34/286.62	Proof:
286.34/286.62	  multiplication(forward_diamond(Z, Y), domain(X))
286.34/286.62	= { by axiom 27 (forward_diamond) }
286.34/286.62	  multiplication(domain(multiplication(Z, domain(Y))), domain(X))
286.34/286.62	= { by lemma 58 }
286.34/286.62	  domain_difference(multiplication(Z, domain(Y)), antidomain(X))
286.34/286.62	= { by lemma 87 }
286.34/286.62	  multiplication(forward_diamond(Z, Y), antidomain(antidomain(X)))
286.34/286.62	= { by lemma 128 }
286.34/286.62	  domain_difference(forward_diamond(Z, Y), antidomain(X))
286.34/286.62	= { by lemma 214 }
286.34/286.62	  domain_difference(X, antidomain(forward_diamond(Z, Y)))
286.34/286.62	= { by lemma 133 }
286.34/286.62	  multiplication(domain(X), forward_diamond(Z, Y))
286.34/286.62	
286.34/286.62	Lemma 232: multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X))) = domain_difference(Y, codomain(antidomain(X))).
286.34/286.62	Proof:
286.34/286.62	  multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X)))
286.34/286.62	= { by lemma 214 }
286.34/286.62	  multiplication(domain_difference(Y, antidomain(X)), coantidomain(antidomain(X)))
286.34/286.62	= { by lemma 131 }
286.34/286.62	  multiplication(domain(Y), multiplication(domain(X), coantidomain(antidomain(X))))
286.34/286.62	= { by lemma 88 }
286.34/286.62	  multiplication(domain(Y), coantidomain(antidomain(X)))
286.34/286.62	= { by lemma 194 }
286.34/286.62	  domain_difference(Y, codomain(antidomain(X)))
286.34/286.62	
286.34/286.62	Lemma 233: forward_diamond(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X))) = forward_diamond(coantidomain(antidomain(X)), Y).
286.34/286.62	Proof:
286.34/286.62	  forward_diamond(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X)))
286.34/286.62	= { by lemma 214 }
286.34/286.62	  forward_diamond(domain_difference(Y, antidomain(X)), coantidomain(antidomain(X)))
286.34/286.62	= { by lemma 202 }
286.34/286.62	  domain(multiplication(domain_difference(Y, antidomain(X)), coantidomain(antidomain(X))))
286.34/286.62	= { by lemma 198 }
286.34/286.62	  domain(domain_difference(Y, codomain(antidomain(X))))
286.34/286.62	= { by lemma 219 }
286.34/286.62	  forward_diamond(coantidomain(antidomain(X)), Y)
286.34/286.62	
286.34/286.62	Lemma 234: forward_diamond(domain(Z), forward_diamond(X, Y)) = forward_diamond(forward_diamond(X, Y), Z).
286.34/286.62	Proof:
286.34/286.62	  forward_diamond(domain(Z), forward_diamond(X, Y))
286.34/286.62	= { by lemma 127 }
286.34/286.62	  domain(multiplication(domain(Z), forward_diamond(X, Y)))
286.34/286.62	= { by lemma 231 }
286.34/286.62	  domain(multiplication(forward_diamond(X, Y), domain(Z)))
286.34/286.62	= { by axiom 27 (forward_diamond) }
286.34/286.62	  forward_diamond(forward_diamond(X, Y), Z)
286.34/286.62	
286.34/286.62	Lemma 235: addition(domain_difference(Z, Y), multiplication(X, domain(Z))) = multiplication(addition(X, antidomain(Y)), domain(Z)).
286.34/286.62	Proof:
286.34/286.62	  addition(domain_difference(Z, Y), multiplication(X, domain(Z)))
286.34/286.62	= { by axiom 5 (additive_commutativity) }
286.34/286.62	  addition(multiplication(X, domain(Z)), domain_difference(Z, Y))
286.34/286.62	= { by lemma 101 }
286.34/286.62	  addition(multiplication(X, domain(Z)), domain_difference(Z, domain(Y)))
286.34/286.62	= { by axiom 23 (domain4) }
286.34/286.62	  addition(multiplication(X, domain(Z)), domain_difference(Z, antidomain(antidomain(Y))))
286.34/286.62	= { by lemma 214 }
286.34/286.62	  addition(multiplication(X, domain(Z)), domain_difference(antidomain(Y), antidomain(Z)))
286.34/286.62	= { by axiom 5 (additive_commutativity) }
286.34/286.62	  addition(domain_difference(antidomain(Y), antidomain(Z)), multiplication(X, domain(Z)))
286.34/286.62	= { by lemma 58 }
286.34/286.62	  addition(multiplication(domain(antidomain(Y)), domain(Z)), multiplication(X, domain(Z)))
286.34/286.62	= { by axiom 4 (left_distributivity) }
286.34/286.62	  multiplication(addition(domain(antidomain(Y)), X), domain(Z))
286.34/286.62	= { by axiom 5 (additive_commutativity) }
286.34/286.62	  multiplication(addition(X, domain(antidomain(Y))), domain(Z))
286.34/286.62	= { by lemma 161 }
286.34/286.62	  multiplication(addition(X, antidomain(Y)), domain(Z))
286.34/286.62	
286.34/286.62	Lemma 236: multiplication(domain_difference(X, Y), antidomain(Z)) = multiplication(antidomain(Y), domain_difference(X, Z)).
286.34/286.62	Proof:
286.34/286.62	  multiplication(domain_difference(X, Y), antidomain(Z))
286.34/286.62	= { by lemma 179 }
286.34/286.62	  multiplication(addition(domain_difference(X, Y), multiplication(antidomain(multiplication(domain(X), antidomain(Z))), domain(X))), antidomain(Z))
286.34/286.62	= { by lemma 235 }
286.34/286.62	  multiplication(multiplication(addition(antidomain(multiplication(domain(X), antidomain(Z))), antidomain(Y)), domain(X)), antidomain(Z))
286.34/286.62	= { by axiom 11 (multiplicative_associativity) }
286.34/286.62	  multiplication(addition(antidomain(multiplication(domain(X), antidomain(Z))), antidomain(Y)), multiplication(domain(X), antidomain(Z)))
286.34/286.62	= { by lemma 82 }
286.34/286.62	  multiplication(antidomain(Y), multiplication(domain(X), antidomain(Z)))
286.34/286.62	= { by axiom 28 (domain_difference) }
286.34/286.62	  multiplication(antidomain(Y), domain_difference(X, Z))
286.34/286.62	
286.34/286.62	Lemma 237: forward_diamond(domain_difference(X, Y), Z) = forward_diamond(domain_difference(Z, Y), X).
286.34/286.62	Proof:
286.34/286.62	  forward_diamond(domain_difference(X, Y), Z)
286.34/286.62	= { by lemma 98 }
286.34/286.62	  domain(forward_diamond(domain_difference(X, Y), Z))
286.34/286.62	= { by axiom 23 (domain4) }
286.34/286.62	  antidomain(antidomain(forward_diamond(domain_difference(X, Y), Z)))
286.34/286.62	= { by lemma 114 }
286.34/286.62	  antidomain(forward_box(domain_difference(X, Y), antidomain(Z)))
286.34/286.62	= { by lemma 225 }
286.34/286.62	  antidomain(forward_box(domain_difference(X, antidomain(Z)), Y))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(antidomain(multiplication(domain_difference(X, antidomain(Z)), antidomain(Y))))
286.34/286.62	= { by lemma 179 }
286.34/286.62	  antidomain(antidomain(multiplication(addition(domain_difference(X, antidomain(Z)), multiplication(antidomain(multiplication(domain(X), antidomain(Y))), domain(X))), antidomain(Y))))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(forward_box(addition(domain_difference(X, antidomain(Z)), multiplication(antidomain(multiplication(domain(X), antidomain(Y))), domain(X))), Y))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(forward_box(addition(domain_difference(X, antidomain(Z)), multiplication(forward_box(domain(X), Y), domain(X))), Y))
286.34/286.62	= { by lemma 235 }
286.34/286.62	  antidomain(forward_box(multiplication(addition(forward_box(domain(X), Y), antidomain(antidomain(Z))), domain(X)), Y))
286.34/286.62	= { by lemma 146 }
286.34/286.62	  antidomain(antidomain(multiplication(addition(forward_box(domain(X), Y), antidomain(antidomain(Z))), multiplication(domain(X), antidomain(Y)))))
286.34/286.62	= { by lemma 115 }
286.34/286.62	  antidomain(antidomain(multiplication(addition(antidomain(domain_difference(X, Y)), antidomain(antidomain(Z))), multiplication(domain(X), antidomain(Y)))))
286.34/286.62	= { by axiom 28 (domain_difference) }
286.34/286.62	  antidomain(antidomain(multiplication(addition(antidomain(domain_difference(X, Y)), antidomain(antidomain(Z))), domain_difference(X, Y))))
286.34/286.62	= { by lemma 82 }
286.34/286.62	  antidomain(antidomain(multiplication(antidomain(antidomain(Z)), domain_difference(X, Y))))
286.34/286.62	= { by axiom 23 (domain4) }
286.34/286.62	  antidomain(antidomain(multiplication(domain(Z), domain_difference(X, Y))))
286.34/286.62	= { by lemma 152 }
286.34/286.62	  antidomain(antidomain(multiplication(domain_difference(Z, antidomain(X)), antidomain(Y))))
286.34/286.62	= { by lemma 117 }
286.34/286.62	  antidomain(forward_box(domain_difference(Z, antidomain(X)), Y))
286.34/286.62	= { by lemma 225 }
286.34/286.62	  antidomain(forward_box(domain_difference(Z, Y), antidomain(X)))
286.34/286.62	= { by lemma 114 }
286.34/286.62	  antidomain(antidomain(forward_diamond(domain_difference(Z, Y), X)))
286.34/286.62	= { by axiom 23 (domain4) }
286.34/286.62	  domain(forward_diamond(domain_difference(Z, Y), X))
286.34/286.62	= { by lemma 98 }
286.34/286.62	  forward_diamond(domain_difference(Z, Y), X)
286.34/286.62	
286.34/286.62	Lemma 238: multiplication(domain(X), multiplication(forward_diamond(Y, Z), W)) = multiplication(forward_diamond(Y, Z), multiplication(domain(X), W)).
286.34/286.62	Proof:
286.34/286.62	  multiplication(domain(X), multiplication(forward_diamond(Y, Z), W))
286.34/286.62	= { by axiom 11 (multiplicative_associativity) }
286.34/286.62	  multiplication(multiplication(domain(X), forward_diamond(Y, Z)), W)
286.34/286.62	= { by lemma 231 }
286.34/286.62	  multiplication(multiplication(forward_diamond(Y, Z), domain(X)), W)
286.34/286.62	= { by axiom 11 (multiplicative_associativity) }
286.34/286.62	  multiplication(forward_diamond(Y, Z), multiplication(domain(X), W))
286.34/286.62	
286.34/286.62	Lemma 239: addition(X, multiplication(antidomain(Y), addition(X, Z))) = addition(X, multiplication(antidomain(Y), Z)).
286.34/286.62	Proof:
286.34/286.62	  addition(X, multiplication(antidomain(Y), addition(X, Z)))
286.34/286.62	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.62	  addition(multiplication(one, X), multiplication(antidomain(Y), addition(X, Z)))
286.34/286.62	= { by axiom 5 (additive_commutativity) }
286.34/286.62	  addition(multiplication(one, X), multiplication(antidomain(Y), addition(Z, X)))
286.34/286.62	= { by lemma 173 }
286.34/286.62	  addition(multiplication(antidomain(Y), Z), addition(multiplication(one, X), multiplication(antidomain(Y), X)))
286.34/286.62	= { by axiom 4 (left_distributivity) }
286.34/286.62	  addition(multiplication(antidomain(Y), Z), multiplication(addition(one, antidomain(Y)), X))
286.34/286.62	= { by lemma 53 }
286.34/286.62	  addition(multiplication(antidomain(Y), Z), multiplication(one, X))
286.34/286.62	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.62	  addition(multiplication(antidomain(Y), Z), X)
286.34/286.62	= { by axiom 5 (additive_commutativity) }
286.34/286.62	  addition(X, multiplication(antidomain(Y), Z))
286.34/286.62	
286.34/286.62	Lemma 240: addition(antidomain(X), domain_difference(X, Y)) = addition(antidomain(X), antidomain(Y)).
286.34/286.62	Proof:
286.34/286.62	  addition(antidomain(X), domain_difference(X, Y))
286.34/286.62	= { by lemma 213 }
286.34/286.62	  addition(antidomain(X), multiplication(antidomain(Y), domain(X)))
286.34/286.62	= { by lemma 239 }
286.34/286.62	  addition(antidomain(X), multiplication(antidomain(Y), addition(antidomain(X), domain(X))))
286.34/286.62	= { by lemma 47 }
286.34/286.62	  addition(antidomain(X), multiplication(antidomain(Y), one))
286.34/286.62	= { by axiom 7 (multiplicative_right_identity) }
286.34/286.64	  addition(antidomain(X), antidomain(Y))
286.34/286.64	
286.34/286.64	Lemma 241: codomain(antidomain(X)) = antidomain(X).
