0.05/0.07	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.05/0.07	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.06/0.26	% Computer   : n014.cluster.edu
0.06/0.26	% Model      : x86_64 x86_64
0.06/0.26	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.06/0.26	% Memory     : 8042.1875MB
0.06/0.26	% OS         : Linux 3.10.0-693.el7.x86_64
0.06/0.26	% CPULimit   : 1200
0.06/0.26	% WCLimit    : 120
0.06/0.26	% DateTime   : Tue Jul 13 15:54:49 EDT 2021
0.06/0.26	% CPUTime    : 
335.89/42.64	% SZS status Theorem
335.89/42.64	
347.18/44.01	% SZS output start Proof
347.18/44.01	Take the following subset of the input axioms:
347.18/44.02	  fof(additive_associativity, axiom, ![A, C, B]: addition(addition(A, B), C)=addition(A, addition(B, C))).
347.18/44.02	  fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)).
347.18/44.02	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
347.18/44.02	  fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
347.18/44.02	  fof(backward_box, axiom, ![X0, X1]: c(backward_diamond(X0, c(X1)))=backward_box(X0, X1)).
347.18/44.02	  fof(backward_diamond, axiom, ![X0, X1]: codomain(multiplication(codomain(X1), X0))=backward_diamond(X0, X1)).
347.18/44.02	  fof(codomain1, axiom, ![X0]: zero=multiplication(X0, coantidomain(X0))).
347.18/44.02	  fof(codomain2, axiom, ![X0, X1]: addition(coantidomain(multiplication(X0, X1)), coantidomain(multiplication(coantidomain(coantidomain(X0)), X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0)), X1))).
347.18/44.02	  fof(codomain3, axiom, ![X0]: addition(coantidomain(coantidomain(X0)), coantidomain(X0))=one).
347.18/44.02	  fof(codomain4, axiom, ![X0]: coantidomain(coantidomain(X0))=codomain(X0)).
347.18/44.02	  fof(complement, axiom, ![X0]: c(X0)=antidomain(domain(X0))).
347.18/44.02	  fof(domain1, axiom, ![X0]: zero=multiplication(antidomain(X0), X0)).
347.18/44.02	  fof(domain2, axiom, ![X0, X1]: antidomain(multiplication(X0, antidomain(antidomain(X1))))=addition(antidomain(multiplication(X0, X1)), antidomain(multiplication(X0, antidomain(antidomain(X1)))))).
347.18/44.02	  fof(domain3, axiom, ![X0]: addition(antidomain(antidomain(X0)), antidomain(X0))=one).
347.18/44.02	  fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))).
347.18/44.02	  fof(domain_difference, axiom, ![X0, X1]: domain_difference(X0, X1)=multiplication(domain(X0), antidomain(X1))).
347.18/44.02	  fof(forward_box, axiom, ![X0, X1]: forward_box(X0, X1)=c(forward_diamond(X0, c(X1)))).
347.18/44.02	  fof(forward_diamond, axiom, ![X0, X1]: forward_diamond(X0, X1)=domain(multiplication(X0, domain(X1)))).
347.18/44.02	  fof(goals, conjecture, ![X0, X1, X2]: (backward_box(X0, domain(X2))=addition(domain(X1), backward_box(X0, domain(X2))) => addition(forward_diamond(X0, domain(X1)), domain(X2))=domain(X2))).
347.18/44.02	  fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)).
347.18/44.02	  fof(left_distributivity, axiom, ![A, C, B]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
347.18/44.02	  fof(multiplicative_associativity, axiom, ![A, C, B]: multiplication(A, multiplication(B, C))=multiplication(multiplication(A, B), C)).
347.18/44.02	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
347.18/44.02	  fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A).
347.18/44.02	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
347.18/44.02	  fof(right_annihilation, axiom, ![A]: multiplication(A, zero)=zero).
347.18/44.02	  fof(right_distributivity, axiom, ![A, C, B]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
347.18/44.02	
347.18/44.02	Now clausify the problem and encode Horn clauses using encoding 3 of
347.18/44.02	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
347.18/44.02	We repeatedly replace C & s=t => u=v by the two clauses:
347.18/44.02	  fresh(y, y, x1...xn) = u
347.18/44.02	  C => fresh(s, t, x1...xn) = v
347.18/44.02	where fresh is a fresh function symbol and x1..xn are the free
347.18/44.02	variables of u and v.
347.18/44.02	A predicate p(X) is encoded as p(X)=true (this is sound, because the
347.18/44.02	input problem has no model of domain size 1).
347.18/44.02	
347.18/44.02	The encoding turns the above axioms into the following unit equations and goals:
347.18/44.02	
347.18/44.02	Axiom 1 (codomain4): coantidomain(coantidomain(X)) = codomain(X).
347.18/44.02	Axiom 2 (complement): c(X) = antidomain(domain(X)).
347.18/44.02	Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
347.18/44.02	Axiom 4 (additive_idempotence): X = addition(X, X).
347.18/44.02	Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
347.18/44.02	Axiom 6 (additive_identity): addition(X, zero) = X.
347.18/44.02	Axiom 7 (multiplicative_right_identity): multiplication(X, one) = X.
347.18/44.02	Axiom 8 (right_annihilation): multiplication(X, zero) = zero.
347.18/44.02	Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
347.18/44.02	Axiom 10 (left_annihilation): zero = multiplication(zero, X).
347.18/44.02	Axiom 11 (codomain1): zero = multiplication(X, coantidomain(X)).
347.18/44.02	Axiom 12 (domain1): zero = multiplication(antidomain(X), X).
347.18/44.02	Axiom 13 (order): fresh(X, X, Y, Z) = true.
347.18/44.02	Axiom 14 (order_1): fresh2(X, X, Y, Z) = Z.
347.18/44.02	Axiom 15 (backward_diamond): codomain(multiplication(codomain(X), Y)) = backward_diamond(Y, X).
347.18/44.02	Axiom 16 (backward_box): c(backward_diamond(X, c(Y))) = backward_box(X, Y).
347.18/44.02	Axiom 17 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
347.18/44.02	Axiom 18 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
347.18/44.02	Axiom 19 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)).
347.18/44.02	Axiom 20 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
347.18/44.02	Axiom 21 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
347.18/44.02	Axiom 22 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
347.18/44.02	Axiom 23 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
347.18/44.02	Axiom 24 (order): fresh(X, addition(Y, X), Y, X) = leq(Y, X).
347.18/44.02	Axiom 25 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
347.18/44.02	Axiom 26 (goals): backward_box(x0, domain(x2)) = addition(domain(x1), backward_box(x0, domain(x2))).
347.18/44.02	Axiom 27 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
347.18/44.02	Axiom 28 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
347.18/44.02	Axiom 29 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
347.18/44.02	Axiom 30 (domain2): antidomain(multiplication(X, antidomain(antidomain(Y)))) = addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))).
347.18/44.02	
347.18/44.02	Lemma 31: addition(domain(X), antidomain(X)) = one.
347.18/44.02	Proof:
347.18/44.02	  addition(domain(X), antidomain(X))
347.18/44.02	= { by axiom 3 (domain4) }
347.18/44.02	  addition(antidomain(antidomain(X)), antidomain(X))
347.18/44.02	= { by axiom 23 (domain3) }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 32: multiplication(domain(X), X) = X.
347.18/44.02	Proof:
347.18/44.02	  multiplication(domain(X), X)
347.18/44.02	= { by axiom 6 (additive_identity) R->L }
347.18/44.02	  addition(multiplication(domain(X), X), zero)
347.18/44.02	= { by axiom 12 (domain1) }
347.18/44.02	  addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
347.18/44.02	= { by axiom 28 (left_distributivity) }
347.18/44.02	  multiplication(addition(domain(X), antidomain(X)), X)
347.18/44.02	= { by lemma 31 }
347.18/44.02	  multiplication(one, X)
347.18/44.02	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.02	  X
347.18/44.02	
347.18/44.02	Lemma 33: domain(antidomain(X)) = c(X).
347.18/44.02	Proof:
347.18/44.02	  domain(antidomain(X))
347.18/44.02	= { by axiom 3 (domain4) }
347.18/44.02	  antidomain(antidomain(antidomain(X)))
347.18/44.02	= { by axiom 3 (domain4) R->L }
347.18/44.02	  antidomain(domain(X))
347.18/44.02	= { by axiom 2 (complement) R->L }
347.18/44.02	  c(X)
347.18/44.02	
347.18/44.02	Lemma 34: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
347.18/44.02	Proof:
347.18/44.02	  multiplication(antidomain(X), addition(X, Y))
347.18/44.02	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.02	  multiplication(antidomain(X), addition(Y, X))
347.18/44.02	= { by axiom 27 (right_distributivity) R->L }
347.18/44.02	  addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
347.18/44.02	= { by axiom 12 (domain1) R->L }
347.18/44.02	  addition(multiplication(antidomain(X), Y), zero)
347.18/44.02	= { by axiom 6 (additive_identity) }
347.18/44.02	  multiplication(antidomain(X), Y)
347.18/44.02	
347.18/44.02	Lemma 35: antidomain(X) = c(X).
347.18/44.02	Proof:
347.18/44.02	  antidomain(X)
347.18/44.02	= { by lemma 32 R->L }
347.18/44.02	  multiplication(domain(antidomain(X)), antidomain(X))
347.18/44.02	= { by lemma 33 }
347.18/44.02	  multiplication(c(X), antidomain(X))
347.18/44.02	= { by axiom 2 (complement) }
347.18/44.02	  multiplication(antidomain(domain(X)), antidomain(X))
347.18/44.02	= { by lemma 34 R->L }
347.18/44.02	  multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
347.18/44.02	= { by lemma 31 }
347.18/44.02	  multiplication(antidomain(domain(X)), one)
347.18/44.02	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.02	  antidomain(domain(X))
347.18/44.02	= { by axiom 2 (complement) R->L }
347.18/44.02	  c(X)
347.18/44.02	
347.18/44.02	Lemma 36: antidomain(one) = zero.
347.18/44.02	Proof:
347.18/44.02	  antidomain(one)
347.18/44.02	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.02	  multiplication(antidomain(one), one)
347.18/44.02	= { by axiom 12 (domain1) R->L }
347.18/44.02	  zero
347.18/44.02	
347.18/44.02	Lemma 37: domain(one) = one.
347.18/44.02	Proof:
347.18/44.02	  domain(one)
347.18/44.02	= { by axiom 6 (additive_identity) R->L }
347.18/44.02	  addition(domain(one), zero)
347.18/44.02	= { by lemma 36 R->L }
347.18/44.02	  addition(domain(one), antidomain(one))
347.18/44.02	= { by lemma 31 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 38: antidomain(zero) = one.
347.18/44.02	Proof:
347.18/44.02	  antidomain(zero)
347.18/44.02	= { by lemma 36 R->L }
347.18/44.02	  antidomain(antidomain(one))
347.18/44.02	= { by axiom 3 (domain4) R->L }
347.18/44.02	  domain(one)
347.18/44.02	= { by lemma 37 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 39: coantidomain(one) = zero.
347.18/44.02	Proof:
347.18/44.02	  coantidomain(one)
347.18/44.02	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.02	  multiplication(one, coantidomain(one))
347.18/44.02	= { by axiom 11 (codomain1) R->L }
347.18/44.02	  zero
347.18/44.02	
347.18/44.02	Lemma 40: addition(codomain(X), coantidomain(X)) = one.
347.18/44.02	Proof:
347.18/44.02	  addition(codomain(X), coantidomain(X))
347.18/44.02	= { by axiom 1 (codomain4) R->L }
347.18/44.02	  addition(coantidomain(coantidomain(X)), coantidomain(X))
347.18/44.02	= { by axiom 22 (codomain3) }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 41: codomain(one) = one.
347.18/44.02	Proof:
347.18/44.02	  codomain(one)
347.18/44.02	= { by axiom 6 (additive_identity) R->L }
347.18/44.02	  addition(codomain(one), zero)
347.18/44.02	= { by lemma 39 R->L }
347.18/44.02	  addition(codomain(one), coantidomain(one))
347.18/44.02	= { by lemma 40 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 42: coantidomain(zero) = one.
347.18/44.02	Proof:
347.18/44.02	  coantidomain(zero)
347.18/44.02	= { by lemma 39 R->L }
347.18/44.02	  coantidomain(coantidomain(one))
347.18/44.02	= { by axiom 1 (codomain4) }
347.18/44.02	  codomain(one)
347.18/44.02	= { by lemma 41 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 43: c(one) = zero.
347.18/44.02	Proof:
347.18/44.02	  c(one)
347.18/44.02	= { by axiom 2 (complement) }
347.18/44.02	  antidomain(domain(one))
347.18/44.02	= { by lemma 37 }
347.18/44.02	  antidomain(one)
347.18/44.02	= { by lemma 36 }
347.18/44.02	  zero
347.18/44.02	
347.18/44.02	Lemma 44: c(zero) = one.
347.18/44.02	Proof:
347.18/44.02	  c(zero)
347.18/44.02	= { by lemma 33 R->L }
347.18/44.02	  domain(antidomain(zero))
347.18/44.02	= { by lemma 38 }
347.18/44.02	  domain(one)
347.18/44.02	= { by lemma 37 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 45: addition(zero, X) = X.
347.18/44.02	Proof:
347.18/44.02	  addition(zero, X)
347.18/44.02	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.02	  addition(X, zero)
347.18/44.02	= { by axiom 6 (additive_identity) }
347.18/44.02	  X
347.18/44.02	
347.18/44.02	Lemma 46: antidomain(c(X)) = domain(domain(X)).
347.18/44.02	Proof:
347.18/44.02	  antidomain(c(X))
347.18/44.02	= { by axiom 2 (complement) }
347.18/44.02	  antidomain(antidomain(domain(X)))
347.18/44.02	= { by axiom 3 (domain4) R->L }
347.18/44.02	  domain(domain(X))
347.18/44.02	
347.18/44.02	Lemma 47: antidomain(c(X)) = c(antidomain(X)).
347.18/44.02	Proof:
347.18/44.02	  antidomain(c(X))
347.18/44.02	= { by lemma 33 R->L }
347.18/44.02	  antidomain(domain(antidomain(X)))
347.18/44.02	= { by axiom 2 (complement) R->L }
347.18/44.02	  c(antidomain(X))
347.18/44.02	
347.18/44.02	Lemma 48: domain(domain(X)) = forward_diamond(one, X).
347.18/44.02	Proof:
347.18/44.02	  domain(domain(X))
347.18/44.02	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.02	  domain(multiplication(one, domain(X)))
347.18/44.02	= { by axiom 18 (forward_diamond) R->L }
347.18/44.02	  forward_diamond(one, X)
347.18/44.02	
347.18/44.02	Lemma 49: c(antidomain(X)) = forward_diamond(one, X).
347.18/44.02	Proof:
347.18/44.02	  c(antidomain(X))
347.18/44.02	= { by lemma 33 R->L }
347.18/44.02	  domain(antidomain(antidomain(X)))
347.18/44.02	= { by axiom 3 (domain4) R->L }
347.18/44.02	  domain(domain(X))
347.18/44.02	= { by lemma 48 }
347.18/44.02	  forward_diamond(one, X)
347.18/44.02	
347.18/44.02	Lemma 50: addition(c(X), domain(X)) = one.
347.18/44.02	Proof:
347.18/44.02	  addition(c(X), domain(X))
347.18/44.02	= { by axiom 3 (domain4) }
347.18/44.02	  addition(c(X), antidomain(antidomain(X)))
347.18/44.02	= { by lemma 33 R->L }
347.18/44.02	  addition(domain(antidomain(X)), antidomain(antidomain(X)))
347.18/44.02	= { by lemma 31 }
347.18/44.02	  one
347.18/44.02	
347.18/44.02	Lemma 51: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
347.18/44.02	Proof:
347.18/44.02	  multiplication(domain(X), domain(Y))
347.18/44.02	= { by axiom 3 (domain4) }
347.18/44.02	  multiplication(domain(X), antidomain(antidomain(Y)))
347.18/44.02	= { by axiom 20 (domain_difference) R->L }
347.18/44.02	  domain_difference(X, antidomain(Y))
347.18/44.02	
347.18/44.02	Lemma 52: multiplication(c(X), domain(Y)) = domain_difference(antidomain(X), antidomain(Y)).
347.18/44.02	Proof:
347.18/44.03	  multiplication(c(X), domain(Y))
347.18/44.03	= { by lemma 33 R->L }
347.18/44.03	  multiplication(domain(antidomain(X)), domain(Y))
347.18/44.03	= { by lemma 51 }
347.18/44.03	  domain_difference(antidomain(X), antidomain(Y))
347.18/44.03	
347.18/44.03	Lemma 53: domain_difference(antidomain(X), X) = antidomain(X).
347.18/44.03	Proof:
347.18/44.03	  domain_difference(antidomain(X), X)
347.18/44.03	= { by axiom 20 (domain_difference) }
347.18/44.03	  multiplication(domain(antidomain(X)), antidomain(X))
347.18/44.03	= { by lemma 32 }
347.18/44.03	  antidomain(X)
347.18/44.03	
347.18/44.03	Lemma 54: forward_diamond(one, X) = domain(X).
347.18/44.03	Proof:
347.18/44.03	  forward_diamond(one, X)
347.18/44.03	= { by lemma 49 R->L }
347.18/44.03	  c(antidomain(X))
347.18/44.03	= { by lemma 47 R->L }
347.18/44.03	  antidomain(c(X))
347.18/44.03	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.03	  multiplication(antidomain(c(X)), one)
347.18/44.03	= { by lemma 50 R->L }
347.18/44.03	  multiplication(antidomain(c(X)), addition(c(X), domain(X)))
347.18/44.03	= { by lemma 34 }
347.18/44.03	  multiplication(antidomain(c(X)), domain(X))
347.18/44.03	= { by lemma 47 }
347.18/44.03	  multiplication(c(antidomain(X)), domain(X))
347.18/44.03	= { by lemma 52 }
347.18/44.03	  domain_difference(antidomain(antidomain(X)), antidomain(X))
347.18/44.03	= { by lemma 53 }
347.18/44.03	  antidomain(antidomain(X))
347.18/44.03	= { by axiom 3 (domain4) R->L }
347.18/44.03	  domain(X)
347.18/44.03	
347.18/44.03	Lemma 55: domain(domain(X)) = domain(X).
347.18/44.03	Proof:
347.18/44.03	  domain(domain(X))
347.18/44.03	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.03	  domain(multiplication(one, domain(X)))
347.18/44.03	= { by axiom 18 (forward_diamond) R->L }
347.18/44.03	  forward_diamond(one, X)
347.18/44.03	= { by lemma 54 }
347.18/44.03	  domain(X)
347.18/44.03	
347.18/44.03	Lemma 56: c(c(X)) = domain(X).
347.18/44.03	Proof:
347.18/44.03	  c(c(X))
347.18/44.03	= { by lemma 35 R->L }
347.18/44.03	  c(antidomain(X))
347.18/44.03	= { by lemma 35 R->L }
347.18/44.03	  antidomain(antidomain(X))
347.18/44.03	= { by axiom 3 (domain4) R->L }
347.18/44.03	  domain(X)
347.18/44.03	
347.18/44.03	Lemma 57: backward_diamond(X, one) = codomain(X).
347.18/44.03	Proof:
347.18/44.03	  backward_diamond(X, one)
347.18/44.03	= { by axiom 15 (backward_diamond) R->L }
347.18/44.03	  codomain(multiplication(codomain(one), X))
347.18/44.03	= { by lemma 41 }
347.18/44.03	  codomain(multiplication(one, X))
347.18/44.03	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.03	  codomain(X)
347.18/44.03	
347.18/44.03	Lemma 58: c(codomain(X)) = backward_box(X, zero).
347.18/44.03	Proof:
347.18/44.03	  c(codomain(X))
347.18/44.03	= { by lemma 57 R->L }
347.18/44.03	  c(backward_diamond(X, one))
347.18/44.03	= { by lemma 44 R->L }
347.18/44.03	  c(backward_diamond(X, c(zero)))
347.18/44.03	= { by axiom 16 (backward_box) }
347.18/44.03	  backward_box(X, zero)
347.18/44.03	
347.18/44.03	Lemma 59: codomain(codomain(X)) = backward_diamond(one, X).
347.18/44.03	Proof:
347.18/44.03	  codomain(codomain(X))
347.18/44.03	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.03	  codomain(multiplication(codomain(X), one))
347.18/44.03	= { by axiom 15 (backward_diamond) }
347.18/44.03	  backward_diamond(one, X)
347.18/44.03	
347.18/44.03	Lemma 60: coantidomain(codomain(X)) = codomain(coantidomain(X)).
347.18/44.03	Proof:
347.18/44.03	  coantidomain(codomain(X))
347.18/44.03	= { by axiom 1 (codomain4) R->L }
347.18/44.03	  coantidomain(coantidomain(coantidomain(X)))
347.18/44.03	= { by axiom 1 (codomain4) }
347.18/44.03	  codomain(coantidomain(X))
347.18/44.03	
347.18/44.03	Lemma 61: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
347.18/44.03	Proof:
347.18/44.03	  multiplication(addition(X, Y), coantidomain(X))
347.18/44.03	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.03	  multiplication(addition(Y, X), coantidomain(X))
347.18/44.03	= { by axiom 28 (left_distributivity) R->L }
347.18/44.03	  addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
347.18/44.03	= { by axiom 11 (codomain1) R->L }
347.18/44.03	  addition(multiplication(Y, coantidomain(X)), zero)
347.18/44.03	= { by axiom 6 (additive_identity) }
347.18/44.03	  multiplication(Y, coantidomain(X))
347.18/44.03	
347.18/44.03	Lemma 62: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
347.18/44.03	Proof:
347.18/44.03	  multiplication(X, addition(Y, coantidomain(X)))
347.18/44.03	= { by axiom 27 (right_distributivity) R->L }
347.18/44.03	  addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
347.18/44.03	= { by axiom 11 (codomain1) R->L }
347.18/44.03	  addition(multiplication(X, Y), zero)
347.18/44.03	= { by axiom 6 (additive_identity) }
347.18/44.03	  multiplication(X, Y)
347.18/44.03	
347.18/44.03	Lemma 63: multiplication(X, codomain(X)) = X.
347.18/44.03	Proof:
347.18/44.03	  multiplication(X, codomain(X))
347.18/44.03	= { by lemma 62 R->L }
347.18/44.03	  multiplication(X, addition(codomain(X), coantidomain(X)))
347.18/44.03	= { by lemma 40 }
347.18/44.03	  multiplication(X, one)
347.18/44.03	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.03	  X
347.18/44.03	
347.18/44.03	Lemma 64: codomain(coantidomain(X)) = coantidomain(X).
347.18/44.03	Proof:
347.18/44.03	  codomain(coantidomain(X))
347.18/44.03	= { by lemma 60 R->L }
347.18/44.03	  coantidomain(codomain(X))
347.18/44.03	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.03	  multiplication(one, coantidomain(codomain(X)))
347.18/44.03	= { by lemma 40 R->L }
347.18/44.03	  multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
347.18/44.03	= { by lemma 61 }
347.18/44.03	  multiplication(coantidomain(X), coantidomain(codomain(X)))
347.18/44.03	= { by lemma 60 }
347.18/44.03	  multiplication(coantidomain(X), codomain(coantidomain(X)))
347.18/44.03	= { by lemma 63 }
347.18/44.03	  coantidomain(X)
347.18/44.03	
347.18/44.03	Lemma 65: backward_diamond(one, X) = codomain(X).
347.18/44.03	Proof:
347.18/44.03	  backward_diamond(one, X)
347.18/44.03	= { by lemma 59 R->L }
347.18/44.03	  codomain(codomain(X))
347.18/44.03	= { by axiom 1 (codomain4) R->L }
347.18/44.03	  codomain(coantidomain(coantidomain(X)))
347.18/44.03	= { by lemma 64 }
347.18/44.03	  coantidomain(coantidomain(X))
347.18/44.03	= { by axiom 1 (codomain4) }
347.18/44.03	  codomain(X)
347.18/44.03	
347.18/44.03	Lemma 66: backward_box(codomain(X), zero) = backward_box(X, zero).
347.18/44.03	Proof:
347.18/44.03	  backward_box(codomain(X), zero)
347.18/44.03	= { by lemma 58 R->L }
347.18/44.03	  c(codomain(codomain(X)))
347.18/44.03	= { by lemma 59 }
347.18/44.03	  c(backward_diamond(one, X))
347.18/44.03	= { by lemma 65 }
347.18/44.03	  c(codomain(X))
347.18/44.03	= { by lemma 58 }
347.18/44.03	  backward_box(X, zero)
347.18/44.03	
347.18/44.03	Lemma 67: multiplication(X, addition(coantidomain(X), Y)) = multiplication(X, Y).
