0.00/0.04	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.02/0.23	% Computer   : n120.star.cs.uiowa.edu
0.02/0.23	% Model      : x86_64 x86_64
0.02/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.23	% Memory     : 32218.625MB
0.02/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.23	% CPULimit   : 300
0.02/0.23	% DateTime   : Sat Jul 14 05:50:10 CDT 2018
0.02/0.24	% CPUTime    : 
33.01/33.22	% SZS status Theorem
33.01/33.22	
34.03/34.25	% SZS output start Proof
34.03/34.25	Take the following subset of the input axioms:
34.63/34.85	  fof(additive_associativity, axiom,
34.63/34.85	      ![A, B, C]:
34.63/34.85	        addition(A, addition(B, C))=addition(addition(A, B), C)).
34.63/34.85	  fof(additive_commutativity, axiom,
34.63/34.85	      ![A, B]: addition(B, A)=addition(A, B)).
34.63/34.85	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
34.63/34.85	  fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
34.63/34.85	  fof(backward_box, axiom,
34.63/34.85	      ![X0, X1]: c(backward_diamond(X0, c(X1)))=backward_box(X0, X1)).
34.63/34.85	  fof(backward_diamond, axiom,
34.63/34.85	      ![X0, X1]:
34.63/34.85	        codomain(multiplication(codomain(X1), X0))=backward_diamond(X0,
34.63/34.85	                                                                    X1)).
34.63/34.85	  fof(codomain1, axiom,
34.63/34.85	      ![X0]: multiplication(X0, coantidomain(X0))=zero).
34.63/34.85	  fof(codomain2, axiom,
34.63/34.85	      ![X0, X1]:
34.63/34.85	        addition(coantidomain(multiplication(X0, X1)),
34.63/34.85	                 coantidomain(multiplication(coantidomain(coantidomain(X0)),
34.63/34.85	                                             X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0)),
34.63/34.85	                                                                               X1))).
34.63/34.85	  fof(codomain3, axiom,
34.63/34.85	      ![X0]:
34.63/34.85	        one=addition(coantidomain(coantidomain(X0)), coantidomain(X0))).
34.63/34.85	  fof(codomain4, axiom,
34.63/34.85	      ![X0]: codomain(X0)=coantidomain(coantidomain(X0))).
34.63/34.85	  fof(complement, axiom, ![X0]: antidomain(domain(X0))=c(X0)).
34.63/34.85	  fof(domain1, axiom,
34.63/34.85	      ![X0]: zero=multiplication(antidomain(X0), X0)).
34.63/34.85	  fof(domain2, axiom,
34.63/34.85	      ![X0, X1]:
34.63/34.85	        antidomain(multiplication(X0,
34.63/34.85	                                  antidomain(antidomain(X1))))=addition(antidomain(multiplication(X0,
34.63/34.85	                                                                                                  X1)),
34.63/34.85	                                                                        antidomain(multiplication(X0,
34.63/34.85	                                                                                                  antidomain(antidomain(X1)))))).
34.63/34.85	  fof(domain3, axiom,
34.63/34.85	      ![X0]: one=addition(antidomain(antidomain(X0)), antidomain(X0))).
34.63/34.85	  fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))).
34.63/34.85	  fof(domain_difference, axiom,
34.63/34.85	      ![X0, X1]:
34.63/34.85	        domain_difference(X0, X1)=multiplication(domain(X0),
34.63/34.85	                                                 antidomain(X1))).
34.63/34.85	  fof(forward_box, axiom,
34.63/34.85	      ![X0, X1]: forward_box(X0, X1)=c(forward_diamond(X0, c(X1)))).
34.63/34.85	  fof(forward_diamond, axiom,
34.63/34.85	      ![X0, X1]:
34.63/34.85	        forward_diamond(X0, X1)=domain(multiplication(X0, domain(X1)))).
34.63/34.85	  fof(goals, conjecture,
34.63/34.85	      ![X0, X1, X2]:
34.63/34.85	        (zero=multiplication(forward_diamond(X0, domain(X1)), domain(X2))
34.63/34.85	           <= zero=multiplication(domain(X1),
34.63/34.85	                                  backward_diamond(X0, domain(X2))))).
34.63/34.85	  fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)).
34.63/34.85	  fof(left_distributivity, axiom,
34.63/34.85	      ![A, B, C]:
34.63/34.85	        addition(multiplication(A, C),
34.63/34.85	                 multiplication(B, C))=multiplication(addition(A, B), C)).