286.34/286.64	Proof:
286.34/286.64	  codomain(antidomain(X))
286.34/286.64	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.64	  multiplication(one, codomain(antidomain(X)))
286.34/286.64	= { by lemma 53 }
286.34/286.64	  multiplication(addition(one, antidomain(X)), codomain(antidomain(X)))
286.34/286.64	= { by lemma 153 }
286.34/286.64	  addition(antidomain(X), codomain(antidomain(X)))
286.34/286.64	= { by lemma 229 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), codomain(antidomain(X)))
286.34/286.64	= { by lemma 185 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), antidomain(coantidomain(antidomain(X))))
286.34/286.64	= { by lemma 240 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), domain_difference(addition(X, domain(X)), coantidomain(antidomain(X))))
286.34/286.64	= { by lemma 136 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), domain_difference(multiplication(domain(X), addition(X, one)), coantidomain(antidomain(X))))
286.34/286.64	= { by lemma 159 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), domain_difference(multiplication(domain(X), addition(X, one)), coantidomain(antidomain(domain(X)))))
286.34/286.64	= { by lemma 218 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), domain(multiplication(domain(X), addition(X, one)))))
286.34/286.64	= { by lemma 81 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(multiplication(domain(X), addition(X, one)))), codomain(multiplication(codomain(antidomain(domain(X))), domain(multiplication(domain(X), addition(X, one)))))))
286.34/286.64	= { by axiom 24 (backward_diamond) }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(multiplication(domain(X), addition(X, one)))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))
286.34/286.64	= { by axiom 11 (multiplicative_associativity) }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X))))))
286.34/286.64	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(one, backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 45 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(zero), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 144 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), addition(X, one)))), multiplication(antidomain(domain(X)), domain(multiplication(domain(X), addition(X, one)))))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 57 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(multiplication(antidomain(zero), multiplication(antidomain(domain(X)), domain(multiplication(domain(X), addition(X, one)))))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 135 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(multiplication(domain_difference(antidomain(zero), domain(X)), domain(multiplication(domain(X), addition(X, one))))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 48 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(multiplication(domain_difference(one, domain(X)), domain(multiplication(domain(X), addition(X, one))))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 76 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), multiplication(coantidomain(multiplication(antidomain(domain(X)), domain(multiplication(domain(X), addition(X, one))))), backward_diamond(domain(multiplication(domain(X), addition(X, one))), antidomain(domain(X)))))))
286.34/286.64	= { by lemma 137 }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(multiplication(domain(X), addition(X, one))), zero)))
286.34/286.64	= { by axiom 10 (right_annihilation) }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), multiplication(codomain(antidomain(domain(X))), zero))
286.34/286.64	= { by axiom 10 (right_annihilation) }
286.34/286.64	  addition(antidomain(addition(X, domain(X))), zero)
286.34/286.64	= { by axiom 9 (additive_identity) }
286.34/286.64	  antidomain(addition(X, domain(X)))
286.34/286.64	= { by lemma 229 }
286.34/286.64	  antidomain(X)
286.34/286.64	
286.34/286.64	Lemma 242: antidomain(forward_diamond(X, Y)) = antidomain(multiplication(X, Y)).
286.34/286.64	Proof:
286.34/286.64	  antidomain(forward_diamond(X, Y))
286.34/286.64	= { by lemma 158 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), antidomain(forward_diamond(X, Y)))
286.34/286.64	= { by lemma 240 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), domain_difference(multiplication(X, Y), forward_diamond(X, Y)))
286.34/286.64	= { by lemma 213 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 164 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), domain_difference(antidomain(forward_diamond(X, Y)), antidomain(multiplication(X, Y))))
286.34/286.64	= { by lemma 58 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain(antidomain(forward_diamond(X, Y))), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 50 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), zero), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 57 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 116 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y))))
286.34/286.64	= { by axiom 11 (multiplicative_associativity) }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, multiplication(domain(Y), Y)))), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 83 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 216 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), forward_diamond(X, Y)), domain(multiplication(X, Y))))
286.34/286.64	= { by lemma 135 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), multiplication(antidomain(forward_diamond(X, Y)), domain(multiplication(X, Y)))))
286.34/286.64	= { by lemma 144 }
286.34/286.64	  addition(antidomain(multiplication(X, Y)), zero)
286.34/286.64	= { by axiom 9 (additive_identity) }
286.34/286.64	  antidomain(multiplication(X, Y))
286.34/286.64	
286.34/286.64	Lemma 243: coantidomain(antidomain(X)) = domain(X).
286.34/286.64	Proof:
286.34/286.64	  coantidomain(antidomain(X))
286.34/286.64	= { by lemma 184 }
286.34/286.64	  antidomain(codomain(antidomain(X)))
286.34/286.64	= { by lemma 241 }
286.34/286.64	  antidomain(antidomain(X))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  domain(X)
286.34/286.64	
286.34/286.64	Lemma 244: domain(multiplication(X, Y)) = forward_diamond(X, Y).
286.34/286.64	Proof:
286.34/286.64	  domain(multiplication(X, Y))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  antidomain(antidomain(multiplication(X, Y)))
286.34/286.64	= { by lemma 242 }
286.34/286.64	  antidomain(antidomain(forward_diamond(X, Y)))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  domain(forward_diamond(X, Y))
286.34/286.64	= { by lemma 98 }
286.34/286.64	  forward_diamond(X, Y)
286.34/286.64	
286.34/286.64	Lemma 245: coantidomain(domain(X)) = antidomain(X).
286.34/286.64	Proof:
286.34/286.64	  coantidomain(domain(X))
286.34/286.64	= { by lemma 184 }
286.34/286.64	  antidomain(codomain(domain(X)))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  antidomain(codomain(antidomain(antidomain(X))))
286.34/286.64	= { by lemma 241 }
286.34/286.64	  antidomain(antidomain(antidomain(X)))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  antidomain(domain(X))
286.34/286.64	= { by lemma 159 }
286.34/286.64	  antidomain(X)
286.34/286.64	
286.34/286.64	Lemma 246: domain_difference(domain_difference(X, Y), Y) = domain(domain_difference(X, Y)).
286.34/286.64	Proof:
286.34/286.64	  domain_difference(domain_difference(X, Y), Y)
286.34/286.64	= { by lemma 105 }
286.34/286.64	  domain_difference(domain(domain_difference(X, Y)), Y)
286.34/286.64	= { by axiom 28 (domain_difference) }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), antidomain(Y))
286.34/286.64	= { by axiom 9 (additive_identity) }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), zero))
286.34/286.64	= { by lemma 144 }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), domain_difference(X, Y))), multiplication(domain(Y), domain(domain_difference(X, Y))))))
286.34/286.64	= { by lemma 139 }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(antidomain(zero), multiplication(domain(Y), domain(domain_difference(X, Y))))))
286.34/286.64	= { by lemma 48 }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(one, multiplication(domain(Y), domain(domain_difference(X, Y))))))
286.34/286.64	= { by lemma 58 }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(one, domain_difference(Y, antidomain(domain_difference(X, Y))))))
286.34/286.64	= { by axiom 6 (multiplicative_left_identity) }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), domain_difference(Y, antidomain(domain_difference(X, Y)))))
286.34/286.64	= { by lemma 240 }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), antidomain(antidomain(domain_difference(X, Y)))))
286.34/286.64	= { by axiom 23 (domain4) }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), domain(domain_difference(X, Y))))
286.34/286.64	= { by axiom 5 (additive_commutativity) }
286.34/286.64	  multiplication(domain(domain(domain_difference(X, Y))), addition(domain(domain_difference(X, Y)), antidomain(Y)))
286.34/286.64	= { by lemma 169 }
286.34/286.64	  addition(domain(domain_difference(X, Y)), domain_difference(domain(domain_difference(X, Y)), Y))
286.34/286.64	= { by lemma 105 }
286.34/286.64	  addition(domain(domain_difference(X, Y)), domain_difference(domain_difference(X, Y), Y))
286.34/286.64	= { by lemma 140 }
286.34/286.66	  domain(domain_difference(X, Y))
286.34/286.66	
286.34/286.66	Lemma 247: antidomain(multiplication(antidomain(X), Y)) = antidomain(domain_difference(Y, X)).
286.34/286.66	Proof:
286.34/286.66	  antidomain(multiplication(antidomain(X), Y))
286.34/286.66	= { by lemma 161 }
286.34/286.66	  antidomain(multiplication(domain(antidomain(X)), Y))
286.34/286.66	= { by lemma 132 }
286.34/286.66	  antidomain(multiplication(domain_difference(antidomain(X), antidomain(Y)), Y))
286.34/286.66	= { by lemma 216 }
286.34/286.66	  antidomain(multiplication(domain_difference(antidomain(antidomain(Y)), X), Y))
286.34/286.66	= { by axiom 23 (domain4) }
286.34/286.66	  antidomain(multiplication(domain_difference(domain(Y), X), Y))
286.34/286.66	= { by lemma 242 }
286.34/286.66	  antidomain(forward_diamond(domain_difference(domain(Y), X), Y))
286.34/286.66	= { by lemma 114 }
286.34/286.66	  forward_box(domain_difference(domain(Y), X), antidomain(Y))
286.34/286.66	= { by lemma 243 }
286.34/286.66	  forward_box(domain_difference(coantidomain(antidomain(Y)), X), antidomain(Y))
286.34/286.66	= { by lemma 241 }
286.34/286.66	  forward_box(domain_difference(coantidomain(antidomain(Y)), X), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 200 }
286.34/286.66	  antidomain(multiplication(domain_difference(coantidomain(antidomain(Y)), X), coantidomain(antidomain(Y))))
286.34/286.66	= { by lemma 71 }
286.34/286.66	  antidomain(multiplication(addition(domain_difference(coantidomain(antidomain(Y)), X), antidomain(coantidomain(antidomain(Y)))), coantidomain(antidomain(Y))))
286.34/286.66	= { by lemma 200 }
286.34/286.66	  forward_box(addition(domain_difference(coantidomain(antidomain(Y)), X), antidomain(coantidomain(antidomain(Y)))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 185 }
286.34/286.66	  forward_box(addition(domain_difference(coantidomain(antidomain(Y)), X), codomain(antidomain(Y))), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 5 (additive_commutativity) }
286.34/286.66	  forward_box(addition(codomain(antidomain(Y)), domain_difference(coantidomain(antidomain(Y)), X)), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 20 (codomain4) }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), domain_difference(coantidomain(antidomain(Y)), X)), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 160 }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), domain_difference(codomain(coantidomain(antidomain(Y))), X)), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 221 }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), multiplication(antidomain(X), codomain(coantidomain(antidomain(Y))))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 239 }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), multiplication(antidomain(X), addition(coantidomain(coantidomain(antidomain(Y))), codomain(coantidomain(antidomain(Y)))))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 44 }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), multiplication(antidomain(X), one)), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 7 (multiplicative_right_identity) }
286.34/286.66	  forward_box(addition(coantidomain(coantidomain(antidomain(Y))), antidomain(X)), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 5 (additive_commutativity) }
286.34/286.66	  forward_box(addition(antidomain(X), coantidomain(coantidomain(antidomain(Y)))), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 20 (codomain4) }
286.34/286.66	  forward_box(addition(antidomain(X), codomain(antidomain(Y))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 201 }
286.34/286.66	  forward_box(addition(antidomain(X), forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y)))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 99 }
286.34/286.66	  forward_box(addition(antidomain(X), domain(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y))))), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 23 (domain4) }
286.34/286.66	  forward_box(addition(antidomain(X), antidomain(antidomain(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y)))))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 97 }
286.34/286.66	  forward_box(addition(antidomain(X), antidomain(c(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y)))))), codomain(antidomain(Y)))
286.34/286.66	= { by lemma 201 }
286.34/286.66	  forward_box(addition(antidomain(X), antidomain(c(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y)))))), forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y))))
286.34/286.66	= { by axiom 25 (forward_box) }
286.34/286.66	  c(forward_diamond(addition(antidomain(X), antidomain(c(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y)))))), c(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y))))))
286.34/286.66	= { by lemma 112 }
286.34/286.66	  c(forward_diamond(antidomain(X), c(forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y))))))
286.34/286.66	= { by axiom 25 (forward_box) }
286.34/286.66	  forward_box(antidomain(X), forward_box(coantidomain(antidomain(Y)), codomain(antidomain(Y))))
286.34/286.66	= { by lemma 201 }
286.34/286.66	  forward_box(antidomain(X), codomain(antidomain(Y)))
286.34/286.66	= { by axiom 20 (codomain4) }
286.34/286.66	  forward_box(antidomain(X), coantidomain(coantidomain(antidomain(Y))))
286.34/286.66	= { by lemma 207 }
286.34/286.66	  antidomain(multiplication(antidomain(X), codomain(coantidomain(antidomain(Y)))))
286.34/286.66	= { by lemma 160 }
286.34/286.66	  antidomain(multiplication(antidomain(X), coantidomain(antidomain(Y))))
286.34/286.66	= { by lemma 199 }
286.34/286.66	  antidomain(multiplication(antidomain(X), backward_box(antidomain(Y), zero)))
286.34/286.66	= { by lemma 149 }
286.34/286.66	  antidomain(forward_diamond(antidomain(X), backward_box(antidomain(Y), zero)))
286.34/286.66	= { by lemma 199 }
286.34/286.66	  antidomain(forward_diamond(antidomain(X), coantidomain(antidomain(Y))))
286.34/286.66	= { by lemma 215 }
286.34/286.66	  antidomain(domain(domain_difference(coantidomain(antidomain(Y)), X)))
286.34/286.66	= { by lemma 159 }
286.34/286.67	  antidomain(domain_difference(coantidomain(antidomain(Y)), X))
286.34/286.67	= { by lemma 199 }
286.34/286.67	  antidomain(domain_difference(backward_box(antidomain(Y), zero), X))
286.34/286.67	= { by lemma 151 }
286.34/286.67	  antidomain(multiplication(backward_box(antidomain(Y), zero), antidomain(X)))
286.34/286.67	= { by lemma 117 }
286.34/286.67	  forward_box(backward_box(antidomain(Y), zero), X)
286.34/286.67	= { by lemma 199 }
286.34/286.67	  forward_box(coantidomain(antidomain(Y)), X)
286.34/286.67	= { by lemma 243 }
286.34/286.67	  forward_box(domain(Y), X)
286.34/286.67	= { by lemma 115 }
286.34/286.67	  antidomain(domain_difference(Y, X))
286.34/286.67	
286.34/286.67	Lemma 248: forward_diamond(multiplication(X, Y), Z) = forward_diamond(X, multiplication(Y, Z)).