347.18/44.03	Proof:
347.18/44.03	  multiplication(X, addition(coantidomain(X), Y))
347.18/44.03	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.03	  multiplication(X, addition(Y, coantidomain(X)))
347.18/44.03	= { by lemma 62 }
347.18/44.03	  multiplication(X, Y)
347.18/44.03	
347.18/44.03	Lemma 68: multiplication(backward_box(X, zero), coantidomain(X)) = backward_box(X, zero).
347.18/44.03	Proof:
347.18/44.03	  multiplication(backward_box(X, zero), coantidomain(X))
347.18/44.03	= { by lemma 58 R->L }
347.18/44.03	  multiplication(c(codomain(X)), coantidomain(X))
347.18/44.03	= { by lemma 35 R->L }
347.18/44.03	  multiplication(antidomain(codomain(X)), coantidomain(X))
347.18/44.03	= { by lemma 34 R->L }
347.18/44.03	  multiplication(antidomain(codomain(X)), addition(codomain(X), coantidomain(X)))
347.18/44.03	= { by lemma 40 }
347.18/44.03	  multiplication(antidomain(codomain(X)), one)
347.18/44.03	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.03	  antidomain(codomain(X))
347.18/44.03	= { by lemma 35 }
347.18/44.03	  c(codomain(X))
347.18/44.03	= { by lemma 58 }
347.18/44.03	  backward_box(X, zero)
347.18/44.03	
347.18/44.03	Lemma 69: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
347.18/44.03	Proof:
347.18/44.03	  multiplication(addition(X, one), Y)
347.18/44.03	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.03	  multiplication(addition(one, X), Y)
347.18/44.03	= { by axiom 28 (left_distributivity) R->L }
347.18/44.03	  addition(multiplication(one, Y), multiplication(X, Y))
347.18/44.03	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.03	  addition(Y, multiplication(X, Y))
347.18/44.03	
347.18/44.03	Lemma 70: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
347.18/44.03	Proof:
347.18/44.03	  multiplication(addition(one, Y), X)
347.18/44.03	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.03	  multiplication(addition(Y, one), X)
347.18/44.03	= { by lemma 69 }
347.18/44.03	  addition(X, multiplication(Y, X))
347.18/44.03	
347.18/44.03	Lemma 71: addition(X, addition(X, Y)) = addition(X, Y).
347.18/44.03	Proof:
347.18/44.03	  addition(X, addition(X, Y))
347.18/44.03	= { by axiom 19 (additive_associativity) R->L }
347.18/44.03	  addition(addition(X, X), Y)
347.18/44.03	= { by axiom 4 (additive_idempotence) R->L }
347.18/44.03	  addition(X, Y)
347.18/44.03	
347.18/44.03	Lemma 72: addition(one, domain(X)) = one.
347.18/44.03	Proof:
347.18/44.03	  addition(one, domain(X))
347.18/44.03	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.03	  addition(domain(X), one)
347.18/44.03	= { by lemma 31 R->L }
347.18/44.03	  addition(domain(X), addition(domain(X), antidomain(X)))
347.18/44.03	= { by lemma 71 }
347.18/44.03	  addition(domain(X), antidomain(X))
347.18/44.03	= { by lemma 31 }
347.18/44.03	  one
347.18/44.03	
347.18/44.03	Lemma 73: addition(one, c(X)) = one.
347.18/44.03	Proof:
347.18/44.03	  addition(one, c(X))
347.18/44.03	= { by lemma 33 R->L }
347.18/44.03	  addition(one, domain(antidomain(X)))
347.18/44.03	= { by lemma 72 }
347.18/44.03	  one
347.18/44.03	
347.18/44.03	Lemma 74: addition(one, backward_box(X, Y)) = one.
347.18/44.03	Proof:
347.18/44.03	  addition(one, backward_box(X, Y))
347.18/44.03	= { by axiom 16 (backward_box) R->L }
347.18/44.03	  addition(one, c(backward_diamond(X, c(Y))))
347.18/44.03	= { by lemma 73 }
347.18/44.03	  one
347.18/44.03	
347.18/44.03	Lemma 75: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
347.18/44.03	Proof:
347.18/44.03	  c(multiplication(X, domain(Y)))
347.18/44.03	= { by axiom 2 (complement) }
347.18/44.03	  antidomain(domain(multiplication(X, domain(Y))))
347.18/44.03	= { by axiom 18 (forward_diamond) R->L }
347.18/44.03	  antidomain(forward_diamond(X, Y))
347.18/44.03	
347.18/44.03	Lemma 76: c(multiplication(X, c(Y))) = forward_box(X, Y).
347.18/44.03	Proof:
347.18/44.03	  c(multiplication(X, c(Y)))
347.18/44.03	= { by lemma 33 R->L }
347.18/44.03	  c(multiplication(X, domain(antidomain(Y))))
347.18/44.03	= { by lemma 75 }
347.18/44.03	  antidomain(forward_diamond(X, antidomain(Y)))
347.18/44.03	= { by lemma 35 }
347.18/44.03	  c(forward_diamond(X, antidomain(Y)))
347.18/44.03	= { by lemma 35 }
347.18/44.03	  c(forward_diamond(X, c(Y)))
347.18/44.03	= { by axiom 17 (forward_box) R->L }
347.18/44.03	  forward_box(X, Y)
347.18/44.03	
347.18/44.03	Lemma 77: multiplication(antidomain(X), multiplication(X, Y)) = zero.
347.18/44.03	Proof:
347.18/44.03	  multiplication(antidomain(X), multiplication(X, Y))
347.18/44.03	= { by axiom 21 (multiplicative_associativity) }
347.18/44.03	  multiplication(multiplication(antidomain(X), X), Y)
347.18/44.03	= { by axiom 12 (domain1) R->L }
347.18/44.03	  multiplication(zero, Y)
347.18/44.03	= { by axiom 10 (left_annihilation) R->L }
347.18/44.03	  zero
347.18/44.03	
347.18/44.03	Lemma 78: multiplication(c(X), X) = zero.
347.18/44.03	Proof:
347.18/44.03	  multiplication(c(X), X)
347.18/44.03	= { by axiom 2 (complement) }
347.18/44.03	  multiplication(antidomain(domain(X)), X)
347.18/44.03	= { by lemma 32 R->L }
347.18/44.03	  multiplication(antidomain(domain(X)), multiplication(domain(X), X))
347.18/44.03	= { by lemma 77 }
347.18/44.03	  zero
347.18/44.03	
347.18/44.03	Lemma 79: multiplication(forward_box(X, Y), multiplication(X, c(Y))) = zero.
347.18/44.03	Proof:
347.18/44.03	  multiplication(forward_box(X, Y), multiplication(X, c(Y)))
347.18/44.03	= { by lemma 76 R->L }
347.18/44.03	  multiplication(c(multiplication(X, c(Y))), multiplication(X, c(Y)))
347.18/44.03	= { by lemma 78 }
347.18/44.03	  zero
347.18/44.03	
347.18/44.03	Lemma 80: c(multiplication(X, domain(Y))) = forward_box(X, c(Y)).
347.18/44.03	Proof:
347.18/44.03	  c(multiplication(X, domain(Y)))
347.18/44.03	= { by lemma 56 R->L }
347.18/44.03	  c(multiplication(X, c(c(Y))))
347.18/44.03	= { by lemma 76 }
347.18/44.03	  forward_box(X, c(Y))
347.18/44.03	
347.18/44.03	Lemma 81: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
347.18/44.03	Proof:
347.18/44.04	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
347.18/44.04	= { by axiom 3 (domain4) }
347.18/44.04	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
347.18/44.04	= { by axiom 30 (domain2) R->L }
347.18/44.04	  antidomain(multiplication(X, antidomain(antidomain(Y))))
347.18/44.04	= { by axiom 3 (domain4) R->L }
347.18/44.04	  antidomain(multiplication(X, domain(Y)))
347.18/44.04	
347.18/44.04	Lemma 82: leq(X, addition(Y, X)) = true.
347.18/44.04	Proof:
347.18/44.04	  leq(X, addition(Y, X))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  leq(X, addition(X, Y))
347.18/44.04	= { by axiom 24 (order) R->L }
347.18/44.04	  fresh(addition(X, Y), addition(X, addition(X, Y)), X, addition(X, Y))
347.18/44.04	= { by lemma 71 }
347.18/44.04	  fresh(addition(X, Y), addition(X, Y), X, addition(X, Y))
347.18/44.04	= { by axiom 13 (order) }
347.18/44.04	  true
347.18/44.04	
347.18/44.04	Lemma 83: addition(one, antidomain(X)) = one.
347.18/44.04	Proof:
347.18/44.04	  addition(one, antidomain(X))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  addition(antidomain(X), one)
347.18/44.04	= { by axiom 25 (order_1) R->L }
347.18/44.04	  fresh2(leq(antidomain(X), one), true, antidomain(X), one)
347.18/44.04	= { by lemma 31 R->L }
347.18/44.04	  fresh2(leq(antidomain(X), addition(domain(X), antidomain(X))), true, antidomain(X), one)
347.18/44.04	= { by lemma 82 }
347.18/44.04	  fresh2(true, true, antidomain(X), one)
347.18/44.04	= { by axiom 14 (order_1) }
347.18/44.04	  one
347.18/44.04	
347.18/44.04	Lemma 84: multiplication(X, multiplication(codomain(X), Y)) = multiplication(X, Y).
347.18/44.04	Proof:
347.18/44.04	  multiplication(X, multiplication(codomain(X), Y))
347.18/44.04	= { by axiom 21 (multiplicative_associativity) }
347.18/44.04	  multiplication(multiplication(X, codomain(X)), Y)
347.18/44.04	= { by lemma 63 }
347.18/44.04	  multiplication(X, Y)
347.18/44.04	
347.18/44.04	Lemma 85: multiplication(X, c(coantidomain(X))) = X.
347.18/44.04	Proof:
347.18/44.04	  multiplication(X, c(coantidomain(X)))
347.18/44.04	= { by axiom 6 (additive_identity) R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), zero))
347.18/44.04	= { by lemma 79 R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(forward_box(codomain(X), c(coantidomain(X))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 80 R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(c(multiplication(codomain(X), domain(coantidomain(X)))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 35 R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 81 R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(codomain(X), coantidomain(X))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by axiom 1 (codomain4) R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), coantidomain(X))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 64 R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), codomain(coantidomain(X)))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by axiom 1 (codomain4) R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X))))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by axiom 11 (codomain1) R->L }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(zero), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 38 }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(addition(one, antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by lemma 83 }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(one, multiplication(codomain(X), c(c(coantidomain(X)))))))
347.18/44.04	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(codomain(X), c(c(coantidomain(X))))))
347.18/44.04	= { by lemma 56 }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), multiplication(codomain(X), domain(coantidomain(X)))))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  multiplication(X, addition(multiplication(codomain(X), domain(coantidomain(X))), c(coantidomain(X))))
347.18/44.04	= { by axiom 27 (right_distributivity) R->L }
347.18/44.04	  addition(multiplication(X, multiplication(codomain(X), domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
347.18/44.04	= { by lemma 84 }
347.18/44.04	  addition(multiplication(X, domain(coantidomain(X))), multiplication(X, c(coantidomain(X))))
347.18/44.04	= { by axiom 27 (right_distributivity) }
347.18/44.04	  multiplication(X, addition(domain(coantidomain(X)), c(coantidomain(X))))
347.18/44.04	= { by axiom 5 (additive_commutativity) }
347.18/44.04	  multiplication(X, addition(c(coantidomain(X)), domain(coantidomain(X))))
347.18/44.04	= { by lemma 50 }
347.18/44.04	  multiplication(X, one)
347.18/44.04	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.04	  X
347.18/44.04	
347.18/44.04	Lemma 86: backward_box(X, zero) = coantidomain(X).
347.18/44.04	Proof:
347.18/44.04	  backward_box(X, zero)
347.18/44.04	= { by lemma 66 R->L }
347.18/44.04	  backward_box(codomain(X), zero)
347.18/44.04	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.04	  multiplication(one, backward_box(codomain(X), zero))
347.18/44.04	= { by lemma 40 R->L }
347.18/44.04	  multiplication(addition(codomain(X), coantidomain(X)), backward_box(codomain(X), zero))
347.18/44.04	= { by axiom 28 (left_distributivity) R->L }
347.18/44.04	  addition(multiplication(codomain(X), backward_box(codomain(X), zero)), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by lemma 67 R->L }
347.18/44.04	  addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), backward_box(codomain(X), zero))), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by lemma 68 R->L }
347.18/44.04	  addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), multiplication(backward_box(codomain(X), zero), coantidomain(codomain(X))))), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by lemma 70 R->L }
347.18/44.04	  addition(multiplication(codomain(X), multiplication(addition(one, backward_box(codomain(X), zero)), coantidomain(codomain(X)))), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by lemma 74 }
347.18/44.04	  addition(multiplication(codomain(X), multiplication(one, coantidomain(codomain(X)))), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.04	  addition(multiplication(codomain(X), coantidomain(codomain(X))), multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by axiom 11 (codomain1) R->L }
347.18/44.04	  addition(zero, multiplication(coantidomain(X), backward_box(codomain(X), zero)))
347.18/44.04	= { by lemma 45 }
347.18/44.04	  multiplication(coantidomain(X), backward_box(codomain(X), zero))
347.18/44.04	= { by lemma 66 }
347.18/44.04	  multiplication(coantidomain(X), backward_box(X, zero))
347.18/44.04	= { by lemma 58 R->L }
347.18/44.04	  multiplication(coantidomain(X), c(codomain(X)))
347.18/44.04	= { by axiom 1 (codomain4) R->L }
347.18/44.04	  multiplication(coantidomain(X), c(coantidomain(coantidomain(X))))
347.18/44.04	= { by lemma 85 }
347.18/44.04	  coantidomain(X)
347.18/44.04	
347.18/44.04	Lemma 87: backward_box(coantidomain(X), zero) = c(coantidomain(X)).
347.18/44.04	Proof:
347.18/44.04	  backward_box(coantidomain(X), zero)
347.18/44.04	= { by lemma 58 R->L }
347.18/44.04	  c(codomain(coantidomain(X)))
347.18/44.04	= { by lemma 64 }
347.18/44.04	  c(coantidomain(X))
347.18/44.04	
347.18/44.04	Lemma 88: c(coantidomain(X)) = codomain(X).
347.18/44.04	Proof:
347.18/44.04	  c(coantidomain(X))
347.18/44.04	= { by lemma 87 R->L }
347.18/44.04	  backward_box(coantidomain(X), zero)
347.18/44.04	= { by lemma 86 }
347.18/44.04	  coantidomain(coantidomain(X))
347.18/44.04	= { by axiom 1 (codomain4) }
347.18/44.04	  codomain(X)
347.18/44.04	
347.18/44.04	Lemma 89: domain(codomain(X)) = codomain(X).
347.18/44.04	Proof:
347.18/44.04	  domain(codomain(X))
347.18/44.04	= { by lemma 56 R->L }
347.18/44.04	  c(c(codomain(X)))
347.18/44.04	= { by lemma 58 }
347.18/44.04	  c(backward_box(X, zero))
347.18/44.04	= { by lemma 86 }
347.18/44.04	  c(coantidomain(X))
347.18/44.04	= { by lemma 88 }
347.18/44.04	  codomain(X)
347.18/44.04	
347.18/44.04	Lemma 90: domain(c(X)) = c(X).
347.18/44.04	Proof:
347.18/44.04	  domain(c(X))
347.18/44.04	= { by lemma 35 R->L }
347.18/44.04	  domain(antidomain(X))
347.18/44.04	= { by lemma 33 }
347.18/44.04	  c(X)
347.18/44.04	
347.18/44.04	Lemma 91: addition(X, multiplication(domain(Y), X)) = X.
347.18/44.04	Proof:
347.18/44.04	  addition(X, multiplication(domain(Y), X))
347.18/44.04	= { by lemma 70 R->L }
347.18/44.04	  multiplication(addition(one, domain(Y)), X)
347.18/44.04	= { by lemma 72 }
347.18/44.04	  multiplication(one, X)
347.18/44.04	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.04	  X
347.18/44.04	
347.18/44.04	Lemma 92: addition(domain(X), codomain(domain(X))) = codomain(domain(X)).
347.18/44.04	Proof:
347.18/44.04	  addition(domain(X), codomain(domain(X)))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  addition(codomain(domain(X)), domain(X))
347.18/44.04	= { by lemma 63 R->L }
347.18/44.04	  addition(codomain(domain(X)), multiplication(domain(X), codomain(domain(X))))
347.18/44.04	= { by lemma 91 }
347.18/44.04	  codomain(domain(X))
347.18/44.04	
347.18/44.04	Lemma 93: multiplication(domain(X), domain(Y)) = domain_difference(X, c(Y)).
347.18/44.04	Proof:
347.18/44.04	  multiplication(domain(X), domain(Y))
347.18/44.04	= { by axiom 3 (domain4) }
347.18/44.04	  multiplication(domain(X), antidomain(antidomain(Y)))
347.18/44.04	= { by axiom 20 (domain_difference) R->L }
347.18/44.04	  domain_difference(X, antidomain(Y))
347.18/44.04	= { by lemma 35 }
347.18/44.04	  domain_difference(X, c(Y))
347.18/44.04	
347.18/44.04	Lemma 94: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
347.18/44.04	Proof:
347.18/44.04	  addition(Y, addition(X, Z))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  addition(addition(X, Z), Y)
347.18/44.04	= { by axiom 19 (additive_associativity) }
347.18/44.04	  addition(X, addition(Z, Y))
347.18/44.04	= { by axiom 5 (additive_commutativity) }
347.18/44.04	  addition(X, addition(Y, Z))
347.18/44.04	
347.18/44.04	Lemma 95: addition(X, multiplication(domain(Y), addition(X, Z))) = addition(X, multiplication(domain(Y), Z)).
347.18/44.04	Proof:
347.18/44.04	  addition(X, multiplication(domain(Y), addition(X, Z)))
347.18/44.04	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.04	  addition(multiplication(domain(Y), addition(X, Z)), X)
347.18/44.04	= { by axiom 27 (right_distributivity) R->L }
347.18/44.04	  addition(addition(multiplication(domain(Y), X), multiplication(domain(Y), Z)), X)
347.18/44.04	= { by axiom 19 (additive_associativity) }
347.18/44.04	  addition(multiplication(domain(Y), X), addition(multiplication(domain(Y), Z), X))
347.18/44.04	= { by lemma 94 R->L }
347.18/44.04	  addition(multiplication(domain(Y), Z), addition(multiplication(domain(Y), X), X))
347.18/44.04	= { by axiom 5 (additive_commutativity) }
347.18/44.05	  addition(multiplication(domain(Y), Z), addition(X, multiplication(domain(Y), X)))
347.18/44.05	= { by lemma 91 }
347.18/44.05	  addition(multiplication(domain(Y), Z), X)
347.18/44.05	= { by axiom 5 (additive_commutativity) }
347.18/44.05	  addition(X, multiplication(domain(Y), Z))
347.18/44.05	
347.18/44.05	Lemma 96: addition(c(X), domain_difference(Y, c(X))) = addition(c(X), domain(Y)).
347.18/44.05	Proof:
347.18/44.05	  addition(c(X), domain_difference(Y, c(X)))
347.18/44.05	= { by lemma 93 R->L }
347.18/44.05	  addition(c(X), multiplication(domain(Y), domain(X)))
347.18/44.05	= { by lemma 95 R->L }
347.18/44.05	  addition(c(X), multiplication(domain(Y), addition(c(X), domain(X))))
347.18/44.05	= { by lemma 50 }
347.18/44.05	  addition(c(X), multiplication(domain(Y), one))
347.18/44.05	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.05	  addition(c(X), domain(Y))
347.18/44.05	
347.18/44.05	Lemma 97: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
347.18/44.05	Proof:
347.18/44.05	  multiplication(c(X), antidomain(Y))
347.18/44.05	= { by lemma 33 R->L }
347.18/44.05	  multiplication(domain(antidomain(X)), antidomain(Y))
347.18/44.05	= { by axiom 20 (domain_difference) R->L }
347.18/44.05	  domain_difference(antidomain(X), Y)
347.18/44.05	
347.18/44.05	Lemma 98: multiplication(c(X), c(Y)) = domain_difference(c(X), Y).
347.18/44.05	Proof:
347.18/44.05	  multiplication(c(X), c(Y))
347.18/44.05	= { by lemma 35 R->L }
347.18/44.05	  multiplication(c(X), antidomain(Y))
347.18/44.05	= { by lemma 97 }
347.18/44.05	  domain_difference(antidomain(X), Y)
347.18/44.05	= { by lemma 35 }
347.18/44.05	  domain_difference(c(X), Y)
347.18/44.05	
347.18/44.05	Lemma 99: domain(multiplication(X, c(Y))) = forward_diamond(X, c(Y)).
347.18/44.05	Proof:
347.18/44.05	  domain(multiplication(X, c(Y)))
347.18/44.05	= { by lemma 33 R->L }
347.18/44.05	  domain(multiplication(X, domain(antidomain(Y))))
347.18/44.05	= { by axiom 18 (forward_diamond) R->L }
347.18/44.05	  forward_diamond(X, antidomain(Y))
347.18/44.05	= { by lemma 35 }
347.18/44.05	  forward_diamond(X, c(Y))
347.18/44.05	
347.18/44.05	Lemma 100: c(domain(X)) = c(X).
347.18/44.05	Proof:
347.18/44.05	  c(domain(X))
347.18/44.05	= { by lemma 35 R->L }
347.18/44.05	  antidomain(domain(X))
347.18/44.05	= { by axiom 2 (complement) R->L }
347.18/44.05	  c(X)
347.18/44.05	
347.18/44.05	Lemma 101: domain(multiplication(X, multiplication(Y, domain(Z)))) = forward_diamond(multiplication(X, Y), Z).
347.18/44.05	Proof:
347.18/44.05	  domain(multiplication(X, multiplication(Y, domain(Z))))
347.18/44.05	= { by axiom 21 (multiplicative_associativity) }
347.18/44.05	  domain(multiplication(multiplication(X, Y), domain(Z)))
347.18/44.05	= { by axiom 18 (forward_diamond) R->L }
347.18/44.05	  forward_diamond(multiplication(X, Y), Z)
347.18/44.05	
347.18/44.05	Lemma 102: multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X))) = multiplication(codomain(X), Y).
347.18/44.05	Proof:
347.18/44.05	  multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X)))
347.18/44.05	= { by axiom 21 (multiplicative_associativity) }
347.18/44.05	  multiplication(multiplication(codomain(X), Y), backward_diamond(Y, X))
347.18/44.05	= { by axiom 15 (backward_diamond) R->L }
347.18/44.05	  multiplication(multiplication(codomain(X), Y), codomain(multiplication(codomain(X), Y)))
347.18/44.05	= { by lemma 63 }
347.18/44.05	  multiplication(codomain(X), Y)
347.18/44.05	
347.18/44.05	Lemma 103: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
347.18/44.05	Proof:
347.18/44.05	  coantidomain(multiplication(codomain(X), Y))
347.18/44.05	= { by lemma 64 R->L }
347.18/44.05	  codomain(coantidomain(multiplication(codomain(X), Y)))
347.18/44.05	= { by lemma 60 R->L }
347.18/44.05	  coantidomain(codomain(multiplication(codomain(X), Y)))
347.18/44.05	= { by axiom 15 (backward_diamond) }
347.18/44.05	  coantidomain(backward_diamond(Y, X))
347.18/44.05	
347.18/44.05	Lemma 104: multiplication(backward_diamond(X, Y), coantidomain(multiplication(Y, X))) = zero.
347.18/44.05	Proof:
347.18/44.05	  multiplication(backward_diamond(X, Y), coantidomain(multiplication(Y, X)))
347.18/44.05	= { by lemma 67 R->L }
347.18/44.05	  multiplication(backward_diamond(X, Y), addition(coantidomain(backward_diamond(X, Y)), coantidomain(multiplication(Y, X))))
347.18/44.05	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.05	  multiplication(backward_diamond(X, Y), addition(coantidomain(multiplication(Y, X)), coantidomain(backward_diamond(X, Y))))
347.18/44.05	= { by lemma 103 R->L }
347.18/44.05	  multiplication(backward_diamond(X, Y), addition(coantidomain(multiplication(Y, X)), coantidomain(multiplication(codomain(Y), X))))
347.18/44.05	= { by axiom 1 (codomain4) R->L }
347.18/44.05	  multiplication(backward_diamond(X, Y), addition(coantidomain(multiplication(Y, X)), coantidomain(multiplication(coantidomain(coantidomain(Y)), X))))
347.18/44.05	= { by axiom 29 (codomain2) }
347.18/44.05	  multiplication(backward_diamond(X, Y), coantidomain(multiplication(coantidomain(coantidomain(Y)), X)))
347.18/44.05	= { by axiom 1 (codomain4) }
347.18/44.05	  multiplication(backward_diamond(X, Y), coantidomain(multiplication(codomain(Y), X)))
347.18/44.05	= { by lemma 103 }
347.18/44.05	  multiplication(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y)))
347.18/44.05	= { by axiom 11 (codomain1) R->L }
347.18/44.05	  zero
347.18/44.05	
347.18/44.05	Lemma 105: multiplication(codomain(c(X)), X) = zero.