34.63/34.85	  fof(multiplicative_associativity, axiom,
34.63/34.85	      ![A, B, C]:
34.63/34.85	        multiplication(multiplication(A, B), C)=multiplication(A,
34.63/34.85	                                                               multiplication(B, C))).
34.63/34.85	  fof(multiplicative_left_identity, axiom,
34.63/34.85	      ![A]: A=multiplication(one, A)).
34.63/34.85	  fof(multiplicative_right_identity, axiom,
34.63/34.85	      ![A]: A=multiplication(A, one)).
34.63/34.85	  fof(order, axiom, ![A, B]: (leq(A, B) <=> B=addition(A, B))).
34.63/34.85	  fof(right_annihilation, axiom, ![A]: multiplication(A, zero)=zero).
34.63/34.85	  fof(right_distributivity, axiom,
34.63/34.85	      ![A, B, C]:
34.63/34.85	        addition(multiplication(A, B),
34.63/34.85	                 multiplication(A, C))=multiplication(A, addition(B, C))).
34.63/34.85	
34.63/34.85	Now clausify the problem and encode Horn clauses using encoding 3 of
34.63/34.85	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
34.63/34.85	We repeatedly replace C & s=t => u=v by the two clauses:
34.63/34.85	  $$fresh(y, y, x1...xn) = u
34.63/34.85	  C => $$fresh(s, t, x1...xn) = v
34.63/34.85	where $$fresh is a fresh function symbol and x1..xn are the free
34.63/34.85	variables of u and v.
34.63/34.85	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
34.63/34.85	input problem has no model of domain size 1).
34.63/34.85	
34.63/34.85	The encoding turns the above axioms into the following unit equations and goals:
34.63/34.85	
34.63/34.85	Axiom 1 (order): $$fresh2(X, X, Y, Z) = $$true.
34.63/34.85	Axiom 2 (order_1): $$fresh(X, X, Y, Z) = Z.
34.63/34.85	Axiom 3 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
34.63/34.85	Axiom 4 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
34.63/34.85	Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
34.63/34.85	Axiom 6 (multiplicative_left_identity): X = multiplication(one, X).
34.63/34.85	Axiom 7 (multiplicative_right_identity): X = multiplication(X, one).
34.63/34.85	Axiom 8 (left_annihilation): zero = multiplication(zero, X).
34.63/34.85	Axiom 9 (additive_identity): addition(X, zero) = X.
34.63/34.85	Axiom 10 (right_annihilation): multiplication(X, zero) = zero.
34.63/34.85	Axiom 11 (multiplicative_associativity): multiplication(multiplication(X, Y), Z) = multiplication(X, multiplication(Y, Z)).
34.63/34.85	Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
34.63/34.85	Axiom 13 (order_1): $$fresh(leq(X, Y), $$true, X, Y) = addition(X, Y).
34.63/34.85	Axiom 14 (order): $$fresh2(X, addition(Y, X), Y, X) = leq(Y, X).
34.63/34.85	Axiom 15 (additive_idempotence): X = addition(X, X).
34.63/34.85	Axiom 16 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
34.63/34.85	Axiom 17 (codomain3): one = addition(coantidomain(coantidomain(X)), coantidomain(X)).
34.63/34.85	Axiom 18 (domain2): antidomain(multiplication(X, antidomain(antidomain(Y)))) = addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))).
34.63/34.85	Axiom 19 (codomain1): multiplication(X, coantidomain(X)) = zero.
34.63/34.85	Axiom 20 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
34.63/34.85	Axiom 21 (domain1): zero = multiplication(antidomain(X), X).
34.63/34.85	Axiom 22 (domain3): one = addition(antidomain(antidomain(X)), antidomain(X)).
34.63/34.85	Axiom 23 (domain4): domain(X) = antidomain(antidomain(X)).
34.63/34.85	Axiom 24 (backward_diamond): codomain(multiplication(codomain(X), Y)) = backward_diamond(Y, X).
34.63/34.85	Axiom 25 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
34.63/34.85	Axiom 26 (backward_box): c(backward_diamond(X, c(Y))) = backward_box(X, Y).
34.63/34.85	Axiom 27 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
34.63/34.85	Axiom 28 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
34.63/34.85	Axiom 29 (complement): antidomain(domain(X)) = c(X).
34.63/34.85	Axiom 30 (goals): zero = multiplication(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))).