286.34/286.67	Proof:
286.34/286.67	  forward_diamond(multiplication(X, Y), Z)
286.34/286.67	= { by lemma 244 }
286.34/286.67	  domain(multiplication(multiplication(X, Y), Z))
286.34/286.67	= { by axiom 11 (multiplicative_associativity) }
286.34/286.67	  domain(multiplication(X, multiplication(Y, Z)))
286.34/286.67	= { by lemma 244 }
286.34/286.67	  forward_diamond(X, multiplication(Y, Z))
286.34/286.67	
286.34/286.67	Lemma 249: forward_diamond(X, forward_diamond(Y, Z)) = forward_diamond(X, multiplication(Y, Z)).
286.34/286.67	Proof:
286.34/286.67	  forward_diamond(X, forward_diamond(Y, Z))
286.34/286.67	= { by lemma 127 }
286.34/286.67	  domain(multiplication(X, forward_diamond(Y, Z)))
286.34/286.67	= { by lemma 244 }
286.34/286.67	  domain(multiplication(X, domain(multiplication(Y, Z))))
286.34/286.67	= { by axiom 27 (forward_diamond) }
286.34/286.67	  forward_diamond(X, multiplication(Y, Z))
286.34/286.67	
286.34/286.67	Lemma 250: domain_difference(forward_diamond(X, Y), Z) = domain_difference(multiplication(X, Y), Z).
286.34/286.67	Proof:
286.34/286.67	  domain_difference(forward_diamond(X, Y), Z)
286.34/286.67	= { by lemma 128 }
286.34/286.67	  multiplication(forward_diamond(X, Y), antidomain(Z))
286.34/286.67	= { by lemma 244 }
286.34/286.67	  multiplication(domain(multiplication(X, Y)), antidomain(Z))
286.34/286.67	= { by axiom 28 (domain_difference) }
286.34/286.67	  domain_difference(multiplication(X, Y), Z)
286.34/286.67	
286.34/286.67	Lemma 251: forward_diamond(domain(Z), multiplication(Y, X)) = forward_diamond(forward_diamond(Y, X), Z).
286.34/286.67	Proof:
286.34/286.67	  forward_diamond(domain(Z), multiplication(Y, X))
286.34/286.67	= { by lemma 77 }
286.34/286.67	  domain(domain_difference(Z, antidomain(multiplication(Y, X))))
286.34/286.67	= { by lemma 242 }
286.34/286.67	  domain(domain_difference(Z, antidomain(forward_diamond(Y, X))))
286.34/286.67	= { by lemma 77 }
286.34/286.67	  forward_diamond(domain(Z), forward_diamond(Y, X))
286.34/286.67	= { by lemma 234 }
286.34/286.67	  forward_diamond(forward_diamond(Y, X), Z)
286.34/286.67	
286.34/286.67	Lemma 252: antidomain(domain_difference(multiplication(X, Y), Z)) = forward_box(forward_diamond(X, Y), Z).
286.34/286.67	Proof:
286.34/286.67	  antidomain(domain_difference(multiplication(X, Y), Z))
286.34/286.67	= { by lemma 247 }
286.34/286.67	  antidomain(multiplication(antidomain(Z), multiplication(X, Y)))
286.34/286.67	= { by axiom 11 (multiplicative_associativity) }
286.34/286.67	  antidomain(multiplication(multiplication(antidomain(Z), X), Y))
286.34/286.67	= { by lemma 242 }
286.34/286.67	  antidomain(forward_diamond(multiplication(antidomain(Z), X), Y))
286.34/286.67	= { by lemma 134 }
286.34/286.67	  antidomain(multiplication(antidomain(Z), multiplication(X, domain(Y))))
286.34/286.67	= { by lemma 247 }
286.34/286.67	  antidomain(domain_difference(multiplication(X, domain(Y)), Z))
286.34/286.67	= { by lemma 166 }
286.34/286.67	  antidomain(domain_difference(forward_diamond(X, Y), Z))
286.34/286.67	= { by lemma 128 }
286.34/286.67	  antidomain(multiplication(forward_diamond(X, Y), antidomain(Z)))
286.34/286.67	= { by lemma 117 }
286.34/286.67	  forward_box(forward_diamond(X, Y), Z)
286.34/286.67	
286.34/286.67	Lemma 253: antidomain(multiplication(X, forward_diamond(Y, Z))) = antidomain(multiplication(X, multiplication(Y, Z))).
286.34/286.67	Proof:
286.34/286.67	  antidomain(multiplication(X, forward_diamond(Y, Z)))
286.34/286.67	= { by lemma 147 }
286.34/286.67	  antidomain(forward_diamond(X, forward_diamond(Y, Z)))
286.34/286.67	= { by lemma 114 }
286.34/286.67	  forward_box(X, antidomain(forward_diamond(Y, Z)))
286.34/286.67	= { by lemma 161 }
286.34/286.67	  forward_box(X, domain(antidomain(forward_diamond(Y, Z))))
286.34/286.67	= { by lemma 97 }
286.34/286.67	  forward_box(X, domain(c(forward_diamond(Y, Z))))
286.34/286.67	= { by lemma 43 }
286.34/286.67	  forward_box(X, c(domain(forward_diamond(Y, Z))))
286.34/286.67	= { by lemma 74 }
286.34/286.67	  forward_box(X, c(forward_diamond(forward_diamond(Y, Z), one)))
286.34/286.67	= { by lemma 49 }
286.34/286.67	  forward_box(X, c(forward_diamond(forward_diamond(Y, Z), c(zero))))
286.34/286.67	= { by axiom 25 (forward_box) }
286.34/286.67	  forward_box(X, forward_box(forward_diamond(Y, Z), zero))
286.34/286.67	= { by lemma 99 }
286.34/286.67	  domain(forward_box(X, forward_box(forward_diamond(Y, Z), zero)))
286.34/286.67	= { by axiom 23 (domain4) }
286.34/286.67	  antidomain(antidomain(forward_box(X, forward_box(forward_diamond(Y, Z), zero))))
286.34/286.67	= { by lemma 113 }
286.34/286.67	  antidomain(forward_diamond(X, antidomain(forward_box(forward_diamond(Y, Z), zero))))
286.34/286.67	= { by lemma 113 }
286.34/286.67	  antidomain(forward_diamond(X, forward_diamond(forward_diamond(Y, Z), antidomain(zero))))
286.34/286.67	= { by lemma 73 }
286.34/286.67	  antidomain(forward_diamond(X, domain(multiplication(forward_diamond(Y, Z), c(zero)))))
286.34/286.67	= { by lemma 104 }
286.34/286.67	  antidomain(forward_diamond(X, multiplication(forward_diamond(Y, Z), c(zero))))
286.34/286.67	= { by lemma 97 }
286.34/286.67	  antidomain(forward_diamond(X, multiplication(forward_diamond(Y, Z), antidomain(zero))))
286.34/286.67	= { by lemma 242 }
286.34/286.67	  antidomain(multiplication(X, multiplication(forward_diamond(Y, Z), antidomain(zero))))
286.34/286.67	= { by lemma 128 }
286.34/286.67	  antidomain(multiplication(X, domain_difference(forward_diamond(Y, Z), zero)))
286.34/286.67	= { by lemma 250 }
286.34/286.67	  antidomain(multiplication(X, domain_difference(multiplication(Y, Z), zero)))
286.34/286.67	= { by lemma 50 }
286.34/286.67	  antidomain(multiplication(X, domain(multiplication(Y, Z))))
286.34/286.67	= { by lemma 116 }
286.34/286.67	  antidomain(forward_diamond(X, multiplication(Y, Z)))
286.34/286.67	= { by lemma 242 }
286.34/286.67	  antidomain(multiplication(X, multiplication(Y, Z)))
286.34/286.67	
286.34/286.67	Lemma 254: multiplication(domain(Z), domain_difference(multiplication(X, Y), W)) = multiplication(forward_diamond(X, Y), domain_difference(Z, W)).
286.34/286.67	Proof:
286.34/286.67	  multiplication(domain(Z), domain_difference(multiplication(X, Y), W))
286.34/286.67	= { by lemma 250 }
286.34/286.67	  multiplication(domain(Z), domain_difference(forward_diamond(X, Y), W))
286.34/286.67	= { by lemma 128 }
286.34/286.67	  multiplication(domain(Z), multiplication(forward_diamond(X, Y), antidomain(W)))
286.34/286.67	= { by lemma 238 }
286.34/286.67	  multiplication(forward_diamond(X, Y), multiplication(domain(Z), antidomain(W)))
286.34/286.67	= { by axiom 28 (domain_difference) }
286.34/286.67	  multiplication(forward_diamond(X, Y), domain_difference(Z, W))
286.34/286.67	
286.34/286.67	Lemma 255: antidomain(forward_box(antidomain(X), Y)) = domain_difference(antidomain(X), Y).
286.34/286.67	Proof:
286.34/286.67	  antidomain(forward_box(antidomain(X), Y))
286.34/286.67	= { by lemma 126 }
286.34/286.67	  domain(domain_difference(antidomain(X), Y))
286.34/286.67	= { by lemma 246 }
286.34/286.67	  domain_difference(domain_difference(antidomain(X), Y), Y)
286.34/286.67	= { by lemma 216 }
286.34/286.67	  domain_difference(domain_difference(antidomain(Y), X), Y)
286.34/286.67	= { by lemma 213 }
286.34/286.67	  multiplication(antidomain(Y), domain(domain_difference(antidomain(Y), X)))
286.34/286.67	= { by lemma 246 }
286.34/286.67	  multiplication(antidomain(Y), domain_difference(domain_difference(antidomain(Y), X), X))
286.34/286.67	= { by lemma 236 }
286.34/286.67	  multiplication(domain_difference(domain_difference(antidomain(Y), X), Y), antidomain(X))
286.34/286.67	= { by lemma 101 }
286.34/286.67	  multiplication(domain_difference(domain_difference(antidomain(Y), X), domain(Y)), antidomain(X))
286.34/286.67	= { by axiom 23 (domain4) }
286.34/286.67	  multiplication(domain_difference(domain_difference(antidomain(Y), X), antidomain(antidomain(Y))), antidomain(X))
286.34/286.67	= { by lemma 152 }
286.34/286.67	  multiplication(domain(domain_difference(antidomain(Y), X)), domain_difference(antidomain(Y), X))
286.34/286.67	= { by lemma 83 }
286.34/286.67	  domain_difference(antidomain(Y), X)
286.34/286.67	= { by lemma 216 }
286.34/286.67	  domain_difference(antidomain(X), Y)
286.34/286.67	
286.34/286.67	Lemma 256: domain(domain_difference(X, Y)) = domain_difference(X, Y).
286.34/286.67	Proof:
286.34/286.67	  domain(domain_difference(X, Y))
286.34/286.67	= { by axiom 23 (domain4) }
286.34/286.67	  antidomain(antidomain(domain_difference(X, Y)))
286.34/286.67	= { by lemma 115 }
286.34/286.67	  antidomain(forward_box(domain(X), Y))
286.34/286.67	= { by axiom 23 (domain4) }
286.34/286.67	  antidomain(forward_box(antidomain(antidomain(X)), Y))
286.34/286.67	= { by lemma 255 }
286.34/286.67	  domain_difference(antidomain(antidomain(X)), Y)
286.34/286.67	= { by axiom 23 (domain4) }
286.34/286.67	  domain_difference(domain(X), Y)
286.34/286.67	= { by lemma 105 }
286.34/286.67	  domain_difference(X, Y)
286.34/286.67	
286.34/286.67	Lemma 257: domain_difference(X, antidomain(Y)) = forward_diamond(domain(X), Y).
286.34/286.67	Proof:
286.34/286.67	  domain_difference(X, antidomain(Y))
286.34/286.67	= { by lemma 256 }
286.34/286.67	  domain(domain_difference(X, antidomain(Y)))
286.34/286.67	= { by lemma 77 }
286.43/286.72	  forward_diamond(domain(X), Y)
286.43/286.72	
286.43/286.72	Lemma 258: domain_difference(X, codomain(Y)) = forward_diamond(coantidomain(Y), X).