347.18/44.05	Proof:
347.18/44.05	  multiplication(codomain(c(X)), X)
347.18/44.05	= { by lemma 102 R->L }
347.18/44.05	  multiplication(codomain(c(X)), multiplication(X, backward_diamond(X, c(X))))
347.18/44.05	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.05	  multiplication(codomain(c(X)), multiplication(X, multiplication(backward_diamond(X, c(X)), one)))
347.18/44.05	= { by lemma 42 R->L }
347.18/44.05	  multiplication(codomain(c(X)), multiplication(X, multiplication(backward_diamond(X, c(X)), coantidomain(zero))))
347.18/44.05	= { by lemma 78 R->L }
347.18/44.05	  multiplication(codomain(c(X)), multiplication(X, multiplication(backward_diamond(X, c(X)), coantidomain(multiplication(c(X), X)))))
347.18/44.05	= { by lemma 104 }
347.18/44.05	  multiplication(codomain(c(X)), multiplication(X, zero))
347.18/44.05	= { by axiom 8 (right_annihilation) }
347.18/44.05	  multiplication(codomain(c(X)), zero)
347.18/44.05	= { by axiom 8 (right_annihilation) }
347.18/44.05	  zero
347.18/44.05	
347.18/44.05	Lemma 106: codomain(domain(X)) = domain(X).
347.18/44.05	Proof:
347.18/44.05	  codomain(domain(X))
347.18/44.05	= { by lemma 56 R->L }
347.18/44.05	  codomain(c(c(X)))
347.18/44.05	= { by lemma 33 R->L }
347.18/44.05	  codomain(domain(antidomain(c(X))))
347.18/44.05	= { by lemma 92 R->L }
347.18/44.05	  addition(domain(antidomain(c(X))), codomain(domain(antidomain(c(X)))))
347.18/44.05	= { by lemma 33 }
347.18/44.05	  addition(c(c(X)), codomain(domain(antidomain(c(X)))))
347.18/44.05	= { by lemma 33 }
347.18/44.05	  addition(c(c(X)), codomain(c(c(X))))
347.18/44.05	= { by lemma 90 R->L }
347.18/44.05	  addition(c(c(X)), codomain(domain(c(c(X)))))
347.18/44.05	= { by lemma 88 R->L }
347.18/44.05	  addition(c(c(X)), c(coantidomain(domain(c(c(X))))))
347.18/44.05	= { by lemma 90 R->L }
347.18/44.05	  addition(c(c(X)), domain(c(coantidomain(domain(c(c(X)))))))
347.18/44.05	= { by lemma 96 R->L }
347.18/44.05	  addition(c(c(X)), domain_difference(c(coantidomain(domain(c(c(X))))), c(c(X))))
347.18/44.05	= { by lemma 56 R->L }
347.18/44.05	  addition(c(c(X)), domain_difference(c(coantidomain(c(c(c(c(X)))))), c(c(X))))
347.18/44.05	= { by lemma 98 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X)))))
347.18/44.05	= { by lemma 32 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(domain(multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 99 }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(c(coantidomain(c(c(c(c(X)))))), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 87 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(backward_box(coantidomain(c(c(c(c(X))))), zero), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 68 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(multiplication(backward_box(coantidomain(c(c(c(c(X))))), zero), coantidomain(coantidomain(c(c(c(c(X))))))), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by axiom 1 (codomain4) }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(multiplication(backward_box(coantidomain(c(c(c(c(X))))), zero), codomain(c(c(c(c(X)))))), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 87 }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(multiplication(c(coantidomain(c(c(c(c(X)))))), codomain(c(c(c(c(X)))))), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 100 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(forward_diamond(multiplication(c(coantidomain(c(c(c(c(X)))))), codomain(c(domain(c(c(c(X))))))), c(c(c(X)))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 101 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(domain(multiplication(c(coantidomain(c(c(c(c(X)))))), multiplication(codomain(c(domain(c(c(c(X)))))), domain(c(c(c(X))))))), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 105 }
347.18/44.05	  addition(c(c(X)), multiplication(domain(multiplication(c(coantidomain(c(c(c(c(X)))))), zero)), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by axiom 8 (right_annihilation) }
347.18/44.05	  addition(c(c(X)), multiplication(domain(zero), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 36 R->L }
347.18/44.05	  addition(c(c(X)), multiplication(domain(antidomain(one)), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 33 }
347.18/44.05	  addition(c(c(X)), multiplication(c(one), multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by lemma 43 }
347.18/44.05	  addition(c(c(X)), multiplication(zero, multiplication(c(coantidomain(c(c(c(c(X)))))), c(c(c(X))))))
347.18/44.05	= { by axiom 10 (left_annihilation) R->L }
347.18/44.05	  addition(c(c(X)), zero)
347.18/44.05	= { by axiom 6 (additive_identity) }
347.18/44.05	  c(c(X))
347.18/44.05	= { by lemma 56 }
347.18/44.05	  domain(X)
347.18/44.05	
347.18/44.05	Lemma 107: forward_diamond(X, one) = domain(X).
347.18/44.05	Proof:
347.18/44.05	  forward_diamond(X, one)
347.18/44.05	= { by axiom 18 (forward_diamond) }
347.18/44.05	  domain(multiplication(X, domain(one)))
347.18/44.05	= { by lemma 37 }
347.18/44.05	  domain(multiplication(X, one))
347.18/44.05	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.06	  domain(X)
347.18/44.06	
347.18/44.06	Lemma 108: forward_box(X, zero) = c(X).
347.18/44.06	Proof:
347.18/44.06	  forward_box(X, zero)
347.18/44.06	= { by axiom 17 (forward_box) }
347.18/44.06	  c(forward_diamond(X, c(zero)))
347.18/44.06	= { by lemma 44 }
347.18/44.06	  c(forward_diamond(X, one))
347.18/44.06	= { by lemma 107 }
347.18/44.06	  c(domain(X))
347.18/44.06	= { by lemma 100 }
347.18/44.06	  c(X)
347.18/44.06	
347.18/44.06	Lemma 109: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
347.18/44.06	Proof:
347.18/44.06	  domain(multiplication(X, c(Y)))
347.18/44.06	= { by lemma 33 R->L }
347.18/44.06	  domain(multiplication(X, domain(antidomain(Y))))
347.18/44.06	= { by axiom 18 (forward_diamond) R->L }
347.18/44.06	  forward_diamond(X, antidomain(Y))
347.18/44.06	
347.18/44.06	Lemma 110: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), c(Y))
347.18/44.06	= { by axiom 2 (complement) }
347.18/44.06	  multiplication(domain(X), antidomain(domain(Y)))
347.18/44.06	= { by axiom 20 (domain_difference) R->L }
347.18/44.06	  domain_difference(X, domain(Y))
347.18/44.06	
347.18/44.06	Lemma 111: multiplication(domain(X), c(Y)) = domain_difference(X, Y).
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), c(Y))
347.18/44.06	= { by lemma 35 R->L }
347.18/44.06	  multiplication(domain(X), antidomain(Y))
347.18/44.06	= { by axiom 20 (domain_difference) R->L }
347.18/44.06	  domain_difference(X, Y)
347.18/44.06	
347.18/44.06	Lemma 112: domain_difference(X, domain(Y)) = domain_difference(X, Y).
347.18/44.06	Proof:
347.18/44.06	  domain_difference(X, domain(Y))
347.18/44.06	= { by lemma 110 R->L }
347.18/44.06	  multiplication(domain(X), c(Y))
347.18/44.06	= { by lemma 111 }
347.18/44.06	  domain_difference(X, Y)
347.18/44.06	
347.18/44.06	Lemma 113: forward_diamond(domain(X), c(Y)) = domain(domain_difference(X, Y)).
347.18/44.06	Proof:
347.18/44.06	  forward_diamond(domain(X), c(Y))
347.18/44.06	= { by lemma 35 R->L }
347.18/44.06	  forward_diamond(domain(X), antidomain(Y))
347.18/44.06	= { by lemma 109 R->L }
347.18/44.06	  domain(multiplication(domain(X), c(Y)))
347.18/44.06	= { by lemma 110 }
347.18/44.06	  domain(domain_difference(X, domain(Y)))
347.18/44.06	= { by lemma 112 }
347.18/44.06	  domain(domain_difference(X, Y))
347.18/44.06	
347.18/44.06	Lemma 114: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y).
347.18/44.06	Proof:
347.18/44.06	  domain(domain_difference(X, antidomain(Y)))
347.18/44.06	= { by lemma 51 R->L }
347.18/44.06	  domain(multiplication(domain(X), domain(Y)))
347.18/44.06	= { by axiom 18 (forward_diamond) R->L }
347.18/44.06	  forward_diamond(domain(X), Y)
347.18/44.06	
347.18/44.06	Lemma 115: domain(domain_difference(X, c(Y))) = forward_diamond(domain(X), Y).
347.18/44.06	Proof:
347.18/44.06	  domain(domain_difference(X, c(Y)))
347.18/44.06	= { by lemma 35 R->L }
347.18/44.06	  domain(domain_difference(X, antidomain(Y)))
347.18/44.06	= { by lemma 114 }
347.18/44.06	  forward_diamond(domain(X), Y)
347.18/44.06	
347.18/44.06	Lemma 116: multiplication(domain(X), antidomain(X)) = zero.
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), antidomain(X))
347.18/44.06	= { by axiom 3 (domain4) }
347.18/44.06	  multiplication(antidomain(antidomain(X)), antidomain(X))
347.18/44.06	= { by axiom 12 (domain1) R->L }
347.18/44.06	  zero
347.18/44.06	
347.18/44.06	Lemma 117: multiplication(domain(X), addition(c(X), Y)) = multiplication(domain(X), Y).
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), addition(c(X), Y))
347.18/44.06	= { by lemma 35 R->L }
347.18/44.06	  multiplication(domain(X), addition(antidomain(X), Y))
347.18/44.06	= { by axiom 27 (right_distributivity) R->L }
347.18/44.06	  addition(multiplication(domain(X), antidomain(X)), multiplication(domain(X), Y))
347.18/44.06	= { by lemma 116 }
347.18/44.06	  addition(zero, multiplication(domain(X), Y))
347.18/44.06	= { by lemma 45 }
347.18/44.06	  multiplication(domain(X), Y)
347.18/44.06	
347.18/44.06	Lemma 118: multiplication(domain(X), domain_difference(Y, c(X))) = domain_difference(X, c(Y)).
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), domain_difference(Y, c(X)))
347.18/44.06	= { by lemma 117 R->L }
347.18/44.06	  multiplication(domain(X), addition(c(X), domain_difference(Y, c(X))))
347.18/44.06	= { by lemma 96 }
347.18/44.06	  multiplication(domain(X), addition(c(X), domain(Y)))
347.18/44.06	= { by lemma 117 }
347.18/44.06	  multiplication(domain(X), domain(Y))
347.18/44.06	= { by lemma 93 }
347.18/44.06	  domain_difference(X, c(Y))
347.18/44.06	
347.18/44.06	Lemma 119: domain(multiplication(X, multiplication(Y, c(Z)))) = forward_diamond(multiplication(X, Y), c(Z)).
347.18/44.06	Proof:
347.18/44.06	  domain(multiplication(X, multiplication(Y, c(Z))))
347.18/44.06	= { by lemma 33 R->L }
347.18/44.06	  domain(multiplication(X, multiplication(Y, domain(antidomain(Z)))))
347.18/44.06	= { by lemma 101 }
347.18/44.06	  forward_diamond(multiplication(X, Y), antidomain(Z))
347.18/44.06	= { by lemma 35 }
347.18/44.06	  forward_diamond(multiplication(X, Y), c(Z))
347.18/44.06	
347.18/44.06	Lemma 120: domain(multiplication(X, domain_difference(c(Y), Z))) = forward_diamond(multiplication(X, c(Y)), c(Z)).
347.18/44.06	Proof:
347.18/44.06	  domain(multiplication(X, domain_difference(c(Y), Z)))
347.18/44.06	= { by lemma 98 R->L }
347.18/44.06	  domain(multiplication(X, multiplication(c(Y), c(Z))))
347.18/44.06	= { by lemma 119 }
347.18/44.06	  forward_diamond(multiplication(X, c(Y)), c(Z))
347.18/44.06	
347.18/44.06	Lemma 121: forward_diamond(X, domain(Y)) = forward_diamond(X, Y).
347.18/44.06	Proof:
347.18/44.06	  forward_diamond(X, domain(Y))
347.18/44.06	= { by lemma 54 R->L }
347.18/44.06	  forward_diamond(X, forward_diamond(one, Y))
347.18/44.06	= { by lemma 49 R->L }
347.18/44.06	  forward_diamond(X, c(antidomain(Y)))
347.18/44.06	= { by lemma 47 R->L }
347.18/44.06	  forward_diamond(X, antidomain(c(Y)))
347.18/44.06	= { by lemma 109 R->L }
347.18/44.06	  domain(multiplication(X, c(c(Y))))
347.18/44.06	= { by lemma 56 }
347.18/44.06	  domain(multiplication(X, domain(Y)))
347.18/44.06	= { by axiom 18 (forward_diamond) R->L }
347.18/44.06	  forward_diamond(X, Y)
347.18/44.06	
347.18/44.06	Lemma 122: domain_difference(domain(X), Y) = domain_difference(X, Y).
347.18/44.06	Proof:
347.18/44.06	  domain_difference(domain(X), Y)
347.18/44.06	= { by lemma 54 R->L }
347.18/44.06	  domain_difference(forward_diamond(one, X), Y)
347.18/44.06	= { by lemma 49 R->L }
347.18/44.06	  domain_difference(c(antidomain(X)), Y)
347.18/44.06	= { by lemma 47 R->L }
347.18/44.06	  domain_difference(antidomain(c(X)), Y)
347.18/44.06	= { by lemma 97 R->L }
347.18/44.06	  multiplication(c(c(X)), antidomain(Y))
347.18/44.06	= { by lemma 56 }
347.18/44.06	  multiplication(domain(X), antidomain(Y))
347.18/44.06	= { by axiom 20 (domain_difference) R->L }
347.18/44.06	  domain_difference(X, Y)
347.18/44.06	
347.18/44.06	Lemma 123: multiplication(domain(X), multiplication(c(Y), Z)) = multiplication(domain_difference(X, Y), Z).
347.18/44.06	Proof:
347.18/44.06	  multiplication(domain(X), multiplication(c(Y), Z))
347.18/44.06	= { by lemma 35 R->L }
347.18/44.06	  multiplication(domain(X), multiplication(antidomain(Y), Z))
347.18/44.06	= { by axiom 21 (multiplicative_associativity) }
347.18/44.06	  multiplication(multiplication(domain(X), antidomain(Y)), Z)
347.18/44.06	= { by axiom 20 (domain_difference) R->L }
347.18/44.06	  multiplication(domain_difference(X, Y), Z)
347.18/44.06	
347.18/44.06	Lemma 124: multiplication(c(X), domain(X)) = zero.
347.18/44.06	Proof:
347.18/44.06	  multiplication(c(X), domain(X))
347.18/44.06	= { by axiom 2 (complement) }
347.18/44.06	  multiplication(antidomain(domain(X)), domain(X))
347.18/44.06	= { by axiom 12 (domain1) R->L }
347.18/44.06	  zero
347.18/44.06	
347.18/44.06	Lemma 125: multiplication(addition(X, c(Y)), domain(Y)) = multiplication(X, domain(Y)).
347.18/44.06	Proof:
347.18/44.06	  multiplication(addition(X, c(Y)), domain(Y))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  multiplication(addition(c(Y), X), domain(Y))
347.18/44.06	= { by axiom 28 (left_distributivity) R->L }
347.18/44.06	  addition(multiplication(c(Y), domain(Y)), multiplication(X, domain(Y)))
347.18/44.06	= { by lemma 124 }
347.18/44.06	  addition(zero, multiplication(X, domain(Y)))
347.18/44.06	= { by lemma 45 }
347.18/44.06	  multiplication(X, domain(Y))
347.18/44.06	
347.18/44.06	Lemma 126: addition(codomain(X), addition(Y, coantidomain(X))) = addition(Y, one).
347.18/44.06	Proof:
347.18/44.06	  addition(codomain(X), addition(Y, coantidomain(X)))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  addition(codomain(X), addition(coantidomain(X), Y))
347.18/44.06	= { by axiom 19 (additive_associativity) R->L }
347.18/44.06	  addition(addition(codomain(X), coantidomain(X)), Y)
347.18/44.06	= { by lemma 40 }
347.18/44.06	  addition(one, Y)
347.18/44.06	= { by axiom 5 (additive_commutativity) }
347.18/44.06	  addition(Y, one)
347.18/44.06	
347.18/44.06	Lemma 127: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
347.18/44.06	Proof:
347.18/44.06	  multiplication(X, addition(Y, one))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  multiplication(X, addition(one, Y))
347.18/44.06	= { by axiom 27 (right_distributivity) R->L }
347.18/44.06	  addition(multiplication(X, one), multiplication(X, Y))
347.18/44.06	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.06	  addition(X, multiplication(X, Y))
347.18/44.06	
347.18/44.06	Lemma 128: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
347.18/44.06	Proof:
347.18/44.06	  multiplication(X, addition(one, Y))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  multiplication(X, addition(Y, one))
347.18/44.06	= { by lemma 127 }
347.18/44.06	  addition(X, multiplication(X, Y))
347.18/44.06	
347.18/44.06	Lemma 129: addition(X, multiplication(X, codomain(Y))) = X.
347.18/44.06	Proof:
347.18/44.06	  addition(X, multiplication(X, codomain(Y)))
347.18/44.06	= { by lemma 128 R->L }
347.18/44.06	  multiplication(X, addition(one, codomain(Y)))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  multiplication(X, addition(codomain(Y), one))
347.18/44.06	= { by lemma 40 R->L }
347.18/44.06	  multiplication(X, addition(codomain(Y), addition(codomain(Y), coantidomain(Y))))
347.18/44.06	= { by lemma 71 }
347.18/44.06	  multiplication(X, addition(codomain(Y), coantidomain(Y)))
347.18/44.06	= { by lemma 40 }
347.18/44.06	  multiplication(X, one)
347.18/44.06	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.06	  X
347.18/44.06	
347.18/44.06	Lemma 130: addition(codomain(X), domain(codomain(X))) = domain(codomain(X)).
347.18/44.06	Proof:
347.18/44.06	  addition(codomain(X), domain(codomain(X)))
347.18/44.06	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.06	  addition(domain(codomain(X)), codomain(X))
347.18/44.06	= { by lemma 32 R->L }
347.18/44.06	  addition(domain(codomain(X)), multiplication(domain(codomain(X)), codomain(X)))
347.18/44.06	= { by lemma 129 }
347.18/44.06	  domain(codomain(X))
347.18/44.06	
347.18/44.06	Lemma 131: backward_box(one, codomain(X)) = codomain(X).
347.18/44.06	Proof:
347.18/44.06	  backward_box(one, codomain(X))
347.18/44.06	= { by axiom 1 (codomain4) R->L }
347.18/44.06	  backward_box(one, coantidomain(coantidomain(X)))
347.18/44.06	= { by axiom 16 (backward_box) R->L }
347.18/44.06	  c(backward_diamond(one, c(coantidomain(coantidomain(X)))))
347.18/44.06	= { by lemma 65 }
347.18/44.06	  c(codomain(c(coantidomain(coantidomain(X)))))
347.18/44.06	= { by lemma 58 }
347.18/44.06	  backward_box(c(coantidomain(coantidomain(X))), zero)
347.18/44.06	= { by lemma 86 }
347.18/44.06	  coantidomain(c(coantidomain(coantidomain(X))))
347.18/44.06	= { by lemma 64 R->L }
347.18/44.06	  codomain(coantidomain(c(coantidomain(coantidomain(X)))))
347.18/44.06	= { by axiom 1 (codomain4) R->L }
347.18/44.06	  coantidomain(coantidomain(coantidomain(c(coantidomain(coantidomain(X))))))
347.18/44.06	= { by axiom 1 (codomain4) }
347.18/44.06	  coantidomain(codomain(c(coantidomain(coantidomain(X)))))
347.18/44.06	= { by axiom 9 (multiplicative_left_identity) R->L }
347.18/44.06	  coantidomain(codomain(multiplication(one, c(coantidomain(coantidomain(X))))))
347.18/44.06	= { by lemma 72 R->L }
347.18/44.06	  coantidomain(codomain(multiplication(addition(one, domain(coantidomain(coantidomain(X)))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(domain(coantidomain(coantidomain(X))), one), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 126 R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), addition(domain(coantidomain(coantidomain(X))), coantidomain(coantidomain(X)))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by axiom 5 (additive_commutativity) }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), addition(coantidomain(coantidomain(X)), domain(coantidomain(coantidomain(X))))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 64 R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), addition(coantidomain(coantidomain(X)), domain(codomain(coantidomain(coantidomain(X)))))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 64 R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), addition(codomain(coantidomain(coantidomain(X))), domain(codomain(coantidomain(coantidomain(X)))))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 130 }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), domain(codomain(coantidomain(coantidomain(X))))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 64 }
347.18/44.07	  coantidomain(codomain(multiplication(addition(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(domain(coantidomain(coantidomain(X))), codomain(coantidomain(X))), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 35 R->L }
347.18/44.07	  coantidomain(codomain(multiplication(addition(domain(coantidomain(coantidomain(X))), codomain(coantidomain(X))), antidomain(coantidomain(coantidomain(X))))))
347.18/44.07	= { by axiom 28 (left_distributivity) R->L }
347.18/44.07	  coantidomain(codomain(addition(multiplication(domain(coantidomain(coantidomain(X))), antidomain(coantidomain(coantidomain(X)))), multiplication(codomain(coantidomain(X)), antidomain(coantidomain(coantidomain(X)))))))
347.18/44.07	= { by lemma 116 }
347.18/44.07	  coantidomain(codomain(addition(zero, multiplication(codomain(coantidomain(X)), antidomain(coantidomain(coantidomain(X)))))))
347.18/44.07	= { by lemma 45 }
347.18/44.07	  coantidomain(codomain(multiplication(codomain(coantidomain(X)), antidomain(coantidomain(coantidomain(X))))))
347.18/44.07	= { by lemma 35 }
347.18/44.07	  coantidomain(codomain(multiplication(codomain(coantidomain(X)), c(coantidomain(coantidomain(X))))))
347.18/44.07	= { by axiom 15 (backward_diamond) }
347.18/44.07	  coantidomain(backward_diamond(c(coantidomain(coantidomain(X))), coantidomain(X)))
347.18/44.07	= { by lemma 103 R->L }
347.18/44.07	  coantidomain(multiplication(codomain(coantidomain(X)), c(coantidomain(coantidomain(X)))))
347.18/44.07	= { by lemma 64 }
347.18/44.07	  coantidomain(multiplication(coantidomain(X), c(coantidomain(coantidomain(X)))))
347.18/44.07	= { by lemma 85 }
347.18/44.07	  coantidomain(coantidomain(X))
347.18/44.07	= { by axiom 1 (codomain4) }
347.18/44.07	  codomain(X)
347.18/44.07	
347.18/44.07	Lemma 132: c(backward_diamond(X, backward_box(Y, Z))) = backward_box(X, backward_diamond(Y, c(Z))).
347.18/44.07	Proof:
347.18/44.07	  c(backward_diamond(X, backward_box(Y, Z)))
347.18/44.07	= { by axiom 16 (backward_box) R->L }
347.18/44.07	  c(backward_diamond(X, c(backward_diamond(Y, c(Z)))))
347.18/44.07	= { by axiom 16 (backward_box) }
347.18/44.07	  backward_box(X, backward_diamond(Y, c(Z)))
347.18/44.07	
347.18/44.07	Lemma 133: multiplication(domain(X), backward_box(Y, Z)) = domain_difference(X, backward_diamond(Y, c(Z))).
347.18/44.07	Proof:
347.18/44.07	  multiplication(domain(X), backward_box(Y, Z))
347.18/44.07	= { by axiom 16 (backward_box) R->L }
347.18/44.07	  multiplication(domain(X), c(backward_diamond(Y, c(Z))))
347.18/44.07	= { by lemma 110 }
347.18/44.07	  domain_difference(X, domain(backward_diamond(Y, c(Z))))
347.18/44.07	= { by lemma 112 }
347.18/44.07	  domain_difference(X, backward_diamond(Y, c(Z)))
347.18/44.07	
347.18/44.07	Lemma 134: multiplication(X, addition(Y, codomain(X))) = addition(X, multiplication(X, Y)).
347.18/44.07	Proof:
347.18/44.07	  multiplication(X, addition(Y, codomain(X)))
347.18/44.07	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.07	  multiplication(X, addition(codomain(X), Y))
347.18/44.07	= { by axiom 27 (right_distributivity) R->L }
347.18/44.07	  addition(multiplication(X, codomain(X)), multiplication(X, Y))
347.18/44.07	= { by lemma 63 }
347.18/44.07	  addition(X, multiplication(X, Y))
347.18/44.07	
347.18/44.07	Lemma 135: multiplication(X, domain(codomain(X))) = X.
347.18/44.07	Proof:
347.18/44.07	  multiplication(X, domain(codomain(X)))
347.18/44.07	= { by lemma 130 R->L }
347.18/44.07	  multiplication(X, addition(codomain(X), domain(codomain(X))))
347.18/44.07	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.07	  multiplication(X, addition(domain(codomain(X)), codomain(X)))
347.18/44.07	= { by lemma 134 }
347.18/44.07	  addition(X, multiplication(X, domain(codomain(X))))
347.18/44.07	= { by lemma 128 R->L }
347.18/44.07	  multiplication(X, addition(one, domain(codomain(X))))
347.18/44.07	= { by lemma 72 }
347.18/44.07	  multiplication(X, one)
347.18/44.07	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.07	  X
347.18/44.07	
347.18/44.07	Lemma 136: c(forward_diamond(X, Y)) = forward_box(X, c(Y)).
347.18/44.07	Proof:
347.18/44.07	  c(forward_diamond(X, Y))
347.18/44.07	= { by lemma 35 R->L }
347.18/44.07	  antidomain(forward_diamond(X, Y))
347.18/44.07	= { by lemma 75 R->L }
347.18/44.07	  c(multiplication(X, domain(Y)))
347.18/44.07	= { by lemma 54 R->L }
347.18/44.07	  c(multiplication(X, forward_diamond(one, Y)))
347.18/44.07	= { by lemma 49 R->L }
347.18/44.07	  c(multiplication(X, c(antidomain(Y))))
347.18/44.07	= { by lemma 76 }
347.18/44.07	  forward_box(X, antidomain(Y))
347.18/44.07	= { by lemma 35 }
347.18/44.07	  forward_box(X, c(Y))
347.18/44.07	
347.18/44.07	Lemma 137: multiplication(X, multiplication(coantidomain(X), Y)) = zero.
347.18/44.07	Proof:
347.18/44.07	  multiplication(X, multiplication(coantidomain(X), Y))
347.18/44.07	= { by axiom 21 (multiplicative_associativity) }
347.18/44.07	  multiplication(multiplication(X, coantidomain(X)), Y)
347.18/44.07	= { by axiom 11 (codomain1) R->L }
347.18/44.07	  multiplication(zero, Y)
347.18/44.07	= { by axiom 10 (left_annihilation) R->L }
347.18/44.07	  zero
347.18/44.07	
347.18/44.07	Lemma 138: forward_box(addition(X, one), codomain(X)) = codomain(X).
347.18/44.07	Proof:
347.18/44.07	  forward_box(addition(X, one), codomain(X))
347.18/44.07	= { by lemma 57 R->L }
347.18/44.07	  forward_box(addition(X, one), backward_diamond(X, one))
347.18/44.07	= { by lemma 44 R->L }
347.18/44.07	  forward_box(addition(X, one), backward_diamond(X, c(zero)))
347.18/44.07	= { by lemma 76 R->L }
347.18/44.07	  c(multiplication(addition(X, one), c(backward_diamond(X, c(zero)))))
347.18/44.07	= { by axiom 16 (backward_box) }
347.18/44.07	  c(multiplication(addition(X, one), backward_box(X, zero)))
347.18/44.07	= { by lemma 86 }
347.18/44.07	  c(multiplication(addition(X, one), coantidomain(X)))
347.18/44.07	= { by lemma 135 R->L }
347.18/44.07	  c(multiplication(addition(X, one), multiplication(coantidomain(X), domain(codomain(coantidomain(X))))))
347.18/44.07	= { by axiom 21 (multiplicative_associativity) }
347.18/44.07	  c(multiplication(multiplication(addition(X, one), coantidomain(X)), domain(codomain(coantidomain(X)))))
347.18/44.07	= { by lemma 75 }
347.18/44.07	  antidomain(forward_diamond(multiplication(addition(X, one), coantidomain(X)), codomain(coantidomain(X))))
347.18/44.07	= { by lemma 35 }
347.18/44.07	  c(forward_diamond(multiplication(addition(X, one), coantidomain(X)), codomain(coantidomain(X))))
347.18/44.07	= { by lemma 136 }
347.18/44.07	  forward_box(multiplication(addition(X, one), coantidomain(X)), c(codomain(coantidomain(X))))
347.18/44.07	= { by lemma 58 }
347.18/44.07	  forward_box(multiplication(addition(X, one), coantidomain(X)), backward_box(coantidomain(X), zero))
347.18/44.07	= { by lemma 86 }
347.18/44.07	  forward_box(multiplication(addition(X, one), coantidomain(X)), coantidomain(coantidomain(X)))
347.18/44.08	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.08	  forward_box(multiplication(addition(X, one), multiplication(coantidomain(X), one)), coantidomain(coantidomain(X)))
347.18/44.08	= { by lemma 69 }
347.18/44.08	  forward_box(addition(multiplication(coantidomain(X), one), multiplication(X, multiplication(coantidomain(X), one))), coantidomain(coantidomain(X)))
347.18/44.08	= { by lemma 137 }
347.18/44.08	  forward_box(addition(multiplication(coantidomain(X), one), zero), coantidomain(coantidomain(X)))
347.18/44.08	= { by axiom 6 (additive_identity) }
347.18/44.08	  forward_box(multiplication(coantidomain(X), one), coantidomain(coantidomain(X)))
347.18/44.08	= { by axiom 7 (multiplicative_right_identity) }
347.18/44.08	  forward_box(coantidomain(X), coantidomain(coantidomain(X)))
347.18/44.08	= { by lemma 76 R->L }
347.18/44.08	  c(multiplication(coantidomain(X), c(coantidomain(coantidomain(X)))))
347.18/44.08	= { by lemma 85 }
347.18/44.08	  c(coantidomain(X))
347.18/44.08	= { by lemma 88 }
347.18/44.08	  codomain(X)
347.18/44.08	
347.18/44.08	Lemma 139: domain_difference(c(X), coantidomain(Y)) = domain_difference(codomain(Y), X).
347.18/44.08	Proof:
347.18/44.08	  domain_difference(c(X), coantidomain(Y))
347.18/44.08	= { by lemma 112 R->L }
347.18/44.08	  domain_difference(c(X), domain(coantidomain(Y)))
347.18/44.08	= { by lemma 56 R->L }
347.18/44.08	  domain_difference(c(X), c(c(coantidomain(Y))))
347.18/44.08	= { by lemma 118 R->L }
347.18/44.08	  multiplication(domain(c(X)), domain_difference(c(coantidomain(Y)), c(c(X))))
347.18/44.08	= { by lemma 98 R->L }
347.18/44.08	  multiplication(domain(c(X)), multiplication(c(coantidomain(Y)), c(c(c(X)))))
347.18/44.08	= { by lemma 123 }
347.18/44.08	  multiplication(domain_difference(c(X), coantidomain(Y)), c(c(c(X))))
347.18/44.08	= { by lemma 56 }
347.18/44.08	  multiplication(domain_difference(c(X), coantidomain(Y)), domain(c(X)))
347.18/44.08	= { by lemma 125 R->L }
347.18/44.08	  multiplication(addition(domain_difference(c(X), coantidomain(Y)), c(c(X))), domain(c(X)))
347.18/44.08	= { by axiom 5 (additive_commutativity) }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), coantidomain(Y))), domain(c(X)))
347.18/44.08	= { by lemma 86 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), backward_box(Y, zero))), domain(c(X)))
347.18/44.08	= { by axiom 16 (backward_box) R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), c(backward_diamond(Y, c(zero))))), domain(c(X)))
347.18/44.08	= { by axiom 15 (backward_diamond) R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), c(codomain(multiplication(codomain(c(zero)), Y))))), domain(c(X)))
347.18/44.08	= { by lemma 131 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), c(backward_box(one, codomain(multiplication(codomain(c(zero)), Y)))))), domain(c(X)))
347.18/44.08	= { by axiom 15 (backward_diamond) }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), c(backward_box(one, backward_diamond(Y, c(zero)))))), domain(c(X)))
347.18/44.08	= { by lemma 35 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), antidomain(backward_box(one, backward_diamond(Y, c(zero)))))), domain(c(X)))
347.18/44.08	= { by lemma 132 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), antidomain(c(backward_diamond(one, backward_box(Y, zero)))))), domain(c(X)))
347.18/44.08	= { by lemma 46 }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), domain(domain(backward_diamond(one, backward_box(Y, zero)))))), domain(c(X)))
347.18/44.08	= { by lemma 48 }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), forward_diamond(one, backward_diamond(one, backward_box(Y, zero))))), domain(c(X)))
347.18/44.08	= { by lemma 54 }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), domain(backward_diamond(one, backward_box(Y, zero))))), domain(c(X)))
347.18/44.08	= { by axiom 15 (backward_diamond) R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), domain(codomain(multiplication(codomain(backward_box(Y, zero)), one))))), domain(c(X)))
347.18/44.08	= { by lemma 89 }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), codomain(multiplication(codomain(backward_box(Y, zero)), one)))), domain(c(X)))
347.18/44.08	= { by axiom 15 (backward_diamond) }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), backward_diamond(one, backward_box(Y, zero)))), domain(c(X)))
347.18/44.08	= { by lemma 65 }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), codomain(backward_box(Y, zero)))), domain(c(X)))
347.18/44.08	= { by lemma 58 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), codomain(c(codomain(Y))))), domain(c(X)))
347.18/44.08	= { by lemma 65 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), domain_difference(c(X), backward_diamond(one, c(codomain(Y))))), domain(c(X)))
347.18/44.08	= { by lemma 133 R->L }
347.18/44.08	  multiplication(addition(c(c(X)), multiplication(domain(c(X)), backward_box(one, codomain(Y)))), domain(c(X)))
347.18/44.08	= { by lemma 131 }
347.18/44.08	  multiplication(addition(c(c(X)), multiplication(domain(c(X)), codomain(Y))), domain(c(X)))
347.18/44.08	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.08	  multiplication(addition(multiplication(domain(c(X)), codomain(Y)), c(c(X))), domain(c(X)))
347.18/44.08	= { by lemma 129 R->L }
347.18/44.08	  multiplication(addition(multiplication(domain(c(X)), codomain(Y)), addition(c(c(X)), multiplication(c(c(X)), codomain(Y)))), domain(c(X)))
347.18/44.08	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.08	  multiplication(addition(multiplication(domain(c(X)), codomain(Y)), addition(multiplication(c(c(X)), codomain(Y)), c(c(X)))), domain(c(X)))
347.18/44.08	= { by lemma 94 }
347.18/44.08	  multiplication(addition(multiplication(c(c(X)), codomain(Y)), addition(multiplication(domain(c(X)), codomain(Y)), c(c(X)))), domain(c(X)))
347.18/44.08	= { by axiom 19 (additive_associativity) R->L }
347.18/44.08	  multiplication(addition(addition(multiplication(c(c(X)), codomain(Y)), multiplication(domain(c(X)), codomain(Y))), c(c(X))), domain(c(X)))
347.18/44.08	= { by axiom 28 (left_distributivity) }
347.18/44.08	  multiplication(addition(multiplication(addition(c(c(X)), domain(c(X))), codomain(Y)), c(c(X))), domain(c(X)))
347.18/44.08	= { by axiom 5 (additive_commutativity) }
347.18/44.08	  multiplication(addition(c(c(X)), multiplication(addition(c(c(X)), domain(c(X))), codomain(Y))), domain(c(X)))
347.18/44.08	= { by lemma 50 }
347.18/44.08	  multiplication(addition(c(c(X)), multiplication(one, codomain(Y))), domain(c(X)))
347.18/44.08	= { by axiom 9 (multiplicative_left_identity) }
347.18/44.08	  multiplication(addition(c(c(X)), codomain(Y)), domain(c(X)))
347.18/44.08	= { by axiom 5 (additive_commutativity) }
347.18/44.08	  multiplication(addition(codomain(Y), c(c(X))), domain(c(X)))
347.18/44.08	= { by lemma 125 }
347.18/44.08	  multiplication(codomain(Y), domain(c(X)))
347.18/44.08	= { by lemma 138 R->L }
347.18/44.08	  multiplication(forward_box(addition(Y, one), codomain(Y)), domain(c(X)))
347.18/44.08	= { by lemma 76 R->L }
347.18/44.08	  multiplication(c(multiplication(addition(Y, one), c(codomain(Y)))), domain(c(X)))
347.18/44.08	= { by lemma 52 }
347.18/44.08	  domain_difference(antidomain(multiplication(addition(Y, one), c(codomain(Y)))), antidomain(c(X)))
347.18/44.08	= { by lemma 35 }
347.18/44.08	  domain_difference(c(multiplication(addition(Y, one), c(codomain(Y)))), antidomain(c(X)))
347.18/44.08	= { by lemma 76 }
347.18/44.08	  domain_difference(forward_box(addition(Y, one), codomain(Y)), antidomain(c(X)))
347.18/44.08	= { by lemma 35 }
347.18/44.08	  domain_difference(forward_box(addition(Y, one), codomain(Y)), c(c(X)))
347.18/44.08	= { by lemma 138 }
347.18/44.08	  domain_difference(codomain(Y), c(c(X)))
347.18/44.08	= { by lemma 56 }
347.18/44.08	  domain_difference(codomain(Y), domain(X))
347.18/44.08	= { by lemma 112 }
347.18/44.08	  domain_difference(codomain(Y), X)
347.18/44.08	
347.18/44.08	Lemma 140: multiplication(codomain(domain(X)), X) = X.
347.18/44.08	Proof:
347.18/44.08	  multiplication(codomain(domain(X)), X)
347.18/44.08	= { by lemma 45 R->L }
347.18/44.08	  addition(zero, multiplication(codomain(domain(X)), X))
347.18/44.08	= { by axiom 12 (domain1) }
347.18/44.08	  addition(multiplication(antidomain(X), X), multiplication(codomain(domain(X)), X))
347.18/44.08	= { by axiom 28 (left_distributivity) }
347.18/44.08	  multiplication(addition(antidomain(X), codomain(domain(X))), X)
347.18/44.08	= { by lemma 35 }
347.18/44.08	  multiplication(addition(c(X), codomain(domain(X))), X)
347.18/44.08	= { by axiom 5 (additive_commutativity) R->L }
347.18/44.08	  multiplication(addition(codomain(domain(X)), c(X)), X)
347.18/44.08	= { by axiom 7 (multiplicative_right_identity) R->L }
347.18/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), one)), X)
347.18/44.08	= { by axiom 14 (order_1) R->L }
347.91/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), fresh2(true, true, coantidomain(domain(X)), one))), X)
347.91/44.08	= { by lemma 82 R->L }
347.91/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), fresh2(leq(coantidomain(domain(X)), addition(codomain(domain(X)), coantidomain(domain(X)))), true, coantidomain(domain(X)), one))), X)
347.91/44.08	= { by lemma 40 }
347.91/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), fresh2(leq(coantidomain(domain(X)), one), true, coantidomain(domain(X)), one))), X)
347.91/44.08	= { by axiom 25 (order_1) }
347.91/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), addition(coantidomain(domain(X)), one))), X)
347.91/44.08	= { by axiom 5 (additive_commutativity) }
347.91/44.08	  multiplication(addition(codomain(domain(X)), multiplication(c(X), addition(one, coantidomain(domain(X))))), X)
347.91/44.08	= { by lemma 128 }
347.91/44.08	  multiplication(addition(codomain(domain(X)), addition(c(X), multiplication(c(X), coantidomain(domain(X))))), X)
347.91/44.08	= { by lemma 35 R->L }
347.91/44.08	  multiplication(addition(codomain(domain(X)), addition(c(X), multiplication(antidomain(X), coantidomain(domain(X))))), X)
347.91/44.08	= { by lemma 61 R->L }
347.91/44.08	  multiplication(addition(codomain(domain(X)), addition(c(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X))))), X)
347.91/44.08	= { by lemma 31 }
347.91/44.08	  multiplication(addition(codomain(domain(X)), addition(c(X), multiplication(one, coantidomain(domain(X))))), X)
347.91/44.08	= { by axiom 9 (multiplicative_left_identity) }
347.91/44.08	  multiplication(addition(codomain(domain(X)), addition(c(X), coantidomain(domain(X)))), X)
347.91/44.08	= { by lemma 126 }
347.91/44.08	  multiplication(addition(c(X), one), X)
347.91/44.08	= { by axiom 5 (additive_commutativity) }
347.91/44.08	  multiplication(addition(one, c(X)), X)
347.91/44.08	= { by lemma 73 }
347.91/44.08	  multiplication(one, X)
347.91/44.08	= { by axiom 9 (multiplicative_left_identity) }
347.91/44.09	  X
347.91/44.09	
347.91/44.09	Lemma 141: multiplication(domain(X), multiplication(domain(Y), Z)) = multiplication(domain_difference(X, c(Y)), Z).
347.91/44.09	Proof:
347.91/44.09	  multiplication(domain(X), multiplication(domain(Y), Z))
347.91/44.09	= { by axiom 21 (multiplicative_associativity) }
347.91/44.09	  multiplication(multiplication(domain(X), domain(Y)), Z)
347.91/44.09	= { by lemma 51 }
347.91/44.09	  multiplication(domain_difference(X, antidomain(Y)), Z)
347.91/44.09	= { by lemma 35 }
347.91/44.09	  multiplication(domain_difference(X, c(Y)), Z)
347.91/44.09	
347.91/44.09	Lemma 142: multiplication(domain(X), multiplication(X, Y)) = multiplication(X, Y).
347.91/44.09	Proof:
347.91/44.09	  multiplication(domain(X), multiplication(X, Y))
347.91/44.09	= { by axiom 21 (multiplicative_associativity) }
347.91/44.09	  multiplication(multiplication(domain(X), X), Y)
347.91/44.09	= { by lemma 32 }
347.91/44.09	  multiplication(X, Y)
347.91/44.09	
347.91/44.09	Lemma 143: multiplication(domain_difference(X, Y), X) = multiplication(c(Y), X).
347.91/44.09	Proof:
347.91/44.09	  multiplication(domain_difference(X, Y), X)
347.91/44.09	= { by lemma 122 R->L }
347.91/44.09	  multiplication(domain_difference(domain(X), Y), X)
347.91/44.09	= { by lemma 106 R->L }
347.91/44.09	  multiplication(domain_difference(codomain(domain(X)), Y), X)
347.91/44.09	= { by lemma 139 R->L }
347.91/44.09	  multiplication(domain_difference(c(Y), coantidomain(domain(X))), X)
347.91/44.09	= { by lemma 86 R->L }
347.94/44.09	  multiplication(domain_difference(c(Y), backward_box(domain(X), zero)), X)
347.94/44.09	= { by lemma 58 R->L }
347.94/44.09	  multiplication(domain_difference(c(Y), c(codomain(domain(X)))), X)
347.94/44.09	= { by lemma 140 R->L }
347.94/44.09	  multiplication(domain_difference(c(Y), c(codomain(domain(X)))), multiplication(codomain(domain(X)), X))
347.94/44.09	= { by lemma 141 R->L }
347.94/44.09	  multiplication(domain(c(Y)), multiplication(domain(codomain(domain(X))), multiplication(codomain(domain(X)), X)))
347.94/44.09	= { by lemma 142 }
347.94/44.09	  multiplication(domain(c(Y)), multiplication(codomain(domain(X)), X))
347.94/44.09	= { by lemma 140 }
347.94/44.09	  multiplication(domain(c(Y)), X)
347.94/44.09	= { by lemma 90 }
347.94/44.09	  multiplication(c(Y), X)
347.94/44.09	
347.94/44.09	Lemma 144: multiplication(domain_difference(X, c(Y)), Y) = multiplication(domain(X), Y).
347.94/44.09	Proof:
347.94/44.09	  multiplication(domain_difference(X, c(Y)), Y)
347.94/44.09	= { by lemma 141 R->L }
347.94/44.09	  multiplication(domain(X), multiplication(domain(Y), Y))
347.94/44.09	= { by lemma 32 }
347.94/44.09	  multiplication(domain(X), Y)
347.94/44.09	
347.94/44.09	Lemma 145: domain_difference(c(X), c(multiplication(X, Y))) = zero.
347.94/44.09	Proof:
347.94/44.09	  domain_difference(c(X), c(multiplication(X, Y)))
347.94/44.09	= { by lemma 98 R->L }
347.94/44.09	  multiplication(c(X), c(c(multiplication(X, Y))))
347.94/44.09	= { by axiom 9 (multiplicative_left_identity) R->L }
347.94/44.09	  multiplication(one, multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 83 R->L }
347.94/44.09	  multiplication(addition(one, antidomain(multiplication(c(X), domain(multiplication(X, Y))))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 38 R->L }
347.94/44.09	  multiplication(addition(antidomain(zero), antidomain(multiplication(c(X), domain(multiplication(X, Y))))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 77 R->L }
347.94/44.09	  multiplication(addition(antidomain(multiplication(antidomain(X), multiplication(X, Y))), antidomain(multiplication(c(X), domain(multiplication(X, Y))))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 35 }
347.94/44.09	  multiplication(addition(antidomain(multiplication(c(X), multiplication(X, Y))), antidomain(multiplication(c(X), domain(multiplication(X, Y))))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 81 }
347.94/44.09	  multiplication(antidomain(multiplication(c(X), domain(multiplication(X, Y)))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 35 }
347.94/44.09	  multiplication(c(multiplication(c(X), domain(multiplication(X, Y)))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 80 }
347.94/44.09	  multiplication(forward_box(c(X), c(multiplication(X, Y))), multiplication(c(X), c(c(multiplication(X, Y)))))
347.94/44.09	= { by lemma 79 }
347.94/44.09	  zero
347.94/44.09	
347.94/44.09	Lemma 146: addition(c(X), c(multiplication(X, Y))) = c(multiplication(X, Y)).
347.94/44.09	Proof:
347.94/44.09	  addition(c(X), c(multiplication(X, Y)))
347.94/44.09	= { by axiom 5 (additive_commutativity) R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), c(X))
347.94/44.09	= { by lemma 90 R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), domain(c(X)))
347.94/44.09	= { by lemma 96 R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), domain_difference(c(X), c(multiplication(X, Y))))
347.94/44.09	= { by lemma 145 }
347.94/44.09	  addition(c(multiplication(X, Y)), zero)
347.94/44.09	= { by axiom 6 (additive_identity) }
347.94/44.09	  c(multiplication(X, Y))
347.94/44.09	
347.94/44.09	Lemma 147: c(domain_difference(X, Y)) = forward_box(domain(X), Y).
347.94/44.09	Proof:
347.94/44.09	  c(domain_difference(X, Y))
347.94/44.09	= { by lemma 112 R->L }
347.94/44.09	  c(domain_difference(X, domain(Y)))
347.94/44.09	= { by lemma 110 R->L }
347.94/44.09	  c(multiplication(domain(X), c(Y)))
347.94/44.09	= { by lemma 76 }
347.94/44.09	  forward_box(domain(X), Y)
347.94/44.09	
347.94/44.09	Lemma 148: addition(forward_box(X, c(Y)), c(multiplication(X, Y))) = forward_box(X, c(Y)).
347.94/44.09	Proof:
347.94/44.09	  addition(forward_box(X, c(Y)), c(multiplication(X, Y)))
347.94/44.09	= { by axiom 5 (additive_commutativity) R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), forward_box(X, c(Y)))
347.94/44.09	= { by lemma 80 R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), c(multiplication(X, domain(Y))))
347.94/44.09	= { by lemma 35 R->L }
347.94/44.09	  addition(c(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
347.94/44.09	= { by lemma 35 R->L }
347.94/44.09	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
347.94/44.09	= { by axiom 3 (domain4) }
347.94/44.09	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
347.94/44.09	= { by axiom 30 (domain2) R->L }
347.94/44.09	  antidomain(multiplication(X, antidomain(antidomain(Y))))
347.94/44.09	= { by axiom 3 (domain4) R->L }
347.94/44.09	  antidomain(multiplication(X, domain(Y)))
347.94/44.09	= { by lemma 35 }
347.94/44.09	  c(multiplication(X, domain(Y)))
347.94/44.09	= { by lemma 80 }
347.94/44.09	  forward_box(X, c(Y))
347.94/44.09	
347.94/44.09	Lemma 149: c(multiplication(domain(X), Y)) = forward_box(domain(X), c(Y)).