286.43/286.72	Proof:
286.43/286.72	  domain_difference(X, codomain(Y))
286.43/286.72	= { by axiom 7 (multiplicative_right_identity) }
286.43/286.72	  domain_difference(X, multiplication(codomain(Y), one))
286.43/286.72	= { by lemma 47 }
286.43/286.72	  domain_difference(X, multiplication(codomain(Y), addition(antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 5 (additive_commutativity) }
286.43/286.72	  domain_difference(X, multiplication(codomain(Y), addition(domain(multiplication(coantidomain(codomain(Y)), addition(?, one))), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 3 (right_distributivity) }
286.43/286.72	  domain_difference(X, addition(multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one)))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 6 (multiplicative_left_identity) }
286.43/286.72	  domain_difference(X, addition(multiplication(one, multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by lemma 48 }
286.43/286.72	  domain_difference(X, addition(multiplication(antidomain(zero), multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 8 (left_annihilation) }
286.43/286.72	  domain_difference(X, addition(multiplication(antidomain(multiplication(zero, addition(?, one))), multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 19 (codomain1) }
286.43/286.72	  domain_difference(X, addition(multiplication(antidomain(multiplication(multiplication(codomain(Y), coantidomain(codomain(Y))), addition(?, one))), multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by axiom 11 (multiplicative_associativity) }
286.43/286.72	  domain_difference(X, addition(multiplication(antidomain(multiplication(codomain(Y), multiplication(coantidomain(codomain(Y)), addition(?, one)))), multiplication(codomain(Y), domain(multiplication(coantidomain(codomain(Y)), addition(?, one))))), multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by lemma 144 }
286.43/286.72	  domain_difference(X, addition(zero, multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one))))))
286.43/286.72	= { by lemma 35 }
286.43/286.72	  domain_difference(X, multiplication(codomain(Y), antidomain(multiplication(coantidomain(codomain(Y)), addition(?, one)))))
286.43/286.72	= { by lemma 93 }
286.43/286.72	  domain_difference(X, multiplication(codomain(Y), antidomain(multiplication(coantidomain(Y), addition(?, one)))))
286.43/286.72	= { by lemma 206 }
286.43/286.72	  domain_difference(X, domain_difference(codomain(Y), multiplication(coantidomain(Y), addition(?, one))))
286.43/286.72	= { by lemma 221 }
286.43/286.72	  domain_difference(X, multiplication(antidomain(multiplication(coantidomain(Y), addition(?, one))), codomain(Y)))
286.43/286.72	= { by lemma 195 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, one))), coantidomain(Y)))
286.43/286.72	= { by lemma 130 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), coantidomain(Y)))
286.43/286.72	= { by lemma 199 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), backward_box(Y, zero)))
286.43/286.72	= { by lemma 100 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), domain(backward_box(Y, zero))))
286.43/286.72	= { by lemma 182 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(domain(backward_box(Y, zero)), addition(?, domain(backward_box(Y, zero))))))
286.43/286.72	= { by lemma 100 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(backward_box(Y, zero), addition(?, domain(backward_box(Y, zero))))))
286.43/286.72	= { by lemma 199 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, domain(backward_box(Y, zero))))))
286.43/286.72	= { by lemma 100 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, backward_box(Y, zero)))))
286.43/286.72	= { by lemma 199 }
286.43/286.72	  domain_difference(X, domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, coantidomain(Y)))))
286.43/286.72	= { by axiom 28 (domain_difference) }
286.43/286.72	  domain_difference(X, multiplication(domain(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y))))), antidomain(forward_diamond(coantidomain(Y), addition(?, coantidomain(Y))))))
286.43/286.72	= { by lemma 158 }
286.43/286.72	  domain_difference(X, multiplication(domain(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y))))), addition(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), antidomain(forward_diamond(coantidomain(Y), addition(?, coantidomain(Y)))))))
286.43/286.72	= { by lemma 169 }
286.43/286.72	  domain_difference(X, addition(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, coantidomain(Y))))))
286.43/286.72	= { by lemma 97 }
286.43/286.72	  domain_difference(X, addition(c(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, coantidomain(Y))))))
286.43/286.72	= { by lemma 39 }
286.43/286.72	  domain_difference(X, addition(domain(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y))))), domain_difference(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))), forward_diamond(coantidomain(Y), addition(?, coantidomain(Y))))))
286.43/286.72	= { by lemma 140 }
286.43/286.72	  domain_difference(X, domain(antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y))))))
286.43/286.72	= { by lemma 39 }
286.43/286.72	  domain_difference(X, c(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))))
286.43/286.72	= { by lemma 97 }
286.43/286.72	  domain_difference(X, antidomain(multiplication(coantidomain(Y), addition(?, coantidomain(Y)))))
286.43/286.72	= { by lemma 130 }
286.43/286.72	  domain_difference(X, antidomain(multiplication(coantidomain(Y), addition(?, one))))
286.43/286.72	= { by lemma 257 }
286.43/286.72	  forward_diamond(domain(X), multiplication(coantidomain(Y), addition(?, one)))
286.43/286.72	= { by lemma 251 }
286.43/286.72	  forward_diamond(forward_diamond(coantidomain(Y), addition(?, one)), X)
286.43/286.72	= { by axiom 27 (forward_diamond) }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), domain(addition(?, one)))), X)
286.43/286.72	= { by axiom 23 (domain4) }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), antidomain(antidomain(addition(?, one))))), X)
286.43/286.72	= { by lemma 35 }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(zero, antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by axiom 21 (domain1) }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(multiplication(antidomain(addition(one, ?)), addition(one, ?)), antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by lemma 51 }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(multiplication(antidomain(addition(one, ?)), addition(one, addition(one, ?))), antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by lemma 96 }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(multiplication(antidomain(addition(one, ?)), one), antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by axiom 7 (multiplicative_right_identity) }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(antidomain(addition(one, ?)), antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by axiom 5 (additive_commutativity) }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), addition(antidomain(addition(?, one)), antidomain(antidomain(addition(?, one)))))), X)
286.43/286.72	= { by lemma 32 }
286.43/286.72	  forward_diamond(domain(multiplication(coantidomain(Y), one)), X)
286.43/286.72	= { by axiom 7 (multiplicative_right_identity) }
286.43/286.72	  forward_diamond(domain(coantidomain(Y)), X)
286.43/286.72	= { by lemma 184 }
286.43/286.72	  forward_diamond(domain(antidomain(codomain(Y))), X)
286.43/286.72	= { by lemma 161 }
286.43/286.72	  forward_diamond(antidomain(codomain(Y)), X)
286.43/286.72	= { by lemma 184 }
286.43/286.73	  forward_diamond(coantidomain(Y), X)
286.43/286.73	
286.43/286.73	Lemma 259: antidomain(forward_box(codomain(X), Y)) = domain_difference(codomain(X), Y).
286.43/286.73	Proof:
286.43/286.73	  antidomain(forward_box(codomain(X), Y))
286.43/286.73	= { by lemma 210 }
286.43/286.73	  domain(domain_difference(codomain(X), Y))
286.43/286.73	= { by lemma 215 }
286.43/286.73	  forward_diamond(antidomain(Y), codomain(X))
286.43/286.73	= { by lemma 208 }
286.43/286.73	  antidomain(forward_box(antidomain(Y), coantidomain(X)))
286.43/286.73	= { by lemma 255 }
286.43/286.73	  domain_difference(antidomain(Y), coantidomain(X))
286.43/286.73	= { by lemma 193 }
286.43/286.73	  multiplication(domain(antidomain(Y)), codomain(X))
286.43/286.73	= { by lemma 161 }
286.43/286.73	  multiplication(antidomain(Y), codomain(X))
286.43/286.73	= { by lemma 221 }
286.43/286.73	  domain_difference(codomain(X), Y)
286.43/286.73	
286.43/286.73	Lemma 260: forward_diamond(domain(X), backward_diamond(Z, Y)) = multiplication(domain(X), backward_diamond(Z, Y)).
286.43/286.73	Proof:
286.43/286.73	  forward_diamond(domain(X), backward_diamond(Z, Y))
286.43/286.73	= { by lemma 77 }
286.43/286.73	  domain(domain_difference(X, antidomain(backward_diamond(Z, Y))))
286.43/286.73	= { by lemma 215 }
286.43/286.73	  forward_diamond(antidomain(antidomain(backward_diamond(Z, Y))), X)
286.43/286.73	= { by lemma 245 }
286.43/286.73	  forward_diamond(coantidomain(domain(antidomain(backward_diamond(Z, Y)))), X)
286.43/286.73	= { by lemma 219 }
286.43/286.73	  domain(domain_difference(X, codomain(domain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by lemma 194 }
286.43/286.73	  domain(multiplication(domain(X), coantidomain(domain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by lemma 122 }
286.43/286.73	  domain(multiplication(domain(X), multiplication(antidomain(antidomain(backward_diamond(Z, Y))), coantidomain(domain(antidomain(backward_diamond(Z, Y)))))))
286.43/286.73	= { by lemma 119 }
286.43/286.73	  domain(multiplication(domain_difference(X, antidomain(backward_diamond(Z, Y))), coantidomain(domain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by axiom 23 (domain4) }
286.43/286.73	  domain(multiplication(domain_difference(X, antidomain(backward_diamond(Z, Y))), coantidomain(antidomain(antidomain(antidomain(backward_diamond(Z, Y)))))))
286.43/286.73	= { by lemma 202 }
286.43/286.73	  forward_diamond(domain_difference(X, antidomain(backward_diamond(Z, Y))), coantidomain(antidomain(antidomain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by lemma 192 }
286.43/286.73	  forward_diamond(multiplication(domain(X), backward_diamond(Z, Y)), coantidomain(antidomain(antidomain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by axiom 23 (domain4) }
286.43/286.73	  forward_diamond(multiplication(domain(X), backward_diamond(Z, Y)), coantidomain(domain(antidomain(backward_diamond(Z, Y)))))
286.43/286.73	= { by lemma 248 }
286.43/286.73	  forward_diamond(domain(X), multiplication(backward_diamond(Z, Y), coantidomain(domain(antidomain(backward_diamond(Z, Y))))))
286.43/286.73	= { by lemma 251 }
286.43/286.73	  forward_diamond(forward_diamond(backward_diamond(Z, Y), coantidomain(domain(antidomain(backward_diamond(Z, Y))))), X)
286.43/286.73	= { by lemma 245 }
286.43/286.73	  forward_diamond(forward_diamond(backward_diamond(Z, Y), antidomain(antidomain(backward_diamond(Z, Y)))), X)
286.43/286.73	= { by lemma 113 }
286.43/286.73	  forward_diamond(antidomain(forward_box(backward_diamond(Z, Y), antidomain(backward_diamond(Z, Y)))), X)
286.43/286.73	= { by lemma 215 }
286.43/286.73	  domain(domain_difference(X, forward_box(backward_diamond(Z, Y), antidomain(backward_diamond(Z, Y)))))
286.43/286.73	= { by lemma 256 }
286.43/286.73	  domain_difference(X, forward_box(backward_diamond(Z, Y), antidomain(backward_diamond(Z, Y))))
286.43/286.73	= { by lemma 114 }
286.43/286.73	  domain_difference(X, antidomain(forward_diamond(backward_diamond(Z, Y), backward_diamond(Z, Y))))
286.43/286.73	= { by lemma 133 }
286.43/286.73	  multiplication(domain(X), forward_diamond(backward_diamond(Z, Y), backward_diamond(Z, Y)))
286.43/286.73	= { by lemma 191 }
286.63/286.94	  multiplication(domain(X), backward_diamond(Z, Y))
286.63/286.94	
286.63/286.94	Lemma 261: forward_diamond(codomain(Y), X) = backward_diamond(domain(X), Y).