347.94/44.09	Proof:
347.94/44.09	  c(multiplication(domain(X), Y))
347.94/44.09	= { by lemma 144 R->L }
347.94/44.09	  c(multiplication(domain_difference(X, c(Y)), Y))
347.94/44.09	= { by lemma 146 R->L }
347.94/44.09	  addition(c(domain_difference(X, c(Y))), c(multiplication(domain_difference(X, c(Y)), Y)))
347.94/44.09	= { by lemma 144 }
347.94/44.09	  addition(c(domain_difference(X, c(Y))), c(multiplication(domain(X), Y)))
347.94/44.09	= { by lemma 147 }
347.94/44.09	  addition(forward_box(domain(X), c(Y)), c(multiplication(domain(X), Y)))
347.94/44.09	= { by lemma 148 }
347.94/44.09	  forward_box(domain(X), c(Y))
347.94/44.09	
347.94/44.09	Lemma 150: c(forward_box(X, Y)) = forward_diamond(X, c(Y)).
347.94/44.09	Proof:
347.94/44.09	  c(forward_box(X, Y))
347.94/44.09	= { by lemma 35 R->L }
347.94/44.09	  antidomain(forward_box(X, Y))
347.94/44.09	= { by lemma 76 R->L }
347.94/44.09	  antidomain(c(multiplication(X, c(Y))))
347.94/44.09	= { by lemma 46 }
347.94/44.09	  domain(domain(multiplication(X, c(Y))))
347.94/44.09	= { by lemma 48 }
347.94/44.09	  forward_diamond(one, multiplication(X, c(Y)))
347.94/44.09	= { by lemma 54 }
347.94/44.09	  domain(multiplication(X, c(Y)))
347.94/44.09	= { by lemma 109 }
347.94/44.09	  forward_diamond(X, antidomain(Y))
347.94/44.09	= { by lemma 35 }
347.94/44.09	  forward_diamond(X, c(Y))
347.94/44.09	
347.94/44.09	Lemma 151: domain(multiplication(domain(X), Y)) = forward_diamond(domain(X), Y).
347.94/44.09	Proof:
347.94/44.09	  domain(multiplication(domain(X), Y))
347.94/44.09	= { by lemma 56 R->L }
347.94/44.09	  c(c(multiplication(domain(X), Y)))
347.94/44.09	= { by lemma 149 }
347.94/44.09	  c(forward_box(domain(X), c(Y)))
347.94/44.09	= { by lemma 150 }
347.94/44.09	  forward_diamond(domain(X), c(c(Y)))
347.94/44.09	= { by lemma 113 }
347.94/44.09	  domain(domain_difference(X, c(Y)))
347.94/44.09	= { by lemma 115 }
347.94/44.09	  forward_diamond(domain(X), Y)
347.94/44.09	
347.94/44.09	Lemma 152: domain(multiplication(c(X), Y)) = forward_diamond(c(X), Y).
347.94/44.09	Proof:
347.94/44.09	  domain(multiplication(c(X), Y))
347.94/44.09	= { by lemma 90 R->L }
347.94/44.09	  domain(multiplication(domain(c(X)), Y))
347.94/44.09	= { by lemma 151 }
347.94/44.09	  forward_diamond(domain(c(X)), Y)
347.94/44.09	= { by lemma 90 }
347.94/44.09	  forward_diamond(c(X), Y)
347.94/44.09	
347.94/44.09	Lemma 153: domain(domain_difference(X, Y)) = forward_diamond(c(Y), X).
347.94/44.09	Proof:
347.94/44.09	  domain(domain_difference(X, Y))
347.94/44.09	= { by lemma 113 R->L }
347.94/44.09	  forward_diamond(domain(X), c(Y))
347.94/44.09	= { by lemma 115 R->L }
347.94/44.09	  domain(domain_difference(X, c(c(Y))))
347.94/44.09	= { by lemma 118 R->L }
347.94/44.09	  domain(multiplication(domain(X), domain_difference(c(Y), c(X))))
347.94/44.09	= { by lemma 120 }
347.94/44.09	  forward_diamond(multiplication(domain(X), c(Y)), c(c(X)))
347.94/44.09	= { by lemma 111 }
347.94/44.09	  forward_diamond(domain_difference(X, Y), c(c(X)))
347.94/44.09	= { by lemma 56 }
347.94/44.09	  forward_diamond(domain_difference(X, Y), domain(X))
347.94/44.09	= { by lemma 121 }
347.94/44.09	  forward_diamond(domain_difference(X, Y), X)
347.94/44.09	= { by lemma 122 R->L }
347.94/44.09	  forward_diamond(domain_difference(domain(X), Y), X)
347.94/44.09	= { by axiom 18 (forward_diamond) }
347.94/44.09	  domain(multiplication(domain_difference(domain(X), Y), domain(X)))
347.94/44.09	= { by lemma 143 }
347.94/44.09	  domain(multiplication(c(Y), domain(X)))
347.94/44.09	= { by lemma 152 }
347.94/44.09	  forward_diamond(c(Y), domain(X))
347.94/44.09	= { by lemma 121 }
347.94/44.09	  forward_diamond(c(Y), X)
347.94/44.09	
347.94/44.09	Lemma 154: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
347.94/44.09	Proof:
347.94/44.09	  domain(forward_diamond(X, Y))
347.94/44.09	= { by lemma 54 R->L }
347.94/44.09	  forward_diamond(one, forward_diamond(X, Y))
347.94/44.09	= { by lemma 49 R->L }
347.94/44.09	  c(antidomain(forward_diamond(X, Y)))
347.94/44.09	= { by lemma 75 R->L }
347.94/44.09	  c(c(multiplication(X, domain(Y))))
347.94/44.09	= { by lemma 56 }
347.94/44.09	  domain(multiplication(X, domain(Y)))
347.94/44.09	= { by axiom 18 (forward_diamond) R->L }
347.94/44.09	  forward_diamond(X, Y)
347.94/44.09	
347.94/44.09	Lemma 155: addition(domain(X), domain_difference(Y, X)) = addition(domain(X), domain(Y)).
347.94/44.09	Proof:
347.94/44.09	  addition(domain(X), domain_difference(Y, X))
347.94/44.09	= { by lemma 112 R->L }
347.94/44.09	  addition(domain(X), domain_difference(Y, domain(X)))
347.94/44.09	= { by lemma 56 R->L }
347.94/44.09	  addition(domain(X), domain_difference(Y, c(c(X))))
347.94/44.09	= { by lemma 56 R->L }
347.94/44.09	  addition(c(c(X)), domain_difference(Y, c(c(X))))
347.94/44.09	= { by lemma 96 }
347.94/44.09	  addition(c(c(X)), domain(Y))
347.94/44.09	= { by lemma 56 }
347.94/44.09	  addition(domain(X), domain(Y))
347.94/44.09	
347.94/44.09	Lemma 156: forward_diamond(addition(X, codomain(c(Y))), Y) = forward_diamond(X, Y).
347.94/44.09	Proof:
347.94/44.09	  forward_diamond(addition(X, codomain(c(Y))), Y)
347.94/44.09	= { by lemma 100 R->L }
347.94/44.09	  forward_diamond(addition(X, codomain(c(domain(Y)))), Y)
347.94/44.09	= { by axiom 18 (forward_diamond) }
347.94/44.09	  domain(multiplication(addition(X, codomain(c(domain(Y)))), domain(Y)))
347.94/44.10	= { by axiom 5 (additive_commutativity) R->L }
347.94/44.10	  domain(multiplication(addition(codomain(c(domain(Y))), X), domain(Y)))
347.94/44.10	= { by axiom 28 (left_distributivity) R->L }
347.94/44.10	  domain(addition(multiplication(codomain(c(domain(Y))), domain(Y)), multiplication(X, domain(Y))))
347.94/44.10	= { by lemma 32 R->L }
347.94/44.10	  domain(addition(multiplication(codomain(c(domain(Y))), multiplication(domain(domain(Y)), domain(Y))), multiplication(X, domain(Y))))
347.94/44.10	= { by axiom 21 (multiplicative_associativity) }
347.94/44.10	  domain(addition(multiplication(multiplication(codomain(c(domain(Y))), domain(domain(Y))), domain(Y)), multiplication(X, domain(Y))))
347.94/44.10	= { by axiom 28 (left_distributivity) }
347.94/44.10	  domain(multiplication(addition(multiplication(codomain(c(domain(Y))), domain(domain(Y))), X), domain(Y)))
347.94/44.10	= { by axiom 5 (additive_commutativity) }
347.94/44.10	  domain(multiplication(addition(X, multiplication(codomain(c(domain(Y))), domain(domain(Y)))), domain(Y)))
347.94/44.10	= { by axiom 18 (forward_diamond) R->L }
347.94/44.10	  forward_diamond(addition(X, multiplication(codomain(c(domain(Y))), domain(domain(Y)))), Y)
347.94/44.10	= { by lemma 55 }
347.94/44.10	  forward_diamond(addition(X, multiplication(codomain(c(domain(Y))), domain(Y))), Y)
347.94/44.10	= { by lemma 105 }
347.94/44.10	  forward_diamond(addition(X, zero), Y)
347.94/44.10	= { by axiom 6 (additive_identity) }
347.94/44.10	  forward_diamond(X, Y)
347.94/44.10	
347.94/44.10	Lemma 157: codomain(forward_diamond(X, Y)) = forward_diamond(X, Y).
347.94/44.10	Proof:
347.94/44.10	  codomain(forward_diamond(X, Y))
347.94/44.10	= { by axiom 18 (forward_diamond) }
347.94/44.10	  codomain(domain(multiplication(X, domain(Y))))
347.94/44.10	= { by lemma 92 R->L }
347.94/44.10	  addition(domain(multiplication(X, domain(Y))), codomain(domain(multiplication(X, domain(Y)))))
347.94/44.10	= { by axiom 18 (forward_diamond) R->L }
347.94/44.10	  addition(forward_diamond(X, Y), codomain(domain(multiplication(X, domain(Y)))))
347.94/44.10	= { by axiom 18 (forward_diamond) R->L }
347.94/44.10	  addition(forward_diamond(X, Y), codomain(forward_diamond(X, Y)))
347.94/44.10	= { by lemma 89 R->L }
347.94/44.10	  addition(forward_diamond(X, Y), domain(codomain(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 154 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), domain(codomain(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 155 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), domain_difference(codomain(forward_diamond(X, Y)), forward_diamond(X, Y)))
347.94/44.10	= { by lemma 131 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), domain_difference(backward_box(one, codomain(forward_diamond(X, Y))), forward_diamond(X, Y)))
347.94/44.10	= { by axiom 16 (backward_box) R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), domain_difference(c(backward_diamond(one, c(codomain(forward_diamond(X, Y))))), forward_diamond(X, Y)))
347.94/44.10	= { by lemma 98 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(c(backward_diamond(one, c(codomain(forward_diamond(X, Y))))), c(forward_diamond(X, Y))))
347.94/44.10	= { by axiom 16 (backward_box) }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(backward_box(one, codomain(forward_diamond(X, Y))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 131 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 156 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(addition(X, codomain(c(Y))), Y)), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 135 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(multiplication(forward_diamond(addition(X, codomain(c(Y))), Y), domain(codomain(forward_diamond(addition(X, codomain(c(Y))), Y))))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 154 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(multiplication(domain(forward_diamond(addition(X, codomain(c(Y))), Y)), domain(codomain(forward_diamond(addition(X, codomain(c(Y))), Y))))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 93 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(addition(X, codomain(c(Y))), Y), c(codomain(forward_diamond(addition(X, codomain(c(Y))), Y))))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 58 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(addition(X, codomain(c(Y))), Y), backward_box(forward_diamond(addition(X, codomain(c(Y))), Y), zero))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 156 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), backward_box(forward_diamond(addition(X, codomain(c(Y))), Y), zero))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 86 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(addition(X, codomain(c(Y))), Y)))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 156 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), c(forward_diamond(X, Y))))
347.94/44.10	= { by lemma 102 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))))))
347.94/44.10	= { by axiom 7 (multiplicative_right_identity) R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), one))))
347.94/44.10	= { by lemma 42 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(zero)))))
347.94/44.10	= { by lemma 116 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), antidomain(forward_diamond(X, Y))))))))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), c(forward_diamond(X, Y))))))))
347.94/44.10	= { by axiom 9 (multiplicative_left_identity) R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), multiplication(one, c(forward_diamond(X, Y)))))))))
347.94/44.10	= { by lemma 73 R->L }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), multiplication(addition(one, c(coantidomain(forward_diamond(X, Y)))), c(forward_diamond(X, Y)))))))))
347.94/44.10	= { by lemma 70 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), addition(c(forward_diamond(X, Y)), multiplication(c(coantidomain(forward_diamond(X, Y))), c(forward_diamond(X, Y))))))))))
347.94/44.10	= { by lemma 117 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain(forward_diamond(X, Y)), multiplication(c(coantidomain(forward_diamond(X, Y))), c(forward_diamond(X, Y)))))))))
347.94/44.10	= { by lemma 123 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), multiplication(backward_diamond(c(forward_diamond(X, Y)), domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), coantidomain(multiplication(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y))), c(forward_diamond(X, Y))))))))
347.94/44.10	= { by lemma 104 }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), multiplication(c(forward_diamond(X, Y)), zero)))
347.94/44.10	= { by axiom 8 (right_annihilation) }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), multiplication(codomain(domain_difference(forward_diamond(X, Y), coantidomain(forward_diamond(X, Y)))), zero))
347.94/44.10	= { by axiom 8 (right_annihilation) }
347.94/44.10	  addition(domain(forward_diamond(X, Y)), zero)
347.94/44.10	= { by axiom 6 (additive_identity) }
347.94/44.10	  domain(forward_diamond(X, Y))
347.94/44.10	= { by lemma 154 }
347.94/44.10	  forward_diamond(X, Y)
347.94/44.10	
347.94/44.10	Lemma 158: codomain(backward_diamond(X, Y)) = backward_diamond(X, Y).
347.94/44.10	Proof:
347.94/44.10	  codomain(backward_diamond(X, Y))
347.94/44.10	= { by axiom 7 (multiplicative_right_identity) R->L }
347.94/44.10	  codomain(multiplication(backward_diamond(X, Y), one))
347.94/44.10	= { by axiom 15 (backward_diamond) R->L }
347.94/44.10	  codomain(multiplication(codomain(multiplication(codomain(Y), X)), one))
347.94/44.10	= { by axiom 15 (backward_diamond) }
347.94/44.10	  backward_diamond(one, multiplication(codomain(Y), X))
347.94/44.10	= { by lemma 65 }
347.94/44.10	  codomain(multiplication(codomain(Y), X))
347.94/44.10	= { by axiom 15 (backward_diamond) }
347.94/44.10	  backward_diamond(X, Y)
347.94/44.10	
347.94/44.10	Lemma 159: c(addition(X, one)) = zero.
347.94/44.10	Proof:
347.94/44.10	  c(addition(X, one))
347.94/44.10	= { by axiom 5 (additive_commutativity) R->L }
347.94/44.10	  c(addition(one, X))
347.94/44.10	= { by axiom 7 (multiplicative_right_identity) R->L }
347.94/44.10	  multiplication(c(addition(one, X)), one)
347.94/44.10	= { by lemma 35 R->L }
347.94/44.10	  multiplication(antidomain(addition(one, X)), one)
347.94/44.10	= { by lemma 34 R->L }
347.94/44.10	  multiplication(antidomain(addition(one, X)), addition(addition(one, X), one))
347.94/44.10	= { by axiom 5 (additive_commutativity) }
347.94/44.10	  multiplication(antidomain(addition(one, X)), addition(one, addition(one, X)))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  multiplication(c(addition(one, X)), addition(one, addition(one, X)))
347.94/44.10	= { by lemma 71 }
347.94/44.10	  multiplication(c(addition(one, X)), addition(one, X))
347.94/44.10	= { by lemma 78 }
347.94/44.10	  zero
347.94/44.10	
347.94/44.10	Lemma 160: c(backward_diamond(X, Y)) = backward_box(X, coantidomain(Y)).
347.94/44.10	Proof:
347.94/44.10	  c(backward_diamond(X, Y))
347.94/44.10	= { by axiom 15 (backward_diamond) R->L }
347.94/44.10	  c(codomain(multiplication(codomain(Y), X)))
347.94/44.10	= { by axiom 1 (codomain4) R->L }
347.94/44.10	  c(codomain(multiplication(coantidomain(coantidomain(Y)), X)))
347.94/44.10	= { by lemma 64 R->L }
347.94/44.10	  c(codomain(multiplication(codomain(coantidomain(coantidomain(Y))), X)))
347.94/44.10	= { by axiom 15 (backward_diamond) }
347.94/44.10	  c(backward_diamond(X, coantidomain(coantidomain(Y))))
347.94/44.10	= { by axiom 1 (codomain4) }
347.94/44.10	  c(backward_diamond(X, codomain(Y)))
347.94/44.10	= { by lemma 131 R->L }
347.94/44.10	  c(backward_diamond(X, backward_box(one, codomain(Y))))
347.94/44.10	= { by lemma 132 }
347.94/44.10	  backward_box(X, backward_diamond(one, c(codomain(Y))))
347.94/44.10	= { by lemma 65 }
347.94/44.10	  backward_box(X, codomain(c(codomain(Y))))
347.94/44.10	= { by lemma 58 }
347.94/44.10	  backward_box(X, codomain(backward_box(Y, zero)))
347.94/44.10	= { by lemma 86 }
347.94/44.10	  backward_box(X, codomain(coantidomain(Y)))
347.94/44.10	= { by lemma 64 }
347.94/44.10	  backward_box(X, coantidomain(Y))
347.94/44.10	
347.94/44.10	Lemma 161: multiplication(c(X), domain(Y)) = domain_difference(c(X), c(Y)).
347.94/44.10	Proof:
347.94/44.10	  multiplication(c(X), domain(Y))
347.94/44.10	= { by axiom 3 (domain4) }
347.94/44.10	  multiplication(c(X), antidomain(antidomain(Y)))
347.94/44.10	= { by lemma 97 }
347.94/44.10	  domain_difference(antidomain(X), antidomain(Y))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  domain_difference(c(X), antidomain(Y))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  domain_difference(c(X), c(Y))
347.94/44.10	
347.94/44.10	Lemma 162: domain_difference(Y, c(X)) = domain_difference(X, c(Y)).
347.94/44.10	Proof:
347.94/44.10	  domain_difference(Y, c(X))
347.94/44.10	= { by lemma 118 R->L }
347.94/44.10	  multiplication(domain(Y), domain_difference(X, c(Y)))
347.94/44.10	= { by lemma 55 R->L }
347.94/44.10	  multiplication(domain(domain(Y)), domain_difference(X, c(Y)))
347.94/44.10	= { by lemma 93 R->L }
347.94/44.10	  multiplication(domain(domain(Y)), multiplication(domain(X), domain(Y)))
347.94/44.10	= { by lemma 141 }
347.94/44.10	  multiplication(domain_difference(domain(Y), c(X)), domain(Y))
347.94/44.10	= { by lemma 143 }
347.94/44.10	  multiplication(c(c(X)), domain(Y))
347.94/44.10	= { by lemma 161 }
347.94/44.10	  domain_difference(c(c(X)), c(Y))
347.94/44.10	= { by lemma 56 }
347.94/44.10	  domain_difference(domain(X), c(Y))
347.94/44.10	= { by lemma 122 }
347.94/44.10	  domain_difference(X, c(Y))
347.94/44.10	
347.94/44.10	Lemma 163: domain_difference(c(X), X) = c(X).
347.94/44.10	Proof:
347.94/44.10	  domain_difference(c(X), X)
347.94/44.10	= { by lemma 35 R->L }
347.94/44.10	  domain_difference(antidomain(X), X)
347.94/44.10	= { by lemma 53 }
347.94/44.10	  antidomain(X)
347.94/44.10	= { by lemma 35 }
347.94/44.10	  c(X)
347.94/44.10	
347.94/44.10	Lemma 164: forward_diamond(domain(X), c(Y)) = forward_diamond(c(Y), X).
347.94/44.10	Proof:
347.94/44.10	  forward_diamond(domain(X), c(Y))
347.94/44.10	= { by lemma 35 R->L }
347.94/44.10	  forward_diamond(domain(X), antidomain(Y))
347.94/44.10	= { by lemma 109 R->L }
347.94/44.10	  domain(multiplication(domain(X), c(Y)))
347.94/44.10	= { by lemma 110 }
347.94/44.10	  domain(domain_difference(X, domain(Y)))
347.94/44.10	= { by lemma 112 }
347.94/44.10	  domain(domain_difference(X, Y))
347.94/44.10	= { by lemma 153 }
347.94/44.10	  forward_diamond(c(Y), X)
347.94/44.10	
347.94/44.10	Lemma 165: multiplication(c(X), domain(Y)) = domain_difference(Y, X).
347.94/44.10	Proof:
347.94/44.10	  multiplication(c(X), domain(Y))
347.94/44.10	= { by axiom 3 (domain4) }
347.94/44.10	  multiplication(c(X), antidomain(antidomain(Y)))
347.94/44.10	= { by lemma 97 }
347.94/44.10	  domain_difference(antidomain(X), antidomain(Y))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  domain_difference(c(X), antidomain(Y))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  domain_difference(c(X), c(Y))
347.94/44.10	= { by lemma 162 R->L }
347.94/44.10	  domain_difference(Y, c(c(X)))
347.94/44.10	= { by lemma 56 }
347.94/44.10	  domain_difference(Y, domain(X))
347.94/44.10	= { by lemma 112 }
347.94/44.10	  domain_difference(Y, X)
347.94/44.10	
347.94/44.10	Lemma 166: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z).
347.94/44.10	Proof:
347.94/44.10	  domain_difference(multiplication(X, domain(Y)), Z)
347.94/44.10	= { by lemma 122 R->L }
347.94/44.10	  domain_difference(domain(multiplication(X, domain(Y))), Z)
347.94/44.10	= { by axiom 18 (forward_diamond) R->L }
347.94/44.10	  domain_difference(forward_diamond(X, Y), Z)
347.94/44.10	
347.94/44.10	Lemma 167: multiplication(c(X), forward_diamond(Y, Z)) = domain_difference(forward_diamond(Y, Z), X).
347.94/44.10	Proof:
347.94/44.10	  multiplication(c(X), forward_diamond(Y, Z))
347.94/44.10	= { by axiom 18 (forward_diamond) }
347.94/44.10	  multiplication(c(X), domain(multiplication(Y, domain(Z))))
347.94/44.10	= { by lemma 165 }
347.94/44.10	  domain_difference(multiplication(Y, domain(Z)), X)
347.94/44.10	= { by lemma 166 }
347.94/44.10	  domain_difference(forward_diamond(Y, Z), X)
347.94/44.10	
347.94/44.10	Lemma 168: domain_difference(forward_diamond(c(X), Y), Z) = domain_difference(domain_difference(Y, X), Z).
347.94/44.10	Proof:
347.94/44.10	  domain_difference(forward_diamond(c(X), Y), Z)
347.94/44.10	= { by lemma 167 R->L }
347.94/44.10	  multiplication(c(Z), forward_diamond(c(X), Y))
347.94/44.10	= { by lemma 164 R->L }
347.94/44.10	  multiplication(c(Z), forward_diamond(domain(Y), c(X)))
347.94/44.10	= { by lemma 167 }
347.94/44.10	  domain_difference(forward_diamond(domain(Y), c(X)), Z)
347.94/44.10	= { by lemma 115 R->L }
347.94/44.10	  domain_difference(domain(domain_difference(Y, c(c(X)))), Z)
347.94/44.10	= { by lemma 122 }
347.94/44.10	  domain_difference(domain_difference(Y, c(c(X))), Z)
347.94/44.10	= { by lemma 56 }
347.94/44.10	  domain_difference(domain_difference(Y, domain(X)), Z)
347.94/44.10	= { by lemma 112 }
347.94/44.10	  domain_difference(domain_difference(Y, X), Z)
347.94/44.10	
347.94/44.10	Lemma 169: forward_box(c(X), X) = domain(X).