286.63/286.94	Proof:
286.63/286.94	  forward_diamond(codomain(Y), X)
286.63/286.94	= { by lemma 186 }
286.63/286.94	  forward_diamond(domain(codomain(Y)), X)
286.63/286.94	= { by lemma 244 }
286.63/286.94	  domain(multiplication(domain(codomain(Y)), X))
286.63/286.94	= { by lemma 243 }
286.63/286.94	  coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 160 }
286.63/286.94	  codomain(coantidomain(antidomain(multiplication(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 184 }
286.63/286.94	  codomain(antidomain(codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 84 }
286.63/286.94	  codomain(domain_difference(antidomain(codomain(antidomain(multiplication(domain(codomain(Y)), X)))), codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 184 }
286.63/286.94	  codomain(domain_difference(coantidomain(antidomain(multiplication(domain(codomain(Y)), X))), codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 198 }
286.63/286.94	  codomain(multiplication(domain_difference(coantidomain(antidomain(multiplication(domain(codomain(Y)), X))), antidomain(multiplication(domain(codomain(Y)), X))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 199 }
286.63/286.94	  codomain(multiplication(domain_difference(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(multiplication(domain(codomain(Y)), X))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 199 }
286.63/286.94	  codomain(multiplication(domain_difference(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(multiplication(domain(codomain(Y)), X))), backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero)))
286.63/286.94	= { by lemma 212 }
286.63/286.94	  codomain(multiplication(domain_difference(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(multiplication(domain(codomain(Y)), X))), addition(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero)))))
286.63/286.94	= { by lemma 91 }
286.63/286.94	  codomain(multiplication(domain_difference(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(multiplication(domain(codomain(Y)), X))), one))
286.63/286.94	= { by axiom 7 (multiplicative_right_identity) }
286.63/286.94	  codomain(domain_difference(backward_box(antidomain(multiplication(domain(codomain(Y)), X)), zero), antidomain(multiplication(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 199 }
286.63/286.94	  codomain(domain_difference(coantidomain(antidomain(multiplication(domain(codomain(Y)), X))), antidomain(multiplication(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 220 }
286.63/286.94	  codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 170 }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 81 }
286.63/286.94	  codomain(multiplication(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 11 (multiplicative_associativity) }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), multiplication(addition(antidomain(multiplication(domain(codomain(Y)), X)), one), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 143 }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(multiplication(addition(antidomain(multiplication(domain(codomain(Y)), X)), one), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)))), multiplication(antidomain(multiplication(domain(codomain(Y)), X)), ?))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 5 (additive_commutativity) }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(multiplication(antidomain(multiplication(domain(codomain(Y)), X)), ?), multiplication(addition(antidomain(multiplication(domain(codomain(Y)), X)), one), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)))))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 66 }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(multiplication(antidomain(multiplication(domain(codomain(Y)), X)), ?), addition(codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))), multiplication(antidomain(multiplication(domain(codomain(Y)), X)), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))))))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 173 }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))), multiplication(antidomain(multiplication(domain(codomain(Y)), X)), addition(?, codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))))))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 5 (additive_commutativity) }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))), multiplication(antidomain(multiplication(domain(codomain(Y)), X)), addition(codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))), ?)))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 143 }
286.63/286.94	  codomain(multiplication(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one)))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 221 }
286.63/286.94	  codomain(multiplication(domain_difference(codomain(multiplication(antidomain(antidomain(multiplication(domain(codomain(Y)), X))), addition(antidomain(multiplication(domain(codomain(Y)), X)), one))), antidomain(multiplication(domain(codomain(Y)), X))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 170 }
286.63/286.94	  codomain(multiplication(domain_difference(codomain(antidomain(antidomain(multiplication(domain(codomain(Y)), X)))), antidomain(multiplication(domain(codomain(Y)), X))), coantidomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 198 }
286.63/286.94	  codomain(domain_difference(codomain(antidomain(antidomain(multiplication(domain(codomain(Y)), X)))), codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 23 (domain4) }
286.63/286.94	  codomain(domain_difference(codomain(domain(multiplication(domain(codomain(Y)), X))), codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 203 }
286.63/286.94	  codomain(domain_difference(forward_box(antidomain(multiplication(domain(codomain(Y)), X)), codomain(domain(multiplication(domain(codomain(Y)), X)))), codomain(antidomain(multiplication(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 150 }
286.63/286.94	  codomain(multiplication(forward_box(antidomain(multiplication(domain(codomain(Y)), X)), codomain(domain(multiplication(domain(codomain(Y)), X)))), antidomain(codomain(antidomain(multiplication(domain(codomain(Y)), X))))))
286.63/286.94	= { by lemma 203 }
286.63/286.94	  codomain(multiplication(codomain(domain(multiplication(domain(codomain(Y)), X))), antidomain(codomain(antidomain(multiplication(domain(codomain(Y)), X))))))
286.63/286.94	= { by axiom 24 (backward_diamond) }
286.63/286.94	  backward_diamond(antidomain(codomain(antidomain(multiplication(domain(codomain(Y)), X)))), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 184 }
286.63/286.94	  backward_diamond(coantidomain(antidomain(multiplication(domain(codomain(Y)), X))), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 243 }
286.63/286.94	  backward_diamond(domain(multiplication(domain(codomain(Y)), X)), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 244 }
286.63/286.94	  backward_diamond(forward_diamond(domain(codomain(Y)), X), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 98 }
286.63/286.94	  backward_diamond(domain(forward_diamond(domain(codomain(Y)), X)), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by axiom 23 (domain4) }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain(multiplication(domain(codomain(Y)), X)))
286.63/286.94	= { by lemma 244 }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), forward_diamond(domain(codomain(Y)), X))
286.63/286.94	= { by lemma 222 }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(forward_diamond(domain(codomain(Y)), X), coantidomain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 213 }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 164 }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 101 }
286.63/286.94	  backward_diamond(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 24 (backward_diamond) }
286.63/286.94	  codomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), antidomain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 35 }
286.63/286.94	  codomain(addition(zero, multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))))))
286.63/286.94	= { by lemma 106 }
286.63/286.94	  codomain(addition(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))))), multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))))))
286.63/286.94	= { by axiom 3 (right_distributivity) }
286.63/286.94	  codomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))), antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))))))
286.63/286.94	= { by axiom 5 (additive_commutativity) }
286.63/286.94	  codomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))
286.63/286.94	= { by axiom 24 (backward_diamond) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 197 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 7 (multiplicative_right_identity) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), one)))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 52 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(one, coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 45 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(coantidomain(zero), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 10 (right_annihilation) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(coantidomain(multiplication(domain(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X)))), zero)), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 21 (domain1) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(coantidomain(multiplication(domain(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X)))), multiplication(antidomain(domain(antidomain(forward_diamond(domain(codomain(Y)), X)))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 119 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(coantidomain(multiplication(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 20 (codomain4) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), addition(coantidomain(multiplication(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 16 (codomain2) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 20 (codomain4) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), coantidomain(multiplication(codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 123 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(multiplication(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X))))), coantidomain(backward_diamond(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))))))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 19 (codomain1) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(zero))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 48 }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), multiplication(domain(antidomain(forward_diamond(domain(codomain(Y)), X))), one)), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by axiom 7 (multiplicative_right_identity) }
286.63/286.94	  backward_diamond(addition(antidomain(antidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))), domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 47 }
286.63/286.94	  backward_diamond(one, domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 162 }
286.63/286.94	  codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(antidomain(forward_diamond(domain(codomain(Y)), X)))))
286.63/286.94	= { by lemma 101 }
286.63/286.94	  codomain(domain_difference(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), antidomain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 164 }
286.63/286.94	  codomain(multiplication(antidomain(coantidomain(forward_diamond(domain(codomain(Y)), X))), domain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 213 }
286.63/286.94	  codomain(domain_difference(forward_diamond(domain(codomain(Y)), X), coantidomain(forward_diamond(domain(codomain(Y)), X))))
286.63/286.94	= { by lemma 222 }
286.63/286.94	  codomain(forward_diamond(domain(codomain(Y)), X))
286.63/286.94	= { by lemma 257 }
286.63/286.94	  codomain(domain_difference(codomain(Y), antidomain(X)))
286.63/286.94	= { by lemma 217 }
286.63/286.94	  backward_diamond(antidomain(antidomain(X)), Y)
286.63/286.94	= { by axiom 23 (domain4) }
286.63/286.95	  backward_diamond(domain(X), Y)
286.63/286.95	
286.63/286.95	Lemma 262: antidomain(forward_box(forward_diamond(addition(Y, one), X), Y)) = domain_difference(X, Y).
286.63/286.95	Proof:
286.63/286.95	  antidomain(forward_box(forward_diamond(addition(Y, one), X), Y))
286.63/286.95	= { by lemma 113 }
286.63/286.95	  forward_diamond(forward_diamond(addition(Y, one), X), antidomain(Y))
286.63/286.95	= { by lemma 234 }
286.63/286.95	  forward_diamond(domain(antidomain(Y)), forward_diamond(addition(Y, one), X))
286.63/286.95	= { by lemma 127 }
286.63/286.95	  domain(multiplication(domain(antidomain(Y)), forward_diamond(addition(Y, one), X)))
286.63/286.95	= { by axiom 23 (domain4) }
286.63/286.95	  domain(multiplication(antidomain(antidomain(antidomain(Y))), forward_diamond(addition(Y, one), X)))
286.63/286.95	= { by lemma 227 }
286.63/286.95	  domain(domain_difference(forward_diamond(addition(Y, one), X), antidomain(antidomain(Y))))
286.63/286.95	= { by lemma 250 }
286.63/286.95	  domain(domain_difference(multiplication(addition(Y, one), X), antidomain(antidomain(Y))))
286.63/286.95	= { by lemma 214 }
286.63/286.95	  domain(domain_difference(antidomain(Y), antidomain(multiplication(addition(Y, one), X))))
286.63/286.95	= { by lemma 126 }
286.63/286.95	  antidomain(forward_box(antidomain(Y), antidomain(multiplication(addition(Y, one), X))))
286.63/286.95	= { by lemma 114 }
286.63/286.95	  antidomain(antidomain(forward_diamond(antidomain(Y), multiplication(addition(Y, one), X))))
286.63/286.95	= { by axiom 23 (domain4) }
286.63/286.95	  domain(forward_diamond(antidomain(Y), multiplication(addition(Y, one), X)))
286.63/286.95	= { by lemma 98 }
286.63/286.95	  forward_diamond(antidomain(Y), multiplication(addition(Y, one), X))
286.63/286.95	= { by lemma 112 }
286.63/286.95	  forward_diamond(addition(antidomain(Y), antidomain(multiplication(addition(Y, one), X))), multiplication(addition(Y, one), X))
286.63/286.95	= { by axiom 5 (additive_commutativity) }
286.63/286.95	  forward_diamond(addition(antidomain(multiplication(addition(Y, one), X)), antidomain(Y)), multiplication(addition(Y, one), X))
286.63/286.95	= { by lemma 242 }
286.63/286.95	  forward_diamond(addition(antidomain(forward_diamond(addition(Y, one), X)), antidomain(Y)), multiplication(addition(Y, one), X))
286.63/286.95	= { by lemma 248 }
286.63/286.95	  forward_diamond(multiplication(addition(antidomain(forward_diamond(addition(Y, one), X)), antidomain(Y)), addition(Y, one)), X)
286.63/286.95	= { by lemma 67 }
286.63/286.95	  forward_diamond(addition(addition(antidomain(forward_diamond(addition(Y, one), X)), antidomain(Y)), multiplication(addition(antidomain(forward_diamond(addition(Y, one), X)), antidomain(Y)), Y)), X)
286.63/286.95	= { by lemma 71 }
286.63/286.95	  forward_diamond(addition(addition(antidomain(forward_diamond(addition(Y, one), X)), antidomain(Y)), multiplication(antidomain(forward_diamond(addition(Y, one), X)), Y)), X)
286.63/286.95	= { by axiom 12 (additive_associativity) }
286.63/286.95	  forward_diamond(addition(antidomain(forward_diamond(addition(Y, one), X)), addition(antidomain(Y), multiplication(antidomain(forward_diamond(addition(Y, one), X)), Y))), X)
286.63/286.95	= { by lemma 65 }
286.63/286.95	  forward_diamond(addition(antidomain(Y), addition(antidomain(forward_diamond(addition(Y, one), X)), multiplication(antidomain(forward_diamond(addition(Y, one), X)), Y))), X)
286.63/286.95	= { by lemma 67 }
286.63/286.95	  forward_diamond(addition(antidomain(Y), multiplication(antidomain(forward_diamond(addition(Y, one), X)), addition(Y, one))), X)
286.63/286.95	= { by lemma 228 }
286.63/286.95	  forward_diamond(antidomain(Y), X)
286.63/286.95	= { by lemma 215 }
286.63/286.95	  domain(domain_difference(X, Y))
286.63/286.95	= { by lemma 256 }
286.63/286.95	  domain_difference(X, Y)
286.63/286.95	
286.63/286.95	Lemma 263: multiplication(antidomain(X), domain_difference(Y, Z)) = domain_difference(domain_difference(Y, X), Z).
286.63/286.95	Proof:
286.63/286.95	  multiplication(antidomain(X), domain_difference(Y, Z))
286.63/286.95	= { by lemma 236 }
286.63/286.95	  multiplication(domain_difference(Y, X), antidomain(Z))
286.63/286.95	= { by lemma 262 }
286.63/286.95	  multiplication(antidomain(forward_box(forward_diamond(addition(X, one), Y), X)), antidomain(Z))
286.63/286.95	= { by lemma 124 }
286.63/286.95	  domain_difference(antidomain(forward_box(forward_diamond(addition(X, one), Y), X)), Z)
286.63/286.95	= { by lemma 262 }
286.73/287.05	  domain_difference(domain_difference(Y, X), Z)
286.73/287.05	
286.73/287.05	Lemma 264: multiplication(domain(X), domain_difference(Y, Z)) = forward_diamond(domain_difference(X, Z), Y).