347.94/44.10	Proof:
347.94/44.10	  forward_box(c(X), X)
347.94/44.10	= { by lemma 90 R->L }
347.94/44.10	  forward_box(domain(c(X)), X)
347.94/44.10	= { by lemma 76 R->L }
347.94/44.10	  c(multiplication(domain(c(X)), c(X)))
347.94/44.10	= { by lemma 32 }
347.94/44.10	  c(c(X))
347.94/44.10	= { by lemma 56 }
347.94/44.10	  domain(X)
347.94/44.10	
347.94/44.10	Lemma 170: forward_box(forward_box(X, c(Y)), forward_diamond(X, Y)) = forward_diamond(X, Y).
347.94/44.10	Proof:
347.94/44.10	  forward_box(forward_box(X, c(Y)), forward_diamond(X, Y))
347.94/44.10	= { by lemma 136 R->L }
347.94/44.10	  forward_box(c(forward_diamond(X, Y)), forward_diamond(X, Y))
347.94/44.10	= { by lemma 169 }
347.94/44.10	  domain(forward_diamond(X, Y))
347.94/44.10	= { by lemma 154 }
347.94/44.10	  forward_diamond(X, Y)
347.94/44.10	
347.94/44.10	Lemma 171: domain_difference(forward_diamond(c(X), c(Y)), Z) = domain_difference(domain_difference(c(X), Y), Z).
347.94/44.10	Proof:
347.94/44.10	  domain_difference(forward_diamond(c(X), c(Y)), Z)
347.94/44.10	= { by lemma 35 R->L }
347.94/44.10	  domain_difference(forward_diamond(c(X), antidomain(Y)), Z)
347.94/44.10	= { by lemma 109 R->L }
347.94/44.10	  domain_difference(domain(multiplication(c(X), c(Y))), Z)
347.94/44.10	= { by lemma 98 }
347.94/44.10	  domain_difference(domain(domain_difference(c(X), Y)), Z)
347.94/44.10	= { by lemma 122 }
347.94/44.10	  domain_difference(domain_difference(c(X), Y), Z)
347.94/44.10	
347.94/44.10	Lemma 172: multiplication(c(X), coantidomain(Y)) = domain_difference(c(X), codomain(Y)).
347.94/44.10	Proof:
347.94/44.10	  multiplication(c(X), coantidomain(Y))
347.94/44.10	= { by lemma 100 R->L }
347.94/44.10	  multiplication(c(domain(X)), coantidomain(Y))
347.94/44.10	= { by lemma 169 R->L }
347.94/44.10	  multiplication(c(forward_box(c(X), X)), coantidomain(Y))
347.94/44.10	= { by lemma 150 }
347.94/44.10	  multiplication(forward_diamond(c(X), c(X)), coantidomain(Y))
347.94/44.10	= { by lemma 86 R->L }
347.94/44.10	  multiplication(forward_diamond(c(X), c(X)), backward_box(Y, zero))
347.94/44.10	= { by lemma 170 R->L }
347.94/44.10	  multiplication(forward_box(forward_box(c(X), c(c(X))), forward_diamond(c(X), c(X))), backward_box(Y, zero))
347.94/44.10	= { by lemma 58 R->L }
347.94/44.10	  multiplication(forward_box(forward_box(c(X), c(c(X))), forward_diamond(c(X), c(X))), c(codomain(Y)))
347.94/44.10	= { by lemma 35 R->L }
347.94/44.10	  multiplication(forward_box(forward_box(c(X), c(c(X))), forward_diamond(c(X), c(X))), antidomain(codomain(Y)))
347.94/44.10	= { by lemma 76 R->L }
347.94/44.10	  multiplication(c(multiplication(forward_box(c(X), c(c(X))), c(forward_diamond(c(X), c(X))))), antidomain(codomain(Y)))
347.94/44.10	= { by lemma 97 }
347.94/44.10	  domain_difference(antidomain(multiplication(forward_box(c(X), c(c(X))), c(forward_diamond(c(X), c(X))))), codomain(Y))
347.94/44.10	= { by lemma 35 }
347.94/44.10	  domain_difference(c(multiplication(forward_box(c(X), c(c(X))), c(forward_diamond(c(X), c(X))))), codomain(Y))
347.94/44.10	= { by lemma 76 }
347.94/44.10	  domain_difference(forward_box(forward_box(c(X), c(c(X))), forward_diamond(c(X), c(X))), codomain(Y))
347.94/44.10	= { by lemma 170 }
347.94/44.10	  domain_difference(forward_diamond(c(X), c(X)), codomain(Y))
347.94/44.10	= { by lemma 171 }
347.94/44.10	  domain_difference(domain_difference(c(X), X), codomain(Y))
347.94/44.10	= { by lemma 163 }
347.94/44.10	  domain_difference(c(X), codomain(Y))
347.94/44.10	
347.94/44.10	Lemma 173: multiplication(c(X), codomain(Y)) = domain_difference(codomain(Y), X).
347.94/44.10	Proof:
347.94/44.10	  multiplication(c(X), codomain(Y))
347.94/44.10	= { by axiom 1 (codomain4) R->L }
347.94/44.10	  multiplication(c(X), coantidomain(coantidomain(Y)))
347.94/44.10	= { by lemma 172 }
347.94/44.10	  domain_difference(c(X), codomain(coantidomain(Y)))
347.94/44.10	= { by lemma 64 }
347.94/44.10	  domain_difference(c(X), coantidomain(Y))
347.94/44.10	= { by lemma 139 }
347.94/44.10	  domain_difference(codomain(Y), X)
347.94/44.10	
347.94/44.10	Lemma 174: addition(domain(X), domain(multiplication(X, Y))) = domain(X).
347.94/44.10	Proof:
347.94/44.10	  addition(domain(X), domain(multiplication(X, Y)))
347.94/44.10	= { by lemma 56 R->L }
347.94/44.10	  addition(c(c(X)), domain(multiplication(X, Y)))
347.94/44.10	= { by axiom 7 (multiplicative_right_identity) R->L }
347.94/44.10	  addition(c(c(X)), multiplication(domain(multiplication(X, Y)), one))
347.94/44.10	= { by lemma 50 R->L }
347.94/44.10	  addition(c(c(X)), multiplication(domain(multiplication(X, Y)), addition(c(c(X)), domain(c(X)))))
347.94/44.11	= { by lemma 95 }
347.94/44.11	  addition(c(c(X)), multiplication(domain(multiplication(X, Y)), domain(c(X))))
347.94/44.11	= { by lemma 93 }
347.94/44.11	  addition(c(c(X)), domain_difference(multiplication(X, Y), c(c(X))))
347.94/44.11	= { by lemma 162 R->L }
347.94/44.11	  addition(c(c(X)), domain_difference(c(X), c(multiplication(X, Y))))
347.94/44.11	= { by lemma 145 }
347.94/44.11	  addition(c(c(X)), zero)
347.94/44.11	= { by axiom 6 (additive_identity) }
347.94/44.11	  c(c(X))
347.94/44.11	= { by lemma 56 }
347.94/44.11	  domain(X)
347.94/44.11	
347.94/44.11	Lemma 175: addition(domain(X), domain_difference(X, Y)) = domain(X).
347.94/44.11	Proof:
347.94/44.11	  addition(domain(X), domain_difference(X, Y))
347.94/44.11	= { by axiom 20 (domain_difference) }
347.94/44.11	  addition(domain(X), multiplication(domain(X), antidomain(Y)))
347.94/44.11	= { by lemma 128 R->L }
347.94/44.11	  multiplication(domain(X), addition(one, antidomain(Y)))
347.94/44.11	= { by lemma 83 }
347.94/44.11	  multiplication(domain(X), one)
347.94/44.11	= { by axiom 7 (multiplicative_right_identity) }
347.94/44.11	  domain(X)
347.94/44.11	
347.94/44.11	Lemma 176: domain_difference(domain_difference(X, Y), Y) = forward_diamond(c(Y), X).
347.94/44.11	Proof:
347.94/44.11	  domain_difference(domain_difference(X, Y), Y)
347.94/44.11	= { by lemma 168 R->L }
347.94/44.11	  domain_difference(forward_diamond(c(Y), X), Y)
347.94/44.11	= { by lemma 157 R->L }
347.94/44.11	  domain_difference(codomain(forward_diamond(c(Y), X)), Y)
347.94/44.11	= { by lemma 173 R->L }
347.94/44.11	  multiplication(c(Y), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by lemma 90 R->L }
347.94/44.11	  multiplication(domain(c(Y)), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by lemma 174 R->L }
347.94/44.11	  multiplication(addition(domain(c(Y)), domain(multiplication(c(Y), X))), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by lemma 90 }
347.94/44.11	  multiplication(addition(c(Y), domain(multiplication(c(Y), X))), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by lemma 152 }
347.94/44.11	  multiplication(addition(c(Y), forward_diamond(c(Y), X)), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by axiom 5 (additive_commutativity) R->L }
347.94/44.11	  multiplication(addition(forward_diamond(c(Y), X), c(Y)), codomain(forward_diamond(c(Y), X)))
347.94/44.11	= { by axiom 28 (left_distributivity) R->L }
347.94/44.11	  addition(multiplication(forward_diamond(c(Y), X), codomain(forward_diamond(c(Y), X))), multiplication(c(Y), codomain(forward_diamond(c(Y), X))))
347.94/44.11	= { by lemma 63 }
347.94/44.11	  addition(forward_diamond(c(Y), X), multiplication(c(Y), codomain(forward_diamond(c(Y), X))))
347.94/44.11	= { by lemma 173 }
347.94/44.11	  addition(forward_diamond(c(Y), X), domain_difference(codomain(forward_diamond(c(Y), X)), Y))
347.94/44.11	= { by lemma 157 }
347.94/44.11	  addition(forward_diamond(c(Y), X), domain_difference(forward_diamond(c(Y), X), Y))
347.94/44.11	= { by lemma 154 R->L }
347.94/44.11	  addition(domain(forward_diamond(c(Y), X)), domain_difference(forward_diamond(c(Y), X), Y))
347.94/44.11	= { by lemma 175 }
347.94/44.11	  domain(forward_diamond(c(Y), X))
347.94/44.11	= { by lemma 154 }
347.94/44.11	  forward_diamond(c(Y), X)
347.94/44.11	
347.94/44.11	Lemma 177: multiplication(c(X), addition(domain(X), Y)) = multiplication(c(X), Y).
347.94/44.11	Proof:
347.94/44.11	  multiplication(c(X), addition(domain(X), Y))
347.94/44.11	= { by axiom 27 (right_distributivity) R->L }
347.94/44.11	  addition(multiplication(c(X), domain(X)), multiplication(c(X), Y))
347.94/44.11	= { by lemma 124 }
347.94/44.11	  addition(zero, multiplication(c(X), Y))
347.94/44.11	= { by lemma 45 }
347.94/44.11	  multiplication(c(X), Y)
347.94/44.11	
347.94/44.11	Lemma 178: multiplication(domain(X), forward_diamond(Y, Z)) = domain_difference(X, forward_box(Y, c(Z))).
347.94/44.11	Proof:
347.94/44.11	  multiplication(domain(X), forward_diamond(Y, Z))
347.94/44.11	= { by axiom 18 (forward_diamond) }
347.94/44.11	  multiplication(domain(X), domain(multiplication(Y, domain(Z))))
347.94/44.11	= { by lemma 51 }
347.94/44.11	  domain_difference(X, antidomain(multiplication(Y, domain(Z))))
347.94/44.11	= { by lemma 35 }
347.94/44.11	  domain_difference(X, c(multiplication(Y, domain(Z))))
347.94/44.11	= { by lemma 80 }
347.94/44.11	  domain_difference(X, forward_box(Y, c(Z)))
347.94/44.11	
347.94/44.11	Lemma 179: multiplication(domain_difference(X, forward_box(Y, c(Z))), W) = multiplication(domain(X), multiplication(forward_diamond(Y, Z), W)).
347.94/44.11	Proof:
347.94/44.11	  multiplication(domain_difference(X, forward_box(Y, c(Z))), W)
347.94/44.11	= { by lemma 178 R->L }
347.94/44.11	  multiplication(multiplication(domain(X), forward_diamond(Y, Z)), W)
347.94/44.11	= { by axiom 21 (multiplicative_associativity) R->L }
347.94/44.11	  multiplication(domain(X), multiplication(forward_diamond(Y, Z), W))
347.94/44.11	
347.94/44.11	Lemma 180: multiplication(domain(X), multiplication(forward_diamond(domain(X), Y), Y)) = multiplication(domain(X), Y).
347.94/44.11	Proof:
347.94/44.11	  multiplication(domain(X), multiplication(forward_diamond(domain(X), Y), Y))
347.94/44.11	= { by lemma 179 R->L }
347.94/44.11	  multiplication(domain_difference(X, forward_box(domain(X), c(Y))), Y)
347.94/44.11	= { by lemma 149 R->L }
347.94/44.11	  multiplication(domain_difference(X, c(multiplication(domain(X), Y))), Y)
347.94/44.11	= { by lemma 162 }
347.94/44.11	  multiplication(domain_difference(multiplication(domain(X), Y), c(X)), Y)
347.94/44.11	= { by lemma 141 R->L }
347.94/44.11	  multiplication(domain(multiplication(domain(X), Y)), multiplication(domain(X), Y))
347.94/44.11	= { by lemma 32 }
347.94/44.11	  multiplication(domain(X), Y)
347.94/44.11	
347.94/44.11	Lemma 181: forward_diamond(c(X), Y) = domain_difference(Y, X).
347.94/44.11	Proof:
347.94/44.11	  forward_diamond(c(X), Y)
347.94/44.11	= { by lemma 164 R->L }
347.94/44.11	  forward_diamond(domain(Y), c(X))
347.94/44.11	= { by lemma 56 R->L }
347.94/44.11	  forward_diamond(c(c(Y)), c(X))
347.94/44.11	= { by lemma 176 R->L }
347.94/44.11	  domain_difference(domain_difference(c(X), c(Y)), c(Y))
347.94/44.11	= { by lemma 112 R->L }
347.94/44.11	  domain_difference(domain_difference(c(X), domain(c(Y))), c(Y))
347.94/44.11	= { by lemma 56 R->L }
347.94/44.11	  domain_difference(domain_difference(c(X), c(c(c(Y)))), c(Y))
347.94/44.11	= { by lemma 161 R->L }
347.94/44.11	  domain_difference(multiplication(c(X), domain(c(c(Y)))), c(Y))
347.94/44.11	= { by lemma 177 R->L }
347.94/44.11	  domain_difference(multiplication(c(X), addition(domain(X), domain(c(c(Y))))), c(Y))
347.94/44.11	= { by lemma 155 R->L }
347.94/44.11	  domain_difference(multiplication(c(X), addition(domain(X), domain_difference(c(c(Y)), X))), c(Y))
347.94/44.11	= { by lemma 177 }
347.94/44.11	  domain_difference(multiplication(c(X), domain_difference(c(c(Y)), X)), c(Y))
347.94/44.11	= { by lemma 98 R->L }
347.94/44.11	  domain_difference(multiplication(c(X), multiplication(c(c(Y)), c(X))), c(Y))
347.94/44.11	= { by lemma 33 R->L }
347.94/44.11	  domain_difference(multiplication(domain(antidomain(X)), multiplication(c(c(Y)), c(X))), c(Y))
347.94/44.11	= { by lemma 123 }
347.94/44.11	  domain_difference(multiplication(domain_difference(antidomain(X), c(Y)), c(X)), c(Y))
347.94/44.11	= { by lemma 35 }
347.94/44.11	  domain_difference(multiplication(domain_difference(c(X), c(Y)), c(X)), c(Y))
347.94/44.11	= { by lemma 143 }
347.94/44.11	  domain_difference(multiplication(c(c(Y)), c(X)), c(Y))
347.94/44.11	= { by lemma 98 }
347.94/44.11	  domain_difference(domain_difference(c(c(Y)), X), c(Y))
347.94/44.11	= { by lemma 171 R->L }
347.94/44.11	  domain_difference(forward_diamond(c(c(Y)), c(X)), c(Y))
347.94/44.11	= { by lemma 167 R->L }
347.94/44.11	  multiplication(c(c(Y)), forward_diamond(c(c(Y)), c(X)))
347.94/44.11	= { by lemma 153 R->L }
347.94/44.11	  multiplication(c(c(Y)), domain(domain_difference(c(X), c(Y))))
347.94/44.11	= { by lemma 98 R->L }
347.94/44.11	  multiplication(c(c(Y)), domain(multiplication(c(X), c(c(Y)))))
347.94/44.11	= { by lemma 109 }
347.94/44.11	  multiplication(c(c(Y)), forward_diamond(c(X), antidomain(c(Y))))
347.94/44.11	= { by lemma 35 }
347.94/44.11	  multiplication(c(c(Y)), forward_diamond(c(X), c(c(Y))))
347.94/44.11	= { by lemma 176 R->L }
347.94/44.11	  multiplication(c(c(Y)), domain_difference(domain_difference(c(c(Y)), X), X))
347.94/44.11	= { by lemma 171 R->L }
347.94/44.11	  multiplication(c(c(Y)), domain_difference(forward_diamond(c(c(Y)), c(X)), X))
347.94/44.11	= { by lemma 111 R->L }
347.94/44.11	  multiplication(c(c(Y)), multiplication(domain(forward_diamond(c(c(Y)), c(X))), c(X)))
347.94/44.11	= { by lemma 154 }
347.94/44.11	  multiplication(c(c(Y)), multiplication(forward_diamond(c(c(Y)), c(X)), c(X)))
347.94/44.11	= { by lemma 90 R->L }
347.94/44.11	  multiplication(c(c(Y)), multiplication(forward_diamond(domain(c(c(Y))), c(X)), c(X)))
347.94/44.11	= { by lemma 90 R->L }
347.94/44.11	  multiplication(domain(c(c(Y))), multiplication(forward_diamond(domain(c(c(Y))), c(X)), c(X)))
347.94/44.11	= { by lemma 180 }
347.94/44.11	  multiplication(domain(c(c(Y))), c(X))
347.94/44.11	= { by lemma 90 }
347.94/44.11	  multiplication(c(c(Y)), c(X))
347.94/44.11	= { by lemma 98 }
347.94/44.11	  domain_difference(c(c(Y)), X)
347.94/44.11	= { by lemma 56 }
347.94/44.11	  domain_difference(domain(Y), X)
347.94/44.11	= { by lemma 122 }
347.94/44.11	  domain_difference(Y, X)
347.94/44.11	
347.94/44.11	Lemma 182: forward_diamond(domain(X), Y) = domain_difference(X, c(Y)).
347.94/44.11	Proof:
347.94/44.11	  forward_diamond(domain(X), Y)
347.94/44.11	= { by lemma 56 R->L }
347.94/44.11	  forward_diamond(c(c(X)), Y)
347.94/44.11	= { by lemma 181 }
347.94/44.11	  domain_difference(Y, c(X))
347.94/44.11	= { by lemma 162 R->L }
347.94/44.11	  domain_difference(X, c(Y))
347.94/44.11	
347.94/44.11	Lemma 183: backward_box(X, domain(Y)) = backward_box(X, Y).
347.94/44.11	Proof:
347.94/44.11	  backward_box(X, domain(Y))
347.94/44.11	= { by axiom 16 (backward_box) R->L }
347.94/44.11	  c(backward_diamond(X, c(domain(Y))))
347.94/44.11	= { by lemma 100 }
347.94/44.11	  c(backward_diamond(X, c(Y)))
347.94/44.11	= { by axiom 16 (backward_box) }
347.94/44.11	  backward_box(X, Y)
347.94/44.11	
347.94/44.11	Lemma 184: multiplication(domain_difference(X, Y), domain(Z)) = multiplication(domain(X), domain_difference(Z, Y)).
347.94/44.11	Proof:
347.94/44.11	  multiplication(domain_difference(X, Y), domain(Z))
347.94/44.11	= { by lemma 123 R->L }
347.94/44.11	  multiplication(domain(X), multiplication(c(Y), domain(Z)))
347.94/44.11	= { by lemma 165 }
347.94/44.12	  multiplication(domain(X), domain_difference(Z, Y))
347.94/44.12	
347.94/44.12	Lemma 185: forward_diamond(domain(X), domain_difference(Y, Z)) = forward_diamond(domain_difference(X, Z), Y).
347.94/44.12	Proof:
347.94/44.12	  forward_diamond(domain(X), domain_difference(Y, Z))
347.94/44.12	= { by lemma 151 R->L }
347.94/44.12	  domain(multiplication(domain(X), domain_difference(Y, Z)))
347.94/44.12	= { by lemma 184 R->L }
347.94/44.12	  domain(multiplication(domain_difference(X, Z), domain(Y)))
347.94/44.12	= { by axiom 18 (forward_diamond) R->L }
347.94/44.12	  forward_diamond(domain_difference(X, Z), Y)
348.11/44.13	
348.11/44.13	Lemma 186: forward_diamond(domain_difference(X, Y), c(Z)) = forward_diamond(c(Y), domain_difference(X, Z)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(domain_difference(X, Y), c(Z))
348.11/44.13	= { by lemma 165 R->L }
348.11/44.13	  forward_diamond(multiplication(c(Y), domain(X)), c(Z))
348.11/44.13	= { by lemma 119 R->L }
348.11/44.13	  domain(multiplication(c(Y), multiplication(domain(X), c(Z))))
348.11/44.13	= { by lemma 111 }
348.11/44.13	  domain(multiplication(c(Y), domain_difference(X, Z)))
348.11/44.13	= { by lemma 152 }
348.11/44.13	  forward_diamond(c(Y), domain_difference(X, Z))
348.11/44.13	
348.11/44.13	Lemma 187: forward_diamond(domain_difference(Z, Y), X) = forward_diamond(domain_difference(X, Y), Z).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(domain_difference(Z, Y), X)
348.11/44.13	= { by lemma 185 R->L }
348.11/44.13	  forward_diamond(domain(Z), domain_difference(X, Y))
348.11/44.13	= { by lemma 151 R->L }
348.11/44.13	  domain(multiplication(domain(Z), domain_difference(X, Y)))
348.11/44.13	= { by lemma 111 R->L }
348.11/44.13	  domain(multiplication(domain(Z), multiplication(domain(X), c(Y))))
348.11/44.13	= { by lemma 141 }
348.11/44.13	  domain(multiplication(domain_difference(Z, c(X)), c(Y)))
348.11/44.13	= { by lemma 99 }
348.11/44.13	  forward_diamond(domain_difference(Z, c(X)), c(Y))
348.11/44.13	= { by lemma 186 }
348.11/44.13	  forward_diamond(c(c(X)), domain_difference(Z, Y))
348.11/44.13	= { by lemma 56 }
348.11/44.13	  forward_diamond(domain(X), domain_difference(Z, Y))
348.11/44.13	= { by lemma 185 }
348.11/44.13	  forward_diamond(domain_difference(X, Y), Z)
348.11/44.13	
348.11/44.13	Lemma 188: backward_box(backward_diamond(X, Y), zero) = c(backward_diamond(X, Y)).
348.11/44.13	Proof:
348.11/44.13	  backward_box(backward_diamond(X, Y), zero)
348.11/44.13	= { by lemma 58 R->L }
348.11/44.13	  c(codomain(backward_diamond(X, Y)))
348.11/44.13	= { by lemma 158 }
348.11/44.13	  c(backward_diamond(X, Y))
348.11/44.13	
348.11/44.13	Lemma 189: multiplication(domain(X), domain_difference(X, Y)) = domain_difference(X, Y).
348.11/44.13	Proof:
348.11/44.13	  multiplication(domain(X), domain_difference(X, Y))
348.11/44.13	= { by lemma 54 R->L }
348.11/44.13	  multiplication(forward_diamond(one, X), domain_difference(X, Y))
348.11/44.13	= { by lemma 48 R->L }
348.11/44.13	  multiplication(domain(domain(X)), domain_difference(X, Y))
348.11/44.13	= { by axiom 20 (domain_difference) }
348.11/44.13	  multiplication(domain(domain(X)), multiplication(domain(X), antidomain(Y)))
348.11/44.13	= { by lemma 142 }
348.11/44.13	  multiplication(domain(X), antidomain(Y))
348.11/44.13	= { by axiom 20 (domain_difference) R->L }
348.11/44.13	  domain_difference(X, Y)
348.11/44.13	
348.11/44.13	Lemma 190: multiplication(backward_box(X, Y), domain(Z)) = domain_difference(Z, backward_diamond(X, c(Y))).
348.11/44.13	Proof:
348.11/44.13	  multiplication(backward_box(X, Y), domain(Z))
348.11/44.13	= { by axiom 16 (backward_box) R->L }
348.11/44.13	  multiplication(c(backward_diamond(X, c(Y))), domain(Z))
348.11/44.13	= { by lemma 161 }
348.11/44.13	  domain_difference(c(backward_diamond(X, c(Y))), c(Z))
348.11/44.13	= { by axiom 16 (backward_box) }
348.11/44.13	  domain_difference(backward_box(X, Y), c(Z))
348.11/44.13	= { by lemma 162 R->L }
348.11/44.13	  domain_difference(Z, c(backward_box(X, Y)))
348.11/44.13	= { by lemma 35 R->L }
348.11/44.13	  domain_difference(Z, antidomain(backward_box(X, Y)))
348.11/44.13	= { by axiom 16 (backward_box) R->L }
348.11/44.13	  domain_difference(Z, antidomain(c(backward_diamond(X, c(Y)))))
348.11/44.13	= { by lemma 46 }
348.11/44.13	  domain_difference(Z, domain(domain(backward_diamond(X, c(Y)))))
348.11/44.13	= { by lemma 48 }
348.11/44.13	  domain_difference(Z, forward_diamond(one, backward_diamond(X, c(Y))))
348.11/44.13	= { by lemma 54 }
348.11/44.13	  domain_difference(Z, domain(backward_diamond(X, c(Y))))
348.11/44.13	= { by lemma 112 }
348.11/44.13	  domain_difference(Z, backward_diamond(X, c(Y)))
348.11/44.13	
348.11/44.13	Lemma 191: domain(multiplication(X, forward_diamond(Y, Z))) = forward_diamond(X, multiplication(Y, domain(Z))).
348.11/44.13	Proof:
348.11/44.13	  domain(multiplication(X, forward_diamond(Y, Z)))
348.11/44.13	= { by axiom 18 (forward_diamond) }
348.11/44.13	  domain(multiplication(X, domain(multiplication(Y, domain(Z)))))
348.11/44.13	= { by axiom 18 (forward_diamond) R->L }
348.11/44.13	  forward_diamond(X, multiplication(Y, domain(Z)))
348.11/44.13	
348.11/44.13	Lemma 192: forward_diamond(X, multiplication(Y, domain(Z))) = forward_diamond(X, forward_diamond(Y, Z)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(X, multiplication(Y, domain(Z)))
348.11/44.13	= { by lemma 191 R->L }
348.11/44.13	  domain(multiplication(X, forward_diamond(Y, Z)))
348.11/44.13	= { by lemma 54 R->L }
348.11/44.13	  forward_diamond(one, multiplication(X, forward_diamond(Y, Z)))
348.11/44.13	= { by lemma 48 R->L }
348.11/44.13	  domain(domain(multiplication(X, forward_diamond(Y, Z))))
348.11/44.13	= { by lemma 46 R->L }
348.11/44.13	  antidomain(c(multiplication(X, forward_diamond(Y, Z))))
348.11/44.13	= { by axiom 2 (complement) }
348.11/44.13	  antidomain(antidomain(domain(multiplication(X, forward_diamond(Y, Z)))))
348.11/44.13	= { by lemma 191 }
348.11/44.13	  antidomain(antidomain(forward_diamond(X, multiplication(Y, domain(Z)))))
348.11/44.13	= { by lemma 35 }
348.11/44.13	  antidomain(c(forward_diamond(X, multiplication(Y, domain(Z)))))
348.11/44.13	= { by lemma 136 }
348.11/44.13	  antidomain(forward_box(X, c(multiplication(Y, domain(Z)))))
348.11/44.13	= { by lemma 80 }
348.11/44.13	  antidomain(forward_box(X, forward_box(Y, c(Z))))
348.11/44.13	= { by lemma 35 }
348.11/44.13	  c(forward_box(X, forward_box(Y, c(Z))))
348.11/44.13	= { by lemma 150 }
348.11/44.13	  forward_diamond(X, c(forward_box(Y, c(Z))))
348.11/44.13	= { by lemma 150 }
348.11/44.13	  forward_diamond(X, forward_diamond(Y, c(c(Z))))
348.11/44.13	= { by lemma 56 }
348.11/44.13	  forward_diamond(X, forward_diamond(Y, domain(Z)))
348.11/44.13	= { by lemma 121 }
348.11/44.13	  forward_diamond(X, forward_diamond(Y, Z))
348.11/44.13	
348.11/44.13	Lemma 193: domain(multiplication(X, forward_diamond(Y, Z))) = forward_diamond(X, forward_diamond(Y, Z)).
348.11/44.13	Proof:
348.11/44.13	  domain(multiplication(X, forward_diamond(Y, Z)))
348.11/44.13	= { by axiom 18 (forward_diamond) }
348.11/44.13	  domain(multiplication(X, domain(multiplication(Y, domain(Z)))))
348.11/44.13	= { by axiom 18 (forward_diamond) R->L }
348.11/44.13	  forward_diamond(X, multiplication(Y, domain(Z)))
348.11/44.13	= { by lemma 192 }
348.11/44.13	  forward_diamond(X, forward_diamond(Y, Z))
348.11/44.13	
348.11/44.13	Lemma 194: domain(multiplication(X, multiplication(Y, multiplication(Z, domain(W))))) = forward_diamond(multiplication(X, multiplication(Y, Z)), W).
348.11/44.13	Proof:
348.11/44.13	  domain(multiplication(X, multiplication(Y, multiplication(Z, domain(W)))))
348.11/44.13	= { by axiom 21 (multiplicative_associativity) }
348.11/44.13	  domain(multiplication(multiplication(X, Y), multiplication(Z, domain(W))))
348.11/44.13	= { by lemma 101 }
348.11/44.13	  forward_diamond(multiplication(multiplication(X, Y), Z), W)
348.11/44.13	= { by axiom 21 (multiplicative_associativity) R->L }
348.11/44.13	  forward_diamond(multiplication(X, multiplication(Y, Z)), W)
348.11/44.13	
348.11/44.13	Lemma 195: multiplication(domain_difference(X, c(Y)), multiplication(X, Z)) = multiplication(domain(Y), multiplication(X, Z)).
348.11/44.13	Proof:
348.11/44.13	  multiplication(domain_difference(X, c(Y)), multiplication(X, Z))
348.11/44.13	= { by lemma 162 }
348.11/44.13	  multiplication(domain_difference(Y, c(X)), multiplication(X, Z))
348.11/44.13	= { by lemma 141 R->L }
348.11/44.13	  multiplication(domain(Y), multiplication(domain(X), multiplication(X, Z)))
348.11/44.13	= { by lemma 142 }
348.11/44.13	  multiplication(domain(Y), multiplication(X, Z))
348.11/44.13	
348.11/44.13	Lemma 196: forward_diamond(domain(X), forward_diamond(Y, Z)) = forward_diamond(forward_diamond(Y, Z), X).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(domain(X), forward_diamond(Y, Z))
348.11/44.13	= { by lemma 193 R->L }
348.11/44.13	  domain(multiplication(domain(X), forward_diamond(Y, Z)))
348.11/44.13	= { by lemma 178 }
348.11/44.13	  domain(domain_difference(X, forward_box(Y, c(Z))))
348.11/44.13	= { by lemma 153 }
348.11/44.13	  forward_diamond(c(forward_box(Y, c(Z))), X)
348.11/44.13	= { by lemma 150 }
348.11/44.13	  forward_diamond(forward_diamond(Y, c(c(Z))), X)
348.11/44.13	= { by lemma 56 }
348.11/44.13	  forward_diamond(forward_diamond(Y, domain(Z)), X)
348.11/44.13	= { by lemma 121 }
348.11/44.13	  forward_diamond(forward_diamond(Y, Z), X)
348.11/44.13	
348.11/44.13	Lemma 197: forward_diamond(multiplication(domain(X), Y), Z) = forward_diamond(forward_diamond(Y, Z), X).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(multiplication(domain(X), Y), Z)
348.11/44.13	= { by lemma 56 R->L }
348.11/44.13	  forward_diamond(multiplication(c(c(X)), Y), Z)
348.11/44.13	= { by lemma 143 R->L }
348.11/44.13	  forward_diamond(multiplication(domain_difference(Y, c(X)), Y), Z)
348.11/44.13	= { by lemma 101 R->L }
348.11/44.13	  domain(multiplication(domain_difference(Y, c(X)), multiplication(Y, domain(Z))))
348.11/44.13	= { by lemma 195 }
348.11/44.13	  domain(multiplication(domain(X), multiplication(Y, domain(Z))))
348.11/44.13	= { by lemma 151 }
348.11/44.13	  forward_diamond(domain(X), multiplication(Y, domain(Z)))
348.11/44.13	= { by lemma 192 }
348.11/44.13	  forward_diamond(domain(X), forward_diamond(Y, Z))
348.11/44.13	= { by lemma 196 }
348.11/44.13	  forward_diamond(forward_diamond(Y, Z), X)
348.11/44.13	
348.11/44.13	Lemma 198: forward_diamond(forward_diamond(Y, Z), c(X)) = forward_diamond(c(X), forward_diamond(Y, Z)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(forward_diamond(Y, Z), c(X))
348.11/44.13	= { by lemma 99 R->L }
348.11/44.13	  domain(multiplication(forward_diamond(Y, Z), c(X)))
348.11/44.13	= { by lemma 35 R->L }
348.11/44.13	  domain(multiplication(forward_diamond(Y, Z), antidomain(X)))
348.11/44.13	= { by axiom 18 (forward_diamond) }
348.11/44.13	  domain(multiplication(domain(multiplication(Y, domain(Z))), antidomain(X)))
348.11/44.13	= { by axiom 20 (domain_difference) R->L }
348.11/44.13	  domain(domain_difference(multiplication(Y, domain(Z)), X))
348.11/44.13	= { by lemma 166 }
348.11/44.13	  domain(domain_difference(forward_diamond(Y, Z), X))
348.11/44.13	= { by lemma 153 }
348.11/44.13	  forward_diamond(c(X), forward_diamond(Y, Z))
348.11/44.13	
348.11/44.13	Lemma 199: forward_diamond(multiplication(c(X), Y), Z) = forward_diamond(c(X), forward_diamond(Y, Z)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(multiplication(c(X), Y), Z)
348.11/44.13	= { by lemma 90 R->L }
348.11/44.13	  forward_diamond(multiplication(domain(c(X)), Y), Z)
348.11/44.13	= { by lemma 197 }
348.11/44.13	  forward_diamond(forward_diamond(Y, Z), c(X))
348.11/44.13	= { by lemma 198 }
348.11/44.13	  forward_diamond(c(X), forward_diamond(Y, Z))
348.11/44.13	
348.11/44.13	Lemma 200: forward_diamond(X, forward_diamond(c(Y), Z)) = forward_diamond(X, domain_difference(Z, Y)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(X, forward_diamond(c(Y), Z))
348.11/44.13	= { by lemma 192 R->L }
348.11/44.13	  forward_diamond(X, multiplication(c(Y), domain(Z)))
348.11/44.13	= { by lemma 161 }
348.11/44.13	  forward_diamond(X, domain_difference(c(Y), c(Z)))
348.11/44.13	= { by lemma 162 R->L }
348.11/44.13	  forward_diamond(X, domain_difference(Z, c(c(Y))))
348.11/44.13	= { by lemma 56 }
348.11/44.13	  forward_diamond(X, domain_difference(Z, domain(Y)))
348.11/44.13	= { by lemma 112 }
348.11/44.13	  forward_diamond(X, domain_difference(Z, Y))
348.11/44.13	
348.11/44.13	Lemma 201: forward_diamond(forward_diamond(c(X), Y), Z) = forward_diamond(domain_difference(Y, X), Z).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(forward_diamond(c(X), Y), Z)
348.11/44.13	= { by lemma 196 R->L }
348.11/44.13	  forward_diamond(domain(Z), forward_diamond(c(X), Y))
348.11/44.13	= { by lemma 200 }
348.11/44.13	  forward_diamond(domain(Z), domain_difference(Y, X))
348.11/44.13	= { by lemma 185 }
348.11/44.13	  forward_diamond(domain_difference(Z, X), Y)
348.11/44.13	= { by lemma 187 R->L }
348.11/44.13	  forward_diamond(domain_difference(Y, X), Z)
348.11/44.13	
348.11/44.13	Lemma 202: forward_diamond(multiplication(domain_difference(X, Y), Z), W) = forward_diamond(domain_difference(X, Y), forward_diamond(Z, W)).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(multiplication(domain_difference(X, Y), Z), W)
348.11/44.13	= { by lemma 101 R->L }
348.11/44.13	  domain(multiplication(domain_difference(X, Y), multiplication(Z, domain(W))))
348.11/44.13	= { by lemma 123 R->L }
348.11/44.13	  domain(multiplication(domain(X), multiplication(c(Y), multiplication(Z, domain(W)))))
348.11/44.13	= { by lemma 194 }
348.11/44.13	  forward_diamond(multiplication(domain(X), multiplication(c(Y), Z)), W)
348.11/44.13	= { by lemma 197 }
348.11/44.13	  forward_diamond(forward_diamond(multiplication(c(Y), Z), W), X)
348.11/44.13	= { by lemma 199 }
348.11/44.13	  forward_diamond(forward_diamond(c(Y), forward_diamond(Z, W)), X)
348.11/44.13	= { by lemma 201 }
348.11/44.13	  forward_diamond(domain_difference(forward_diamond(Z, W), Y), X)
348.11/44.13	= { by lemma 187 R->L }
348.11/44.13	  forward_diamond(domain_difference(X, Y), forward_diamond(Z, W))
348.11/44.13	
348.11/44.13	Lemma 203: domain(multiplication(domain_difference(X, Y), Z)) = forward_diamond(domain_difference(X, Y), Z).
348.11/44.13	Proof:
348.11/44.13	  domain(multiplication(domain_difference(X, Y), Z))
348.11/44.13	= { by lemma 107 R->L }
348.11/44.13	  forward_diamond(multiplication(domain_difference(X, Y), Z), one)
348.11/44.13	= { by lemma 202 }
348.11/44.13	  forward_diamond(domain_difference(X, Y), forward_diamond(Z, one))
348.11/44.13	= { by lemma 107 }
348.11/44.13	  forward_diamond(domain_difference(X, Y), domain(Z))
348.11/44.13	= { by lemma 121 }
348.11/44.13	  forward_diamond(domain_difference(X, Y), Z)
348.11/44.13	
348.11/44.13	Lemma 204: domain_difference(X, c(multiplication(X, Y))) = domain(multiplication(X, Y)).
348.11/44.13	Proof:
348.11/44.13	  domain_difference(X, c(multiplication(X, Y)))
348.11/44.13	= { by lemma 162 }
348.11/44.13	  domain_difference(multiplication(X, Y), c(X))
348.11/44.13	= { by lemma 122 R->L }
348.11/44.13	  domain_difference(domain(multiplication(X, Y)), c(X))
348.11/44.13	= { by lemma 93 R->L }
348.11/44.13	  multiplication(domain(domain(multiplication(X, Y))), domain(X))
348.11/44.13	= { by lemma 174 R->L }
348.11/44.13	  multiplication(domain(domain(multiplication(X, Y))), addition(domain(X), domain(multiplication(X, Y))))
348.11/44.13	= { by axiom 5 (additive_commutativity) R->L }
348.11/44.13	  multiplication(domain(domain(multiplication(X, Y))), addition(domain(multiplication(X, Y)), domain(X)))
348.11/44.13	= { by axiom 27 (right_distributivity) R->L }
348.11/44.13	  addition(multiplication(domain(domain(multiplication(X, Y))), domain(multiplication(X, Y))), multiplication(domain(domain(multiplication(X, Y))), domain(X)))
348.11/44.13	= { by lemma 32 }
348.11/44.13	  addition(domain(multiplication(X, Y)), multiplication(domain(domain(multiplication(X, Y))), domain(X)))
348.11/44.13	= { by lemma 93 }
348.11/44.13	  addition(domain(multiplication(X, Y)), domain_difference(domain(multiplication(X, Y)), c(X)))
348.11/44.13	= { by lemma 122 }
348.11/44.13	  addition(domain(multiplication(X, Y)), domain_difference(multiplication(X, Y), c(X)))
348.11/44.13	= { by lemma 175 }
348.11/44.13	  domain(multiplication(X, Y))
348.11/44.13	
348.11/44.13	Lemma 205: domain_difference(backward_box(X, Y), backward_diamond(X, c(Y))) = backward_box(X, Y).
348.11/44.13	Proof:
348.11/44.13	  domain_difference(backward_box(X, Y), backward_diamond(X, c(Y)))
348.11/44.13	= { by lemma 133 R->L }
348.11/44.13	  multiplication(domain(backward_box(X, Y)), backward_box(X, Y))
348.11/44.13	= { by lemma 32 }
348.11/44.13	  backward_box(X, Y)
348.11/44.13	
348.11/44.13	Lemma 206: domain_difference(X, backward_diamond(Y, c(Z))) = forward_diamond(backward_box(Y, Z), X).
348.11/44.13	Proof:
348.11/44.13	  domain_difference(X, backward_diamond(Y, c(Z)))
348.11/44.13	= { by lemma 181 R->L }
348.11/44.13	  forward_diamond(c(backward_diamond(Y, c(Z))), X)
348.11/44.13	= { by lemma 153 R->L }
348.11/44.13	  domain(domain_difference(X, backward_diamond(Y, c(Z))))
348.11/44.13	= { by lemma 190 R->L }
348.11/44.13	  domain(multiplication(backward_box(Y, Z), domain(X)))
348.11/44.13	= { by lemma 205 R->L }
348.11/44.13	  domain(multiplication(domain_difference(backward_box(Y, Z), backward_diamond(Y, c(Z))), domain(X)))
348.11/44.13	= { by lemma 203 }
348.11/44.13	  forward_diamond(domain_difference(backward_box(Y, Z), backward_diamond(Y, c(Z))), domain(X))
348.11/44.13	= { by lemma 205 }
348.11/44.13	  forward_diamond(backward_box(Y, Z), domain(X))
348.11/44.13	= { by lemma 121 }
348.11/44.13	  forward_diamond(backward_box(Y, Z), X)
348.11/44.13	
348.11/44.13	Lemma 207: forward_diamond(multiplication(X, domain(Y)), Y) = forward_diamond(X, Y).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(multiplication(X, domain(Y)), Y)
348.11/44.13	= { by lemma 54 R->L }
348.11/44.13	  forward_diamond(multiplication(X, forward_diamond(one, Y)), Y)
348.11/44.13	= { by lemma 48 R->L }
348.11/44.13	  forward_diamond(multiplication(X, domain(domain(Y))), Y)
348.11/44.13	= { by lemma 101 R->L }
348.11/44.13	  domain(multiplication(X, multiplication(domain(domain(Y)), domain(Y))))
348.11/44.13	= { by lemma 32 }
348.11/44.13	  domain(multiplication(X, domain(Y)))
348.11/44.13	= { by axiom 18 (forward_diamond) R->L }
348.11/44.13	  forward_diamond(X, Y)
348.11/44.13	
348.11/44.13	Lemma 208: multiplication(c(X), multiplication(domain(Y), Z)) = multiplication(domain_difference(Y, X), Z).
348.11/44.13	Proof:
348.11/44.13	  multiplication(c(X), multiplication(domain(Y), Z))
348.11/44.13	= { by axiom 21 (multiplicative_associativity) }
348.11/44.13	  multiplication(multiplication(c(X), domain(Y)), Z)
348.11/44.13	= { by lemma 161 }
348.11/44.13	  multiplication(domain_difference(c(X), c(Y)), Z)
348.11/44.13	= { by lemma 162 R->L }
348.11/44.13	  multiplication(domain_difference(Y, c(c(X))), Z)
348.11/44.13	= { by lemma 56 }
348.11/44.13	  multiplication(domain_difference(Y, domain(X)), Z)
348.11/44.13	= { by lemma 112 }
348.11/44.13	  multiplication(domain_difference(Y, X), Z)
348.11/44.13	
348.11/44.13	Lemma 209: forward_diamond(forward_diamond(X, Y), forward_diamond(X, Y)) = forward_diamond(X, Y).
348.11/44.13	Proof:
348.11/44.13	  forward_diamond(forward_diamond(X, Y), forward_diamond(X, Y))
348.11/44.13	= { by lemma 154 R->L }
348.11/44.13	  forward_diamond(domain(forward_diamond(X, Y)), forward_diamond(X, Y))
348.11/44.13	= { by lemma 54 R->L }
348.11/44.13	  forward_diamond(forward_diamond(one, forward_diamond(X, Y)), forward_diamond(X, Y))
348.11/44.13	= { by lemma 49 R->L }
348.11/44.13	  forward_diamond(c(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
348.11/44.13	= { by lemma 90 R->L }
348.11/44.13	  forward_diamond(domain(c(antidomain(forward_diamond(X, Y)))), forward_diamond(X, Y))
348.27/44.13	= { by lemma 114 R->L }
348.27/44.13	  domain(domain_difference(c(antidomain(forward_diamond(X, Y))), antidomain(forward_diamond(X, Y))))
348.27/44.13	= { by lemma 163 }
348.27/44.13	  domain(c(antidomain(forward_diamond(X, Y))))
348.27/44.13	= { by lemma 90 }
348.27/44.13	  c(antidomain(forward_diamond(X, Y)))
348.27/44.13	= { by lemma 49 }
348.27/44.13	  forward_diamond(one, forward_diamond(X, Y))
348.27/44.13	= { by lemma 54 }
348.27/44.14	  domain(forward_diamond(X, Y))
348.27/44.14	= { by lemma 154 }
348.27/44.14	  forward_diamond(X, Y)
348.27/44.14	
348.27/44.14	Lemma 210: backward_diamond(domain_difference(X, Y), domain_difference(X, Y)) = domain_difference(X, Y).
348.27/44.14	Proof:
348.27/44.14	  backward_diamond(domain_difference(X, Y), domain_difference(X, Y))
348.27/44.14	= { by lemma 181 R->L }
348.27/44.14	  backward_diamond(domain_difference(X, Y), forward_diamond(c(Y), X))
348.27/44.14	= { by lemma 181 R->L }
348.27/44.14	  backward_diamond(forward_diamond(c(Y), X), forward_diamond(c(Y), X))
348.27/44.14	= { by lemma 154 R->L }
348.27/44.14	  backward_diamond(forward_diamond(c(Y), X), domain(forward_diamond(c(Y), X)))
348.27/44.14	= { by axiom 15 (backward_diamond) R->L }
348.27/44.14	  codomain(multiplication(codomain(domain(forward_diamond(c(Y), X))), forward_diamond(c(Y), X)))
348.27/44.14	= { by lemma 140 }
348.27/44.14	  codomain(forward_diamond(c(Y), X))
348.27/44.14	= { by lemma 157 }
348.27/44.14	  forward_diamond(c(Y), X)
348.27/44.14	= { by lemma 181 }
348.27/44.14	  domain_difference(X, Y)
348.27/44.14	
348.27/44.14	Lemma 211: multiplication(domain_difference(multiplication(c(X), Y), X), Y) = multiplication(c(X), Y).
348.27/44.14	Proof:
348.27/44.14	  multiplication(domain_difference(multiplication(c(X), Y), X), Y)
348.27/44.14	= { by lemma 123 R->L }
348.27/44.14	  multiplication(domain(multiplication(c(X), Y)), multiplication(c(X), Y))
348.27/44.14	= { by lemma 32 }
348.27/44.14	  multiplication(c(X), Y)
348.27/44.14	
348.27/44.14	Lemma 212: forward_diamond(domain_difference(forward_diamond(c(X), Y), X), Y) = forward_diamond(c(X), Y).
348.27/44.14	Proof:
348.27/44.14	  forward_diamond(domain_difference(forward_diamond(c(X), Y), X), Y)
348.27/44.14	= { by lemma 166 R->L }
348.27/44.14	  forward_diamond(domain_difference(multiplication(c(X), domain(Y)), X), Y)
348.27/44.14	= { by axiom 18 (forward_diamond) }
348.27/44.14	  domain(multiplication(domain_difference(multiplication(c(X), domain(Y)), X), domain(Y)))
348.27/44.14	= { by lemma 211 }
348.27/44.14	  domain(multiplication(c(X), domain(Y)))
348.27/44.14	= { by axiom 18 (forward_diamond) R->L }
348.27/44.14	  forward_diamond(c(X), Y)
348.27/44.14	
348.27/44.14	Lemma 213: forward_diamond(domain_difference(X, Y), addition(Z, domain_difference(X, Y))) = domain_difference(X, Y).