286.73/287.05	Proof:
286.73/287.05	  multiplication(domain(X), domain_difference(Y, Z))
286.73/287.05	= { by axiom 23 (domain4) }
286.73/287.05	  multiplication(antidomain(antidomain(X)), domain_difference(Y, Z))
286.73/287.05	= { by lemma 236 }
286.73/287.05	  multiplication(domain_difference(Y, antidomain(X)), antidomain(Z))
286.73/287.05	= { by lemma 212 }
286.73/287.05	  multiplication(domain_difference(Y, antidomain(X)), addition(antidomain(Z), antidomain(Y)))
286.73/287.05	= { by lemma 131 }
286.73/287.05	  multiplication(domain(Y), multiplication(domain(X), addition(antidomain(Z), antidomain(Y))))
286.73/287.05	= { by lemma 175 }
286.73/287.05	  multiplication(domain(Y), addition(domain_difference(X, Z), domain_difference(X, Y)))
286.73/287.05	= { by lemma 168 }
286.73/287.05	  multiplication(domain(Y), domain_difference(X, Z))
286.73/287.05	= { by axiom 23 (domain4) }
286.73/287.05	  multiplication(antidomain(antidomain(Y)), domain_difference(X, Z))
286.73/287.05	= { by lemma 263 }
286.73/287.05	  domain_difference(domain_difference(X, antidomain(Y)), Z)
286.73/287.05	= { by lemma 129 }
286.73/287.05	  domain_difference(forward_diamond(domain(X), Y), Z)
286.73/287.05	= { by lemma 250 }
286.73/287.05	  domain_difference(multiplication(domain(X), Y), Z)
286.73/287.05	= { by lemma 243 }
286.73/287.05	  domain_difference(multiplication(coantidomain(antidomain(X)), Y), Z)
286.73/287.05	= { by lemma 250 }
286.73/287.05	  domain_difference(forward_diamond(coantidomain(antidomain(X)), Y), Z)
286.73/287.05	= { by lemma 128 }
286.73/287.05	  multiplication(forward_diamond(coantidomain(antidomain(X)), Y), antidomain(Z))
286.73/287.05	= { by lemma 233 }
286.73/287.05	  multiplication(forward_diamond(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X))), antidomain(Z))
286.73/287.05	= { by lemma 128 }
286.73/287.05	  domain_difference(forward_diamond(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X))), Z)
286.73/287.05	= { by lemma 199 }
286.73/287.05	  domain_difference(forward_diamond(domain_difference(X, antidomain(Y)), backward_box(antidomain(X), zero)), Z)
286.73/287.05	= { by lemma 128 }
286.73/287.05	  multiplication(forward_diamond(domain_difference(X, antidomain(Y)), backward_box(antidomain(X), zero)), antidomain(Z))
286.73/287.05	= { by lemma 148 }
286.73/287.05	  multiplication(domain(multiplication(domain_difference(X, antidomain(Y)), backward_box(antidomain(X), zero))), antidomain(Z))
286.73/287.05	= { by axiom 28 (domain_difference) }
286.73/287.05	  domain_difference(multiplication(domain_difference(X, antidomain(Y)), backward_box(antidomain(X), zero)), Z)
286.73/287.05	= { by lemma 199 }
286.73/287.05	  domain_difference(multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(X))), Z)
286.73/287.05	= { by lemma 232 }
286.73/287.05	  domain_difference(domain_difference(Y, codomain(antidomain(X))), Z)
286.73/287.05	= { by lemma 241 }
286.73/287.05	  domain_difference(domain_difference(Y, antidomain(X)), Z)
286.73/287.05	= { by lemma 129 }
286.73/287.05	  domain_difference(forward_diamond(domain(Y), X), Z)
286.73/287.05	= { by lemma 227 }
286.73/287.05	  multiplication(antidomain(Z), forward_diamond(domain(Y), X))
286.73/287.05	= { by lemma 161 }
286.73/287.05	  multiplication(domain(antidomain(Z)), forward_diamond(domain(Y), X))
286.73/287.05	= { by lemma 77 }
286.73/287.05	  multiplication(domain(antidomain(Z)), domain(domain_difference(Y, antidomain(X))))
286.73/287.05	= { by lemma 214 }
286.73/287.05	  multiplication(domain(antidomain(Z)), domain(domain_difference(X, antidomain(Y))))
286.73/287.05	= { by lemma 77 }
286.73/287.05	  multiplication(domain(antidomain(Z)), forward_diamond(domain(X), Y))
286.73/287.05	= { by lemma 231 }
286.73/287.05	  multiplication(forward_diamond(domain(X), Y), domain(antidomain(Z)))
286.73/287.05	= { by lemma 257 }
286.73/287.05	  multiplication(domain_difference(X, antidomain(Y)), domain(antidomain(Z)))
286.73/287.05	= { by lemma 243 }
286.73/287.05	  multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by lemma 35 }
286.73/287.05	  addition(zero, multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by axiom 10 (right_annihilation) }
286.73/287.05	  addition(multiplication(domain(X), zero), multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by axiom 19 (codomain1) }
286.73/287.05	  addition(multiplication(domain(X), multiplication(antidomain(antidomain(Z)), coantidomain(antidomain(antidomain(Z))))), multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 119 }
286.73/287.05	  addition(multiplication(domain_difference(X, antidomain(Z)), coantidomain(antidomain(antidomain(Z)))), multiplication(domain_difference(X, antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by axiom 4 (left_distributivity) }
286.73/287.05	  multiplication(addition(domain_difference(X, antidomain(Z)), domain_difference(X, antidomain(Y))), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by axiom 5 (additive_commutativity) }
286.73/287.05	  multiplication(addition(domain_difference(X, antidomain(Y)), domain_difference(X, antidomain(Z))), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by lemma 175 }
286.73/287.05	  multiplication(multiplication(domain(X), addition(antidomain(antidomain(Y)), antidomain(antidomain(Z)))), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by axiom 23 (domain4) }
286.73/287.05	  multiplication(multiplication(domain(X), addition(domain(Y), antidomain(antidomain(Z)))), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by axiom 5 (additive_commutativity) }
286.73/287.05	  multiplication(multiplication(domain(X), addition(antidomain(antidomain(Z)), domain(Y))), coantidomain(antidomain(antidomain(Z))))
286.73/287.05	= { by axiom 11 (multiplicative_associativity) }
286.73/287.05	  multiplication(domain(X), multiplication(addition(antidomain(antidomain(Z)), domain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 157 }
286.73/287.05	  multiplication(domain(X), multiplication(addition(domain(Y), addition(antidomain(antidomain(Z)), domain_difference(Y, ?))), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 167 }
286.73/287.05	  multiplication(domain(X), multiplication(addition(domain(Y), domain_difference(Y, ?)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 140 }
286.73/287.05	  multiplication(domain(X), multiplication(domain(Y), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 194 }
286.73/287.05	  multiplication(domain(X), domain_difference(Y, codomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 258 }
286.73/287.05	  multiplication(domain(X), forward_diamond(coantidomain(antidomain(antidomain(Z))), Y))
286.73/287.05	= { by lemma 243 }
286.73/287.05	  multiplication(domain(X), forward_diamond(domain(antidomain(Z)), Y))
286.73/287.05	= { by lemma 161 }
286.73/287.05	  multiplication(domain(X), forward_diamond(antidomain(Z), Y))
286.73/287.05	= { by lemma 215 }
286.73/287.05	  multiplication(domain(X), domain(domain_difference(Y, Z)))
286.73/287.05	= { by lemma 58 }
286.73/287.05	  domain_difference(X, antidomain(domain_difference(Y, Z)))
286.73/287.05	= { by lemma 257 }
286.73/287.05	  forward_diamond(domain(X), domain_difference(Y, Z))
286.73/287.05	= { by lemma 74 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), domain_difference(Y, Z))
286.73/287.05	= { by axiom 28 (domain_difference) }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(domain(Y), antidomain(Z)))
286.73/287.05	= { by lemma 249 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), forward_diamond(domain(Y), antidomain(Z)))
286.73/287.05	= { by lemma 77 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), domain(domain_difference(Y, antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 104 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), domain_difference(Y, antidomain(antidomain(Z))))
286.73/287.05	= { by lemma 241 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), domain_difference(Y, codomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 232 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(domain_difference(antidomain(Z), antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 199 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(domain_difference(antidomain(Z), antidomain(Y)), backward_box(antidomain(antidomain(Z)), zero)))
286.73/287.05	= { by axiom 27 (forward_diamond) }
286.73/287.05	  domain(multiplication(forward_diamond(X, one), domain(multiplication(domain_difference(antidomain(Z), antidomain(Y)), backward_box(antidomain(antidomain(Z)), zero)))))
286.73/287.05	= { by lemma 148 }
286.73/287.05	  domain(multiplication(forward_diamond(X, one), forward_diamond(domain_difference(antidomain(Z), antidomain(Y)), backward_box(antidomain(antidomain(Z)), zero))))
286.73/287.05	= { by lemma 127 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), forward_diamond(domain_difference(antidomain(Z), antidomain(Y)), backward_box(antidomain(antidomain(Z)), zero)))
286.73/287.05	= { by lemma 199 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), forward_diamond(domain_difference(antidomain(Z), antidomain(Y)), coantidomain(antidomain(antidomain(Z)))))
286.73/287.05	= { by lemma 127 }
286.73/287.05	  domain(multiplication(forward_diamond(X, one), forward_diamond(domain_difference(antidomain(Z), antidomain(Y)), coantidomain(antidomain(antidomain(Z))))))
286.73/287.05	= { by lemma 233 }
286.73/287.05	  domain(multiplication(forward_diamond(X, one), forward_diamond(coantidomain(antidomain(antidomain(Z))), Y)))
286.73/287.05	= { by lemma 127 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), forward_diamond(coantidomain(antidomain(antidomain(Z))), Y))
286.73/287.05	= { by lemma 249 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(coantidomain(antidomain(antidomain(Z))), Y))
286.73/287.05	= { by lemma 243 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(domain(antidomain(Z)), Y))
286.73/287.05	= { by lemma 161 }
286.73/287.05	  forward_diamond(forward_diamond(X, one), multiplication(antidomain(Z), Y))
286.73/287.05	= { by lemma 248 }
286.73/287.05	  forward_diamond(multiplication(forward_diamond(X, one), antidomain(Z)), Y)
286.73/287.05	= { by lemma 118 }
286.73/287.05	  domain(multiplication(forward_diamond(X, one), multiplication(antidomain(Z), domain(Y))))
286.73/287.05	= { by axiom 11 (multiplicative_associativity) }
286.73/287.05	  domain(multiplication(multiplication(forward_diamond(X, one), antidomain(Z)), domain(Y)))
286.73/287.05	= { by lemma 128 }
286.73/287.05	  domain(multiplication(domain_difference(forward_diamond(X, one), Z), domain(Y)))
286.73/287.05	= { by lemma 250 }
286.73/287.05	  domain(multiplication(domain_difference(multiplication(X, one), Z), domain(Y)))
286.73/287.05	= { by axiom 27 (forward_diamond) }
286.73/287.05	  forward_diamond(domain_difference(multiplication(X, one), Z), Y)
286.73/287.05	= { by lemma 237 }
286.73/287.05	  forward_diamond(domain_difference(Y, Z), multiplication(X, one))
286.73/287.05	= { by axiom 7 (multiplicative_right_identity) }
286.73/287.05	  forward_diamond(domain_difference(Y, Z), X)
286.73/287.05	= { by lemma 237 }
286.73/287.05	  forward_diamond(domain_difference(X, Z), Y)
286.73/287.05	
286.73/287.05	Lemma 265: forward_diamond(coantidomain(X), domain_difference(Z, Y)) = multiplication(domain_difference(Z, Y), coantidomain(X)).
286.73/287.05	Proof:
286.73/287.05	  forward_diamond(coantidomain(X), domain_difference(Z, Y))
286.73/287.05	= { by lemma 258 }
286.73/287.05	  domain_difference(domain_difference(Z, Y), codomain(X))
286.73/287.05	= { by lemma 263 }
286.73/287.05	  multiplication(antidomain(Y), domain_difference(Z, codomain(X)))
286.73/287.05	= { by lemma 211 }
286.73/287.05	  multiplication(addition(antidomain(Z), antidomain(Y)), domain_difference(Z, codomain(X)))
286.73/287.05	= { by lemma 240 }
286.73/287.05	  multiplication(addition(antidomain(Z), domain_difference(Z, Y)), domain_difference(Z, codomain(X)))
286.73/287.05	= { by lemma 211 }
286.73/287.05	  multiplication(domain_difference(Z, Y), domain_difference(Z, codomain(X)))
286.73/287.05	= { by lemma 194 }
286.73/287.05	  multiplication(domain_difference(Z, Y), multiplication(domain(Z), coantidomain(X)))
286.73/287.05	= { by lemma 243 }
286.73/287.05	  multiplication(domain_difference(Z, Y), multiplication(coantidomain(antidomain(Z)), coantidomain(X)))
286.73/287.05	= { by lemma 172 }
286.73/287.05	  multiplication(addition(X, multiplication(domain_difference(Z, Y), coantidomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by lemma 101 }
286.73/287.05	  multiplication(addition(X, multiplication(domain_difference(Z, domain(Y)), coantidomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by axiom 23 (domain4) }
286.73/287.05	  multiplication(addition(X, multiplication(domain_difference(Z, antidomain(antidomain(Y))), coantidomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by lemma 232 }
286.73/287.05	  multiplication(addition(X, domain_difference(antidomain(Y), codomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by lemma 194 }
286.73/287.05	  multiplication(addition(X, multiplication(domain(antidomain(Y)), coantidomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by lemma 161 }
286.73/287.05	  multiplication(addition(X, multiplication(antidomain(Y), coantidomain(antidomain(Z)))), coantidomain(X))
286.73/287.05	= { by lemma 220 }
286.73/287.05	  multiplication(addition(X, domain_difference(coantidomain(antidomain(Z)), Y)), coantidomain(X))
286.73/287.05	= { by axiom 28 (domain_difference) }
286.73/287.05	  multiplication(addition(X, multiplication(domain(coantidomain(antidomain(Z))), antidomain(Y))), coantidomain(X))
286.73/287.05	= { by lemma 172 }
286.73/287.05	  multiplication(domain(coantidomain(antidomain(Z))), multiplication(antidomain(Y), coantidomain(X)))
286.73/287.05	= { by lemma 119 }
286.73/287.05	  multiplication(domain_difference(coantidomain(antidomain(Z)), Y), coantidomain(X))
286.73/287.05	= { by lemma 243 }
286.73/287.05	  multiplication(domain_difference(domain(Z), Y), coantidomain(X))
286.73/287.05	= { by lemma 105 }
286.73/287.05	  multiplication(domain_difference(Z, Y), coantidomain(X))
286.73/287.05	
286.73/287.05	Lemma 266: forward_diamond(coantidomain(X), coantidomain(Y)) = multiplication(coantidomain(X), coantidomain(Y)).
286.73/287.05	Proof:
286.73/287.05	  forward_diamond(coantidomain(X), coantidomain(Y))
286.73/287.05	= { by lemma 224 }
286.73/287.05	  forward_diamond(coantidomain(X), domain_difference(addition(?, coantidomain(Y)), codomain(Y)))
286.73/287.05	= { by lemma 265 }
286.73/287.05	  multiplication(domain_difference(addition(?, coantidomain(Y)), codomain(Y)), coantidomain(X))
286.73/287.05	= { by lemma 224 }
286.73/287.05	  multiplication(coantidomain(Y), coantidomain(X))
286.73/287.05	= { by lemma 226 }
286.73/287.05	  domain_difference(coantidomain(Y), codomain(X))
286.73/287.05	= { by lemma 220 }
286.73/287.05	  multiplication(antidomain(codomain(X)), coantidomain(Y))
286.73/287.05	= { by lemma 196 }
286.73/287.05	  domain_difference(antidomain(codomain(X)), codomain(Y))
286.73/287.05	= { by lemma 184 }
286.73/287.05	  domain_difference(coantidomain(X), codomain(Y))
286.73/287.05	= { by lemma 226 }
286.73/287.05	  multiplication(coantidomain(X), coantidomain(Y))
286.73/287.05	
286.73/287.05	Lemma 267: multiplication(domain_difference(X, Y), codomain(Z)) = backward_diamond(domain_difference(X, Y), Z).
286.73/287.05	Proof:
286.73/287.05	  multiplication(domain_difference(X, Y), codomain(Z))
286.73/287.05	= { by axiom 20 (codomain4) }
286.73/287.05	  multiplication(domain_difference(X, Y), coantidomain(coantidomain(Z)))
286.73/287.05	= { by lemma 265 }
286.73/287.05	  forward_diamond(coantidomain(coantidomain(Z)), domain_difference(X, Y))
286.73/287.05	= { by axiom 20 (codomain4) }
286.73/287.05	  forward_diamond(codomain(Z), domain_difference(X, Y))
286.73/287.05	= { by lemma 261 }
286.73/287.05	  backward_diamond(domain(domain_difference(X, Y)), Z)
286.73/287.05	= { by lemma 256 }
286.73/287.05	  backward_diamond(domain_difference(X, Y), Z)
286.73/287.05	
286.73/287.05	Lemma 268: forward_diamond(domain_difference(Y, X), coantidomain(Z)) = multiplication(domain_difference(Y, X), coantidomain(Z)).
286.73/287.05	Proof:
286.73/287.05	  forward_diamond(domain_difference(Y, X), coantidomain(Z))
286.73/287.05	= { by lemma 204 }
286.73/287.05	  domain(multiplication(domain_difference(Y, X), coantidomain(Z)))
286.73/287.05	= { by lemma 265 }
286.73/287.05	  domain(forward_diamond(coantidomain(Z), domain_difference(Y, X)))
286.73/287.05	= { by lemma 98 }
286.73/287.05	  forward_diamond(coantidomain(Z), domain_difference(Y, X))
286.73/287.05	= { by lemma 265 }
286.73/287.05	  multiplication(domain_difference(Y, X), coantidomain(Z))
286.73/287.05	
286.73/287.05	Lemma 270: multiplication(coantidomain(X), domain_difference(Y, Z)) = multiplication(domain_difference(Y, Z), coantidomain(X)).
286.73/287.05	Proof:
286.73/287.05	  multiplication(coantidomain(X), domain_difference(Y, Z))
286.73/287.05	= { by lemma 125 }
286.73/287.05	  multiplication(coantidomain(X), multiplication(coantidomain(X), domain_difference(Y, Z)))
286.73/287.05	= { by axiom 11 (multiplicative_associativity) }
286.73/287.05	  multiplication(multiplication(coantidomain(X), coantidomain(X)), domain_difference(Y, Z))
286.73/287.05	= { by lemma 266 }
286.73/287.05	  multiplication(forward_diamond(coantidomain(X), coantidomain(X)), domain_difference(Y, Z))
286.73/287.05	= { by lemma 254 }
286.73/287.05	  multiplication(domain(Y), domain_difference(multiplication(coantidomain(X), coantidomain(X)), Z))
286.73/287.05	= { by lemma 69 }
286.73/287.05	  multiplication(domain(Y), domain_difference(coantidomain(X), Z))
286.73/287.05	= { by lemma 264 }
286.73/287.05	  forward_diamond(domain_difference(Y, Z), coantidomain(X))
286.73/287.05	= { by lemma 268 }
286.73/287.05	  multiplication(domain_difference(Y, Z), coantidomain(X))
286.73/287.05	
286.73/287.05	Lemma 270: multiplication(domain_difference(Y, Z), coantidomain(X)) = multiplication(coantidomain(X), domain_difference(Y, Z)).
286.73/287.05	Proof:
286.73/287.05	  multiplication(domain_difference(Y, Z), coantidomain(X))
286.73/287.05	= { by lemma 268 }
286.73/287.05	  forward_diamond(domain_difference(Y, Z), coantidomain(X))
286.73/287.05	= { by lemma 264 }
286.73/287.05	  multiplication(domain(Y), domain_difference(coantidomain(X), Z))
286.73/287.05	= { by lemma 69 }
286.73/287.05	  multiplication(domain(Y), domain_difference(multiplication(coantidomain(X), coantidomain(X)), Z))
286.73/287.05	= { by lemma 254 }
286.73/287.05	  multiplication(forward_diamond(coantidomain(X), coantidomain(X)), domain_difference(Y, Z))
286.73/287.05	= { by lemma 266 }
286.73/287.05	  multiplication(multiplication(coantidomain(X), coantidomain(X)), domain_difference(Y, Z))
286.73/287.05	= { by axiom 11 (multiplicative_associativity) }
286.73/287.05	  multiplication(coantidomain(X), multiplication(coantidomain(X), domain_difference(Y, Z)))
286.73/287.05	= { by lemma 125 }
286.73/287.05	  multiplication(coantidomain(X), domain_difference(Y, Z))
286.73/287.05	
286.73/287.05	Lemma 271: multiplication(codomain(Z), domain_difference(X, Y)) = backward_diamond(domain_difference(X, Y), Z).
286.73/287.05	Proof:
286.73/287.05	  multiplication(codomain(Z), domain_difference(X, Y))
286.73/287.05	= { by axiom 20 (codomain4) }
286.73/287.05	  multiplication(coantidomain(coantidomain(Z)), domain_difference(X, Y))
286.73/287.05	= { by lemma 270 }
286.73/287.05	  multiplication(domain_difference(X, Y), coantidomain(coantidomain(Z)))
286.73/287.05	= { by axiom 20 (codomain4) }
286.73/287.05	  multiplication(domain_difference(X, Y), codomain(Z))
286.73/287.05	= { by lemma 267 }
286.73/287.06	  backward_diamond(domain_difference(X, Y), Z)
286.73/287.06	
286.73/287.06	Lemma 272: forward_diamond(domain_difference(Y, X), codomain(Z)) = backward_diamond(domain_difference(Y, X), Z).
286.73/287.06	Proof:
286.73/287.06	  forward_diamond(domain_difference(Y, X), codomain(Z))
286.73/287.06	= { by lemma 264 }
286.73/287.06	  multiplication(domain(Y), domain_difference(codomain(Z), X))
286.73/287.06	= { by lemma 70 }
286.73/287.06	  multiplication(domain(Y), domain_difference(multiplication(codomain(Z), codomain(Z)), X))
286.73/287.06	= { by lemma 254 }
286.73/287.06	  multiplication(forward_diamond(codomain(Z), codomain(Z)), domain_difference(Y, X))
286.73/287.06	= { by lemma 185 }
286.73/287.06	  multiplication(forward_diamond(antidomain(coantidomain(Z)), codomain(Z)), domain_difference(Y, X))
286.73/287.06	= { by lemma 161 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), codomain(Z)), domain_difference(Y, X))
286.73/287.06	= { by lemma 94 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), codomain(codomain(Z))), domain_difference(Y, X))
286.73/287.06	= { by lemma 185 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), codomain(antidomain(coantidomain(Z)))), domain_difference(Y, X))
286.73/287.06	= { by lemma 161 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), codomain(domain(antidomain(coantidomain(Z))))), domain_difference(Y, X))
286.73/287.06	= { by axiom 6 (multiplicative_left_identity) }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), multiplication(one, codomain(domain(antidomain(coantidomain(Z)))))), domain_difference(Y, X))
286.73/287.06	= { by lemma 55 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), multiplication(addition(one, domain(antidomain(coantidomain(Z)))), codomain(domain(antidomain(coantidomain(Z)))))), domain_difference(Y, X))
286.73/287.06	= { by lemma 153 }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), addition(domain(antidomain(coantidomain(Z))), codomain(domain(antidomain(coantidomain(Z)))))), domain_difference(Y, X))
286.73/287.06	= { by axiom 5 (additive_commutativity) }
286.73/287.06	  multiplication(forward_diamond(domain(antidomain(coantidomain(Z))), addition(codomain(domain(antidomain(coantidomain(Z)))), domain(antidomain(coantidomain(Z))))), domain_difference(Y, X))
286.73/287.06	= { by lemma 182 }
286.73/287.06	  multiplication(domain(antidomain(coantidomain(Z))), domain_difference(Y, X))
286.73/287.06	= { by lemma 161 }
286.73/287.06	  multiplication(antidomain(coantidomain(Z)), domain_difference(Y, X))
286.73/287.06	= { by lemma 185 }
286.73/287.06	  multiplication(codomain(Z), domain_difference(Y, X))
286.73/287.06	= { by lemma 271 }
286.73/287.06	  backward_diamond(domain_difference(Y, X), Z)
286.73/287.06	
286.73/287.06	Lemma 273: forward_diamond(antidomain(X), Y) = domain_difference(Y, X).
286.73/287.06	Proof:
286.73/287.06	  forward_diamond(antidomain(X), Y)
286.73/287.06	= { by axiom 27 (forward_diamond) }
286.73/287.06	  domain(multiplication(antidomain(X), domain(Y)))
286.73/287.06	= { by lemma 213 }
286.73/287.06	  domain(domain_difference(Y, X))
286.73/287.06	= { by lemma 256 }
286.73/287.06	  domain_difference(Y, X)
286.73/287.06	
286.73/287.06	Lemma 274: multiplication(domain(X), backward_diamond(Y, Z)) = forward_diamond(backward_diamond(Y, Z), X).
286.73/287.06	Proof:
286.73/287.06	  multiplication(domain(X), backward_diamond(Y, Z))
286.73/287.06	= { by lemma 260 }
286.73/287.06	  forward_diamond(domain(X), backward_diamond(Y, Z))
286.73/287.06	= { by lemma 244 }
286.73/287.06	  domain(multiplication(domain(X), backward_diamond(Y, Z)))
286.73/287.06	= { by axiom 23 (domain4) }
286.73/287.06	  antidomain(antidomain(multiplication(domain(X), backward_diamond(Y, Z))))
286.73/287.06	= { by lemma 108 }
286.73/287.06	  antidomain(antidomain(multiplication(domain(X), multiplication(backward_diamond(Y, Z), backward_diamond(Y, Z)))))
286.73/287.06	= { by lemma 253 }
286.73/287.06	  antidomain(antidomain(multiplication(domain(X), forward_diamond(backward_diamond(Y, Z), backward_diamond(Y, Z)))))
286.73/287.06	= { by lemma 231 }
286.73/287.06	  antidomain(antidomain(multiplication(forward_diamond(backward_diamond(Y, Z), backward_diamond(Y, Z)), domain(X))))
286.73/287.06	= { by lemma 116 }
286.73/287.06	  antidomain(antidomain(forward_diamond(forward_diamond(backward_diamond(Y, Z), backward_diamond(Y, Z)), X)))
286.73/287.06	= { by lemma 242 }
286.73/287.06	  antidomain(antidomain(multiplication(forward_diamond(backward_diamond(Y, Z), backward_diamond(Y, Z)), X)))
286.73/287.06	= { by lemma 191 }
286.73/287.06	  antidomain(antidomain(multiplication(backward_diamond(Y, Z), X)))
286.73/287.06	= { by axiom 23 (domain4) }
286.73/287.06	  domain(multiplication(backward_diamond(Y, Z), X))
286.73/287.06	= { by lemma 244 }
286.73/287.06	  forward_diamond(backward_diamond(Y, Z), X)
286.73/287.06	
286.73/287.06	Lemma 275: forward_diamond(domain_difference(Z, W), backward_diamond(X, Y)) = multiplication(backward_diamond(X, Y), domain_difference(Z, W)).