348.27/44.14	Proof:
348.27/44.14	  forward_diamond(domain_difference(X, Y), addition(Z, domain_difference(X, Y)))
348.27/44.14	= { by lemma 210 R->L }
348.27/44.14	  forward_diamond(domain_difference(X, Y), addition(Z, backward_diamond(domain_difference(X, Y), domain_difference(X, Y))))
348.27/44.14	= { by lemma 158 R->L }
348.27/44.14	  forward_diamond(domain_difference(X, Y), addition(Z, codomain(backward_diamond(domain_difference(X, Y), domain_difference(X, Y)))))
348.27/44.14	= { by lemma 210 }
348.27/44.14	  forward_diamond(domain_difference(X, Y), addition(Z, codomain(domain_difference(X, Y))))
348.27/44.14	= { by lemma 203 R->L }
348.27/44.14	  domain(multiplication(domain_difference(X, Y), addition(Z, codomain(domain_difference(X, Y)))))
348.27/44.14	= { by lemma 134 }
348.27/44.14	  domain(addition(domain_difference(X, Y), multiplication(domain_difference(X, Y), Z)))
348.27/44.14	= { by lemma 56 R->L }
348.27/44.14	  c(c(addition(domain_difference(X, Y), multiplication(domain_difference(X, Y), Z))))
348.27/44.14	= { by lemma 134 R->L }
348.27/44.14	  c(c(multiplication(domain_difference(X, Y), addition(Z, codomain(domain_difference(X, Y))))))
348.27/44.14	= { by lemma 146 R->L }
348.27/44.14	  c(addition(c(domain_difference(X, Y)), c(multiplication(domain_difference(X, Y), addition(Z, codomain(domain_difference(X, Y)))))))
348.27/44.14	= { by lemma 134 }
348.27/44.14	  c(addition(c(domain_difference(X, Y)), c(addition(domain_difference(X, Y), multiplication(domain_difference(X, Y), Z)))))
348.27/44.14	= { by lemma 108 R->L }
348.27/44.14	  c(addition(forward_box(domain_difference(X, Y), zero), c(addition(domain_difference(X, Y), multiplication(domain_difference(X, Y), Z)))))
348.27/44.14	= { by lemma 159 R->L }
348.27/44.14	  c(addition(forward_box(domain_difference(X, Y), c(addition(Z, one))), c(addition(domain_difference(X, Y), multiplication(domain_difference(X, Y), Z)))))
348.27/44.14	= { by lemma 127 R->L }
348.27/44.14	  c(addition(forward_box(domain_difference(X, Y), c(addition(Z, one))), c(multiplication(domain_difference(X, Y), addition(Z, one)))))
348.27/44.14	= { by lemma 148 }
348.27/44.14	  c(forward_box(domain_difference(X, Y), c(addition(Z, one))))
348.27/44.14	= { by lemma 159 }
348.27/44.14	  c(forward_box(domain_difference(X, Y), zero))
348.27/44.14	= { by lemma 108 }
348.27/44.14	  c(c(domain_difference(X, Y)))
348.27/44.14	= { by lemma 56 }
348.27/44.14	  domain(domain_difference(X, Y))
348.27/44.14	= { by lemma 153 }
348.27/44.14	  forward_diamond(c(Y), X)
348.27/44.14	= { by lemma 181 }
348.27/44.14	  domain_difference(X, Y)
348.27/44.14	
348.27/44.14	Goal 1 (goals_1): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
348.27/44.14	Proof:
348.27/44.14	  addition(forward_diamond(x0, domain(x1)), domain(x2))
348.27/44.14	= { by lemma 121 }
348.27/44.14	  addition(forward_diamond(x0, x1), domain(x2))
348.27/44.14	= { by axiom 5 (additive_commutativity) }
348.27/44.14	  addition(domain(x2), forward_diamond(x0, x1))
348.27/44.14	= { by lemma 154 R->L }
348.27/44.14	  addition(domain(x2), domain(forward_diamond(x0, x1)))
348.27/44.14	= { by lemma 155 R->L }
348.27/44.14	  addition(domain(x2), domain_difference(forward_diamond(x0, x1), x2))
348.27/44.14	= { by lemma 166 R->L }
348.27/44.14	  addition(domain(x2), domain_difference(multiplication(x0, domain(x1)), x2))
348.27/44.14	= { by lemma 181 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(c(x2), multiplication(x0, domain(x1))))
348.27/44.14	= { by lemma 152 R->L }
348.27/44.14	  addition(domain(x2), domain(multiplication(c(x2), multiplication(x0, domain(x1)))))
348.27/44.14	= { by lemma 84 R->L }
348.27/44.14	  addition(domain(x2), domain(multiplication(c(x2), multiplication(codomain(c(x2)), multiplication(x0, domain(x1))))))
348.27/44.14	= { by lemma 135 R->L }
348.27/44.14	  addition(domain(x2), domain(multiplication(c(x2), multiplication(multiplication(codomain(c(x2)), multiplication(x0, domain(x1))), domain(codomain(multiplication(codomain(c(x2)), multiplication(x0, domain(x1)))))))))
348.27/44.14	= { by lemma 101 }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(codomain(c(x2)), multiplication(x0, domain(x1)))), codomain(multiplication(codomain(c(x2)), multiplication(x0, domain(x1))))))
348.27/44.14	= { by lemma 84 }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), codomain(multiplication(codomain(c(x2)), multiplication(x0, domain(x1))))))
348.27/44.14	= { by axiom 15 (backward_diamond) }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain(x1)), c(x2))))
348.27/44.14	= { by lemma 54 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(one, x1)), c(x2))))
348.27/44.14	= { by lemma 49 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, c(antidomain(x1))), c(x2))))
348.27/44.14	= { by lemma 47 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, antidomain(c(x1))), c(x2))))
348.27/44.14	= { by axiom 7 (multiplicative_right_identity) R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), one)), c(x2))))
348.27/44.14	= { by lemma 74 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(one, backward_box(x0, x2)))), c(x2))))
348.27/44.14	= { by axiom 5 (additive_commutativity) R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(backward_box(x0, x2), one))), c(x2))))
348.27/44.14	= { by lemma 183 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(backward_box(x0, domain(x2)), one))), c(x2))))
348.27/44.14	= { by axiom 5 (additive_commutativity) R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(one, backward_box(x0, domain(x2))))), c(x2))))
348.27/44.14	= { by lemma 31 R->L }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(addition(domain(x1), antidomain(x1)), backward_box(x0, domain(x2))))), c(x2))))
348.27/44.14	= { by axiom 19 (additive_associativity) }
348.27/44.14	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(domain(x1), addition(antidomain(x1), backward_box(x0, domain(x2)))))), c(x2))))
348.27/44.14	= { by lemma 35 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(domain(x1), addition(c(x1), backward_box(x0, domain(x2)))))), c(x2))))
348.27/44.15	= { by lemma 94 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(c(x1), addition(domain(x1), backward_box(x0, domain(x2)))))), c(x2))))
348.27/44.15	= { by axiom 26 (goals) R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(c(x1), backward_box(x0, domain(x2))))), c(x2))))
348.27/44.15	= { by lemma 183 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), addition(c(x1), backward_box(x0, x2)))), c(x2))))
348.27/44.15	= { by lemma 34 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(antidomain(c(x1)), backward_box(x0, x2))), c(x2))))
348.27/44.15	= { by lemma 47 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(c(antidomain(x1)), backward_box(x0, x2))), c(x2))))
348.27/44.15	= { by lemma 49 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(forward_diamond(one, x1), backward_box(x0, x2))), c(x2))))
348.27/44.15	= { by lemma 54 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(domain(x1), backward_box(x0, x2))), c(x2))))
348.27/44.15	= { by lemma 133 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, c(x2)))), c(x2))))
348.27/44.15	= { by lemma 158 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, codomain(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 88 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(coantidomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.15	= { by axiom 2 (complement) }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, antidomain(domain(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by axiom 9 (multiplicative_left_identity) R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, antidomain(multiplication(one, domain(coantidomain(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.15	= { by lemma 81 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(multiplication(one, coantidomain(backward_diamond(x0, c(x2))))), antidomain(multiplication(one, domain(coantidomain(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.15	= { by axiom 9 (multiplicative_left_identity) }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(coantidomain(backward_diamond(x0, c(x2)))), antidomain(multiplication(one, domain(coantidomain(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.15	= { by axiom 9 (multiplicative_left_identity) }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(coantidomain(backward_diamond(x0, c(x2)))), antidomain(domain(coantidomain(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.15	= { by axiom 2 (complement) R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(coantidomain(backward_diamond(x0, c(x2)))), c(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by axiom 5 (additive_commutativity) }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(c(coantidomain(backward_diamond(x0, c(x2)))), antidomain(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 63 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(c(multiplication(coantidomain(backward_diamond(x0, c(x2))), codomain(coantidomain(backward_diamond(x0, c(x2)))))), antidomain(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 35 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(multiplication(coantidomain(backward_diamond(x0, c(x2))), codomain(coantidomain(backward_diamond(x0, c(x2)))))), antidomain(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 135 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, addition(antidomain(multiplication(coantidomain(backward_diamond(x0, c(x2))), codomain(coantidomain(backward_diamond(x0, c(x2)))))), antidomain(multiplication(coantidomain(backward_diamond(x0, c(x2))), domain(codomain(coantidomain(backward_diamond(x0, c(x2)))))))))), c(x2))))
348.27/44.15	= { by lemma 81 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, antidomain(multiplication(coantidomain(backward_diamond(x0, c(x2))), domain(codomain(coantidomain(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.15	= { by lemma 35 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(multiplication(coantidomain(backward_diamond(x0, c(x2))), domain(codomain(coantidomain(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.15	= { by lemma 80 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(coantidomain(backward_diamond(x0, c(x2))), c(codomain(coantidomain(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.15	= { by lemma 58 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(coantidomain(backward_diamond(x0, c(x2))), backward_box(coantidomain(backward_diamond(x0, c(x2))), zero)))), c(x2))))
348.27/44.15	= { by lemma 87 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(coantidomain(backward_diamond(x0, c(x2))), c(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 86 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(backward_box(backward_diamond(x0, c(x2)), zero), c(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 188 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(c(backward_diamond(x0, c(x2))), c(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 160 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(backward_box(x0, coantidomain(c(x2))), c(coantidomain(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 88 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(backward_box(x0, coantidomain(c(x2))), codomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.15	= { by lemma 158 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(backward_box(x0, coantidomain(c(x2))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by axiom 17 (forward_box) }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(forward_diamond(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by lemma 207 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(forward_diamond(multiplication(backward_box(x0, coantidomain(c(x2))), domain(c(backward_diamond(x0, c(x2))))), c(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.15	= { by axiom 17 (forward_box) R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), domain(c(backward_diamond(x0, c(x2))))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 90 }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 32 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 56 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(multiplication(c(c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 143 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.15	= { by lemma 76 R->L }
348.27/44.15	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(multiplication(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.16	= { by axiom 21 (multiplicative_associativity) R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), multiplication(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.16	= { by lemma 195 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), multiplication(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.16	= { by lemma 149 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(multiplication(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2)))))))), c(x2))))
348.27/44.16	= { by lemma 76 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, forward_box(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 147 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, c(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.16	= { by lemma 162 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), c(x1))), c(x2))))
348.27/44.16	= { by lemma 181 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(c(c(x1)), domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 186 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)), c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 181 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(c(c(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 198 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), forward_diamond(c(c(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.16	= { by lemma 181 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(c(c(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 153 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(domain(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 113 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(c(x1))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 115 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(domain(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(c(c(x1))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 118 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(domain(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), domain_difference(c(c(x1)), c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 120 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(c(x1))), c(c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 111 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)), c(c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 56 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)), domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 121 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.16	= { by lemma 181 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), c(x2))))
348.27/44.16	= { by lemma 199 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 189 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1)))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 208 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(x1))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 162 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), domain_difference(x1, c(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 100 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), domain_difference(x1, c(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 101 R->L }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), multiplication(domain_difference(x1, c(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))), domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.16	= { by lemma 144 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), multiplication(domain(x1), domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))))))), c(x2))))
348.27/44.16	= { by lemma 101 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), domain(x1)), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 184 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), domain_difference(x1, forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.16	= { by lemma 197 }
348.27/44.16	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(domain_difference(x1, forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 187 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), x1), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 197 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2))))), x1)), c(x2))))
348.27/44.17	= { by lemma 189 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), forward_box(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), backward_diamond(x0, c(x2)))), x1)), c(x2))))
348.27/44.17	= { by lemma 76 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(multiplication(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2)))))), x1)), c(x2))))
348.27/44.17	= { by lemma 204 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain(multiplication(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2))))), x1)), c(x2))))
348.27/44.17	= { by lemma 99 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2)))), x1)), c(x2))))
348.27/44.17	= { by lemma 197 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain(x1), multiplication(backward_box(x0, coantidomain(c(x2))), c(backward_diamond(x0, c(x2))))), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by axiom 16 (backward_box) R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain(x1), multiplication(c(backward_diamond(x0, c(coantidomain(c(x2))))), c(backward_diamond(x0, c(x2))))), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 123 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), c(backward_diamond(x0, c(x2)))), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 35 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), c(backward_diamond(x0, c(x2)))), antidomain(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 33 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), domain(antidomain(backward_diamond(x0, c(x2))))), antidomain(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 207 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), antidomain(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 35 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 201 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 209 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1), forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1)), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 198 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(c(backward_diamond(x0, c(x2))), forward_diamond(forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1), forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1)))), c(x2))))
348.27/44.17	= { by lemma 181 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1), forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1)), backward_diamond(x0, c(x2)))), c(x2))))
348.27/44.17	= { by lemma 209 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(c(backward_diamond(x0, c(coantidomain(c(x2))))), x1), backward_diamond(x0, c(x2)))), c(x2))))
348.27/44.17	= { by lemma 168 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2))))), backward_diamond(x0, c(x2)))), c(x2))))
348.27/44.17	= { by lemma 206 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), backward_diamond(x0, c(x2)))), c(x2))))
348.27/44.17	= { by lemma 181 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(c(backward_diamond(x0, c(x2))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 199 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), x1)), c(x2))))
348.27/44.17	= { by lemma 211 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), x1)), c(x2))))
348.27/44.17	= { by lemma 202 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 165 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(backward_diamond(x0, c(x2))), domain(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 204 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(backward_diamond(x0, c(x2))), domain_difference(c(backward_diamond(x0, c(x2))), c(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2))))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 111 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(backward_diamond(x0, c(x2))), multiplication(domain(c(backward_diamond(x0, c(x2)))), c(c(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 208 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain_difference(c(backward_diamond(x0, c(x2))), backward_diamond(x0, c(x2))), c(c(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2))))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 163 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(c(backward_diamond(x0, c(x2))), c(c(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2))))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 98 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain_difference(c(backward_diamond(x0, c(x2))), c(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 204 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain(multiplication(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2))))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 152 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 212 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(domain_difference(forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), forward_diamond(backward_box(x0, coantidomain(c(x2))), x1))), c(x2))))
348.27/44.17	= { by lemma 192 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(domain_difference(forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), multiplication(backward_box(x0, coantidomain(c(x2))), domain(x1)))), c(x2))))
348.27/44.17	= { by lemma 196 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(domain(multiplication(backward_box(x0, coantidomain(c(x2))), domain(x1))), forward_diamond(domain_difference(forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))), c(x2))))
348.27/44.17	= { by axiom 18 (forward_diamond) R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), forward_diamond(domain_difference(forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))), backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 212 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), forward_diamond(c(backward_diamond(x0, c(x2))), backward_box(x0, coantidomain(c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 200 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), domain_difference(backward_box(x0, coantidomain(c(x2))), backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 158 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), domain_difference(backward_box(x0, coantidomain(c(x2))), codomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 160 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), domain_difference(c(backward_diamond(x0, c(x2))), codomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 172 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), multiplication(c(backward_diamond(x0, c(x2))), coantidomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 188 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), multiplication(backward_box(backward_diamond(x0, c(x2)), zero), coantidomain(backward_diamond(x0, c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 68 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), backward_box(backward_diamond(x0, c(x2)), zero))), c(x2))))
348.27/44.17	= { by lemma 188 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), c(backward_diamond(x0, c(x2))))), c(x2))))
348.27/44.17	= { by lemma 160 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(forward_diamond(backward_box(x0, coantidomain(c(x2))), x1), backward_box(x0, coantidomain(c(x2))))), c(x2))))
348.27/44.17	= { by lemma 197 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(multiplication(domain(backward_box(x0, coantidomain(c(x2)))), backward_box(x0, coantidomain(c(x2)))), x1)), c(x2))))
348.27/44.17	= { by lemma 32 }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, coantidomain(c(x2))), x1)), c(x2))))
348.27/44.17	= { by lemma 206 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, c(coantidomain(c(x2)))))), c(x2))))
348.27/44.17	= { by lemma 190 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(backward_box(x0, coantidomain(c(x2))), domain(x1))), c(x2))))
348.27/44.17	= { by lemma 160 R->L }
348.27/44.17	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(c(backward_diamond(x0, c(x2))), domain(x1))), c(x2))))
348.27/44.18	= { by axiom 15 (backward_diamond) R->L }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(c(codomain(multiplication(codomain(c(x2)), x0))), domain(x1))), c(x2))))
348.27/44.18	= { by lemma 58 }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(backward_box(multiplication(codomain(c(x2)), x0), zero), domain(x1))), c(x2))))
348.27/44.18	= { by lemma 86 }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), backward_diamond(multiplication(x0, multiplication(coantidomain(multiplication(codomain(c(x2)), x0)), domain(x1))), c(x2))))
348.27/44.18	= { by axiom 15 (backward_diamond) R->L }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), codomain(multiplication(codomain(c(x2)), multiplication(x0, multiplication(coantidomain(multiplication(codomain(c(x2)), x0)), domain(x1)))))))
348.27/44.18	= { by axiom 21 (multiplicative_associativity) }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), codomain(multiplication(multiplication(codomain(c(x2)), x0), multiplication(coantidomain(multiplication(codomain(c(x2)), x0)), domain(x1))))))
348.27/44.18	= { by lemma 137 }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), codomain(zero)))
348.27/44.18	= { by axiom 1 (codomain4) R->L }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), coantidomain(coantidomain(zero))))
348.27/44.18	= { by lemma 42 }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), coantidomain(one)))
348.27/44.18	= { by lemma 39 }
348.27/44.18	  addition(domain(x2), forward_diamond(multiplication(c(x2), multiplication(x0, domain(x1))), zero))
348.27/44.18	= { by lemma 199 }
348.27/44.18	  addition(domain(x2), forward_diamond(c(x2), forward_diamond(multiplication(x0, domain(x1)), zero)))
348.27/44.18	= { by lemma 181 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(multiplication(x0, domain(x1)), zero), x2))
348.27/44.18	= { by lemma 121 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(multiplication(x0, domain(x1)), domain(zero)), x2))
348.27/44.18	= { by lemma 107 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(multiplication(x0, domain(x1)), forward_diamond(zero, one)), x2))
348.27/44.18	= { by lemma 193 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(domain(multiplication(multiplication(x0, domain(x1)), forward_diamond(zero, one))), x2))
348.27/44.18	= { by axiom 21 (multiplicative_associativity) R->L }
348.27/44.18	  addition(domain(x2), domain_difference(domain(multiplication(x0, multiplication(domain(x1), forward_diamond(zero, one)))), x2))
348.27/44.18	= { by lemma 178 }
348.27/44.18	  addition(domain(x2), domain_difference(domain(multiplication(x0, domain_difference(x1, forward_box(zero, c(one))))), x2))
348.27/44.18	= { by lemma 213 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(domain(multiplication(x0, forward_diamond(domain_difference(x1, forward_box(zero, c(one))), addition(X, domain_difference(x1, forward_box(zero, c(one))))))), x2))
348.27/44.18	= { by lemma 193 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(domain_difference(x1, forward_box(zero, c(one))), addition(X, domain_difference(x1, forward_box(zero, c(one)))))), x2))
348.27/44.18	= { by lemma 213 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(x1, forward_box(zero, c(one)))), x2))
348.27/44.18	= { by lemma 80 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(x1, c(multiplication(zero, domain(one))))), x2))
348.27/44.18	= { by lemma 182 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(domain(x1), multiplication(zero, domain(one)))), x2))
348.27/44.18	= { by lemma 151 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain(multiplication(domain(x1), multiplication(zero, domain(one))))), x2))
348.27/44.18	= { by lemma 180 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain(multiplication(domain(x1), multiplication(forward_diamond(domain(x1), multiplication(zero, domain(one))), multiplication(zero, domain(one)))))), x2))
348.27/44.18	= { by lemma 194 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(multiplication(domain(x1), multiplication(forward_diamond(domain(x1), multiplication(zero, domain(one))), zero)), one)), x2))
348.27/44.18	= { by lemma 197 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(forward_diamond(domain(x1), multiplication(zero, domain(one))), zero), one), x1)), x2))
348.27/44.18	= { by lemma 182 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(domain_difference(x1, c(multiplication(zero, domain(one)))), zero), one), x1)), x2))
348.27/44.18	= { by lemma 80 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(domain_difference(x1, forward_box(zero, c(one))), zero), one), x1)), x2))
348.27/44.18	= { by lemma 179 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(domain(x1), multiplication(forward_diamond(zero, one), zero)), one), x1)), x2))
348.27/44.18	= { by lemma 106 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(codomain(domain(x1)), multiplication(forward_diamond(zero, one), zero)), one), x1)), x2))
348.27/44.18	= { by lemma 197 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(multiplication(domain(x1), multiplication(codomain(domain(x1)), multiplication(forward_diamond(zero, one), zero))), one)), x2))
348.27/44.18	= { by lemma 84 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(multiplication(domain(x1), multiplication(forward_diamond(zero, one), zero)), one)), x2))
348.27/44.18	= { by lemma 197 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(forward_diamond(zero, one), zero), one), x1)), x2))
348.27/44.18	= { by axiom 18 (forward_diamond) }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(multiplication(domain(multiplication(zero, domain(one))), zero), one), x1)), x2))
348.27/44.18	= { by lemma 101 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(domain(multiplication(domain(multiplication(zero, domain(one))), multiplication(zero, domain(one)))), x1)), x2))
348.27/44.18	= { by lemma 32 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(domain(multiplication(zero, domain(one))), x1)), x2))
348.27/44.18	= { by axiom 18 (forward_diamond) R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(forward_diamond(zero, one), x1)), x2))
348.27/44.18	= { by lemma 107 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, forward_diamond(domain(zero), x1)), x2))
348.27/44.18	= { by lemma 192 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, multiplication(domain(zero), domain(x1))), x2))
348.27/44.18	= { by lemma 51 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(zero, antidomain(x1))), x2))
348.27/44.18	= { by lemma 35 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(zero, c(x1))), x2))
348.27/44.18	= { by lemma 162 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(x1, c(zero))), x2))
348.27/44.18	= { by lemma 44 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, domain_difference(x1, one)), x2))
348.27/44.18	= { by axiom 20 (domain_difference) }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, multiplication(domain(x1), antidomain(one))), x2))
348.27/44.18	= { by lemma 36 }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, multiplication(domain(x1), zero)), x2))
348.27/44.18	= { by axiom 8 (right_annihilation) }
348.27/44.18	  addition(domain(x2), domain_difference(forward_diamond(x0, zero), x2))
348.27/44.18	= { by axiom 9 (multiplicative_left_identity) R->L }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(one, forward_diamond(x0, zero)), x2))
348.27/44.18	= { by lemma 44 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(c(zero), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by axiom 8 (right_annihilation) R->L }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(c(multiplication(x0, zero)), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by lemma 43 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(c(multiplication(x0, c(one))), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by lemma 33 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(c(multiplication(x0, domain(antidomain(one)))), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by lemma 36 }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(c(multiplication(x0, domain(zero))), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by lemma 75 }
348.27/44.18	  addition(domain(x2), domain_difference(multiplication(antidomain(forward_diamond(x0, zero)), forward_diamond(x0, zero)), x2))
348.27/44.18	= { by axiom 12 (domain1) R->L }
348.27/44.18	  addition(domain(x2), domain_difference(zero, x2))
348.27/44.18	= { by lemma 36 R->L }
348.27/44.18	  addition(domain(x2), domain_difference(antidomain(one), x2))
348.27/44.18	= { by lemma 97 R->L }
348.27/44.18	  addition(domain(x2), multiplication(c(one), antidomain(x2)))
348.27/44.18	= { by lemma 43 }
348.27/44.18	  addition(domain(x2), multiplication(zero, antidomain(x2)))
348.27/44.18	= { by axiom 10 (left_annihilation) R->L }
348.27/44.18	  addition(domain(x2), zero)
348.27/44.18	= { by axiom 6 (additive_identity) }
348.27/44.18	  domain(x2)
348.27/44.18	% SZS output end Proof
348.27/44.18	
348.27/44.18	RESULT: Theorem (the conjecture is true).
348.27/44.21	EOF