286.73/287.06	Proof:
286.73/287.06	  forward_diamond(domain_difference(Z, W), backward_diamond(X, Y))
286.73/287.06	= { by lemma 108 }
286.73/287.06	  forward_diamond(domain_difference(Z, W), multiplication(backward_diamond(X, Y), backward_diamond(X, Y)))
286.73/287.06	= { by lemma 264 }
286.73/287.06	  multiplication(domain(Z), domain_difference(multiplication(backward_diamond(X, Y), backward_diamond(X, Y)), W))
286.73/287.06	= { by lemma 250 }
286.73/287.06	  multiplication(domain(Z), domain_difference(forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)), W))
286.73/287.06	= { by lemma 128 }
286.73/287.06	  multiplication(domain(Z), multiplication(forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)), antidomain(W)))
286.73/287.06	= { by lemma 238 }
286.73/287.06	  multiplication(forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)), multiplication(domain(Z), antidomain(W)))
286.73/287.06	= { by lemma 191 }
286.73/287.06	  multiplication(backward_diamond(X, Y), multiplication(domain(Z), antidomain(W)))
286.73/287.06	= { by axiom 28 (domain_difference) }
287.34/287.63	  multiplication(backward_diamond(X, Y), domain_difference(Z, W))
287.34/287.63	
287.34/287.63	Goal 1 (goals_1): zero = multiplication(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))).
287.34/287.63	Proof:
287.34/287.63	  zero
287.34/287.63	= { by lemma 80 }
287.34/287.63	  multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))))
287.34/287.63	= { by axiom 16 (codomain2) }
287.34/287.63	  multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), addition(coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))), coantidomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), addition(coantidomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1)))))
287.34/287.63	= { by axiom 3 (right_distributivity) }
287.34/287.63	  addition(multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1)))), multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 80 }
287.34/287.63	  addition(zero, multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 35 }
287.34/287.63	  multiplication(codomain(multiplication(coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), domain(sK2_goals_X1))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))))
287.34/287.63	= { by lemma 102 }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), coantidomain(coantidomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0)))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))))
287.34/287.63	= { by axiom 20 (codomain4) }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), codomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))))
287.34/287.63	= { by lemma 95 }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0)), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))))
287.34/287.63	= { by lemma 165 }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(multiplication(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0), domain(sK2_goals_X1))))
287.34/287.63	= { by axiom 11 (multiplicative_associativity) }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(multiplication(codomain(domain(sK1_goals_X2)), multiplication(sK3_goals_X0, domain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 197 }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 76 }
287.34/287.63	  multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain_difference(one, backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 275 }
287.34/287.63	  forward_diamond(domain_difference(one, backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))), backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 76 }
287.34/287.63	  forward_diamond(antidomain(backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))), backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 273 }
287.34/287.63	  domain_difference(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 261 }
287.34/287.63	  domain_difference(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 186 }
287.34/287.63	  domain_difference(forward_diamond(domain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 243 }
287.34/287.63	  domain_difference(forward_diamond(coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), sK2_goals_X1), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 258 }
287.34/287.63	  domain_difference(domain_difference(sK2_goals_X1, codomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 232 }
287.34/287.63	  domain_difference(multiplication(domain_difference(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1)), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 259 }
287.34/287.63	  domain_difference(multiplication(antidomain(forward_box(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1))), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 241 }
287.34/287.63	  domain_difference(multiplication(codomain(antidomain(forward_box(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1)))), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 259 }
287.34/287.63	  domain_difference(multiplication(codomain(domain_difference(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1))), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 217 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(antidomain(antidomain(sK2_goals_X1)), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 184 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 20 (codomain4) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 185 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 97 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 83 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(domain(multiplication(sK3_goals_X0, domain(sK2_goals_X1))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 27 (forward_diamond) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 7 (multiplicative_right_identity) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), one), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 47 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), addition(antidomain(sK1_goals_X2), domain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), addition(domain(sK1_goals_X2), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 3 (right_distributivity) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), domain(sK1_goals_X2)), multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 104 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)), multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 7 (multiplicative_right_identity) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), multiplication(domain(sK1_goals_X2), one)), multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 45 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), multiplication(domain(sK1_goals_X2), coantidomain(zero))), multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 30 (goals) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), multiplication(domain(sK1_goals_X2), coantidomain(multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))))), multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 106 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(addition(zero, multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 35 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2)), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 128 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2)))
287.34/287.63	= { by axiom 6 (multiplicative_left_identity) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(one, backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 45 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(zero), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by axiom 8 (left_annihilation) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(zero, multiplication(sK3_goals_X0, domain(sK2_goals_X1)))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 46 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(domain_difference(sK1_goals_X2, sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1)))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 105 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(domain_difference(domain(sK1_goals_X2), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1)))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(domain_difference(antidomain(antidomain(sK1_goals_X2)), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1)))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 135 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 138 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2)), multiplication(sK3_goals_X0, domain(sK2_goals_X1))))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 154 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), multiplication(coantidomain(multiplication(domain(sK1_goals_X2), multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))))), backward_diamond(multiplication(domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2), multiplication(sK3_goals_X0, domain(sK2_goals_X1))), domain(sK1_goals_X2))))
287.34/287.63	= { by lemma 137 }
287.34/287.63	  domain_difference(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), zero)
287.34/287.63	= { by lemma 105 }
287.34/287.63	  domain_difference(domain(multiplication(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), c(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), zero)
287.34/287.63	= { by lemma 73 }
287.34/287.63	  domain_difference(forward_diamond(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), zero)
287.34/287.63	= { by lemma 113 }
287.34/287.63	  domain_difference(antidomain(forward_box(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), zero)
287.34/287.63	= { by lemma 216 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(backward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 261 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 104 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 252 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain_difference(multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 247 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 154 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain(sK2_goals_X1)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 97 }
287.34/287.63	  domain_difference(antidomain(zero), c(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain(sK2_goals_X1)))))
287.34/287.63	= { by axiom 11 (multiplicative_associativity) }
287.34/287.63	  domain_difference(antidomain(zero), c(multiplication(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), domain(sK2_goals_X1))))
287.34/287.63	= { by lemma 59 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(forward_diamond(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), sK2_goals_X1)))
287.34/287.63	= { by lemma 242 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), sK2_goals_X1)))
287.34/287.63	= { by axiom 11 (multiplicative_associativity) }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(antidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1))))
287.34/287.63	= { by lemma 247 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain_difference(multiplication(addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 252 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(forward_diamond(addition(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(forward_diamond(addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 44 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(forward_diamond(one, sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 163 }
287.34/287.63	  domain_difference(antidomain(zero), forward_box(domain(sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 115 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain_difference(sK2_goals_X1, coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 184 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain_difference(sK2_goals_X1, antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))))
287.34/287.63	= { by lemma 257 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(forward_diamond(domain(sK2_goals_X1), codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 77 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain(domain_difference(sK2_goals_X1, antidomain(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))))
287.34/287.63	= { by lemma 214 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain(domain_difference(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 210 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(antidomain(forward_box(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), antidomain(sK2_goals_X1)))))
287.34/287.63	= { by lemma 114 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(antidomain(antidomain(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1)))))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(domain(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1))))
287.34/287.63	= { by lemma 98 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1)))
287.34/287.63	= { by lemma 242 }
287.34/287.63	  domain_difference(antidomain(zero), antidomain(multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1)))
287.34/287.63	= { by lemma 257 }
287.34/287.63	  forward_diamond(domain(antidomain(zero)), multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1))
287.34/287.63	= { by lemma 251 }
287.34/287.63	  forward_diamond(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), antidomain(zero))
287.34/287.63	= { by lemma 113 }
287.34/287.63	  antidomain(forward_box(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero))
287.34/287.63	= { by lemma 142 }
287.34/287.63	  antidomain(forward_box(multiplication(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), antidomain(zero)), zero))
287.34/287.63	= { by lemma 128 }
287.34/287.63	  antidomain(forward_box(domain_difference(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero), zero))
287.34/287.63	= { by lemma 121 }
287.34/287.63	  antidomain(antidomain(domain_difference(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero)))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  domain(domain_difference(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero))
287.34/287.63	= { by lemma 256 }
287.34/287.63	  domain_difference(forward_diamond(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero)
287.34/287.63	= { by lemma 250 }
287.34/287.63	  domain_difference(multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), sK2_goals_X1), zero)
287.34/287.63	= { by axiom 20 (codomain4) }
287.34/287.63	  domain_difference(multiplication(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), zero)
287.34/287.63	= { by lemma 250 }
287.34/287.63	  domain_difference(forward_diamond(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1), zero)
287.34/287.63	= { by lemma 166 }
287.34/287.63	  domain_difference(multiplication(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain(sK2_goals_X1)), zero)
287.34/287.63	= { by lemma 213 }
287.34/287.63	  multiplication(antidomain(zero), domain(multiplication(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain(sK2_goals_X1))))
287.34/287.63	= { by axiom 27 (forward_diamond) }
287.34/287.63	  multiplication(antidomain(zero), forward_diamond(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), sK2_goals_X1))
287.34/287.63	= { by lemma 258 }
287.34/287.63	  multiplication(antidomain(zero), domain_difference(sK2_goals_X1, codomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 194 }
287.34/287.63	  multiplication(antidomain(zero), multiplication(domain(sK2_goals_X1), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 140 }
287.34/287.63	  multiplication(antidomain(zero), multiplication(addition(domain(sK2_goals_X1), domain_difference(sK2_goals_X1, ?)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 167 }
287.34/287.63	  multiplication(antidomain(zero), multiplication(addition(domain(sK2_goals_X1), addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain_difference(sK2_goals_X1, ?))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 157 }
287.34/287.63	  multiplication(antidomain(zero), multiplication(addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by axiom 11 (multiplicative_associativity) }
287.34/287.63	  multiplication(multiplication(antidomain(zero), addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 161 }
287.34/287.63	  multiplication(multiplication(domain(antidomain(zero)), addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain(sK2_goals_X1))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  multiplication(multiplication(domain(antidomain(zero)), addition(domain(sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 3 (right_distributivity) }
287.34/287.63	  multiplication(addition(multiplication(domain(antidomain(zero)), domain(sK2_goals_X1)), multiplication(domain(antidomain(zero)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 23 (domain4) }
287.34/287.63	  multiplication(addition(multiplication(domain(antidomain(zero)), antidomain(antidomain(sK2_goals_X1))), multiplication(domain(antidomain(zero)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 28 (domain_difference) }
287.34/287.63	  multiplication(addition(domain_difference(antidomain(zero), antidomain(sK2_goals_X1)), multiplication(domain(antidomain(zero)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 257 }
287.34/287.63	  multiplication(addition(forward_diamond(domain(antidomain(zero)), sK2_goals_X1), multiplication(domain(antidomain(zero)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  multiplication(addition(multiplication(domain(antidomain(zero)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), forward_diamond(domain(antidomain(zero)), sK2_goals_X1)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 161 }
287.34/287.63	  multiplication(addition(multiplication(antidomain(zero), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), forward_diamond(domain(antidomain(zero)), sK2_goals_X1)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 161 }
287.34/287.63	  multiplication(addition(multiplication(antidomain(zero), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), forward_diamond(antidomain(zero), sK2_goals_X1)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 273 }
287.34/287.63	  multiplication(addition(multiplication(antidomain(zero), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain_difference(sK2_goals_X1, zero)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by axiom 5 (additive_commutativity) }
287.34/287.63	  multiplication(addition(domain_difference(sK2_goals_X1, zero), multiplication(antidomain(zero), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 171 }
287.34/287.63	  addition(multiplication(antidomain(zero), multiplication(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), multiplication(domain_difference(sK2_goals_X1, zero), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by axiom 19 (codomain1) }
287.34/287.63	  addition(multiplication(antidomain(zero), zero), multiplication(domain_difference(sK2_goals_X1, zero), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by axiom 10 (right_annihilation) }
287.34/287.63	  addition(zero, multiplication(domain_difference(sK2_goals_X1, zero), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))
287.34/287.63	= { by lemma 35 }
287.34/287.63	  multiplication(domain_difference(sK2_goals_X1, zero), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))
287.34/287.63	= { by lemma 270 }
287.34/287.63	  multiplication(coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), domain_difference(sK2_goals_X1, zero))
287.34/287.63	= { by axiom 20 (codomain4) }
287.34/287.63	  multiplication(codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), domain_difference(sK2_goals_X1, zero))
287.34/287.63	= { by lemma 271 }
287.34/287.63	  backward_diamond(domain_difference(sK2_goals_X1, zero), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 165 }
287.34/287.63	  backward_diamond(domain_difference(sK2_goals_X1, zero), multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0))
287.34/287.63	= { by lemma 272 }
287.34/287.63	  forward_diamond(domain_difference(sK2_goals_X1, zero), codomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0)))
287.34/287.63	= { by lemma 50 }
287.34/287.63	  forward_diamond(domain(sK2_goals_X1), codomain(multiplication(codomain(domain(sK1_goals_X2)), sK3_goals_X0)))
287.34/287.63	= { by axiom 24 (backward_diamond) }
287.34/287.63	  forward_diamond(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))
287.34/287.63	= { by lemma 260 }
287.34/287.63	  multiplication(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))
287.34/287.63	% SZS output end Proof
287.34/287.63	
287.34/287.63	RESULT: Theorem (the conjecture is true).
287.34/287.69	EOF