34.63/34.85	
34.63/34.85	Lemma 31: addition(coantidomain(X), coantidomain(coantidomain(X))) = one.
34.63/34.85	Proof:
34.63/34.85	  addition(coantidomain(X), coantidomain(coantidomain(X)))
34.63/34.85	= { by axiom 5 (additive_commutativity) }
34.63/34.85	  addition(coantidomain(coantidomain(X)), coantidomain(X))
34.63/34.85	= { by axiom 17 (codomain3) }
34.63/34.85	  one
34.63/34.85	
34.63/34.85	Lemma 32: addition(antidomain(X), antidomain(antidomain(X))) = one.
34.63/34.85	Proof:
34.63/34.85	  addition(antidomain(X), antidomain(antidomain(X)))
34.63/34.85	= { by axiom 5 (additive_commutativity) }
34.63/34.85	  addition(antidomain(antidomain(X)), antidomain(X))
34.63/34.85	= { by axiom 22 (domain3) }
34.63/34.85	  one
34.63/34.85	
34.63/34.85	Lemma 33: coantidomain(one) = zero.
34.63/34.85	Proof:
34.63/34.85	  coantidomain(one)
34.63/34.85	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.85	  multiplication(one, coantidomain(one))
34.63/34.85	= { by axiom 19 (codomain1) }
34.63/34.85	  zero
34.63/34.85	
34.63/34.85	Lemma 34: antidomain(one) = zero.
34.63/34.85	Proof:
34.63/34.85	  antidomain(one)
34.63/34.85	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.85	  multiplication(antidomain(one), one)
34.63/34.85	= { by axiom 21 (domain1) }
34.63/34.85	  zero
34.63/34.85	
34.63/34.85	Lemma 35: addition(zero, X) = X.
34.63/34.85	Proof:
34.63/34.85	  addition(zero, X)
34.63/34.85	= { by axiom 5 (additive_commutativity) }
34.63/34.85	  addition(X, zero)
34.63/34.85	= { by axiom 9 (additive_identity) }
34.63/34.85	  X
34.63/34.85	
34.63/34.85	Lemma 36: codomain(one) = coantidomain(zero).
34.63/34.85	Proof:
34.63/34.85	  codomain(one)
34.63/34.85	= { by axiom 20 (codomain4) }
34.63/34.85	  coantidomain(coantidomain(one))
34.63/34.85	= { by lemma 33 }
34.63/34.85	  coantidomain(zero)
34.63/34.85	
34.63/34.85	Lemma 37: domain(one) = antidomain(zero).
34.63/34.85	Proof:
34.63/34.85	  domain(one)
34.63/34.85	= { by axiom 23 (domain4) }
34.63/34.85	  antidomain(antidomain(one))
34.63/34.85	= { by lemma 34 }
34.63/34.85	  antidomain(zero)
34.63/34.85	
34.63/34.85	Lemma 38: codomain(coantidomain(X)) = coantidomain(codomain(X)).
34.63/34.85	Proof:
34.63/34.85	  codomain(coantidomain(X))
34.63/34.85	= { by axiom 20 (codomain4) }
34.63/34.85	  coantidomain(coantidomain(coantidomain(X)))
34.63/34.85	= { by axiom 20 (codomain4) }
34.63/34.85	  coantidomain(codomain(X))
34.63/34.85	
34.63/34.85	Lemma 39: domain(antidomain(X)) = c(X).
34.63/34.85	Proof:
34.63/34.85	  domain(antidomain(X))
34.63/34.85	= { by axiom 23 (domain4) }
34.63/34.85	  antidomain(antidomain(antidomain(X)))
34.63/34.85	= { by axiom 23 (domain4) }
34.63/34.85	  antidomain(domain(X))
34.63/34.85	= { by axiom 29 (complement) }
34.63/34.85	  c(X)
34.63/34.85	
34.63/34.85	Lemma 40: backward_diamond(one, X) = codomain(codomain(X)).
34.63/34.85	Proof:
34.63/34.85	  backward_diamond(one, X)
34.63/34.85	= { by axiom 24 (backward_diamond) }
34.63/34.85	  codomain(multiplication(codomain(X), one))
34.63/34.85	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.85	  codomain(codomain(X))
34.63/34.85	
34.63/34.85	Lemma 41: forward_diamond(one, X) = domain(domain(X)).
34.63/34.85	Proof:
34.63/34.85	  forward_diamond(one, X)
34.63/34.85	= { by axiom 27 (forward_diamond) }
34.63/34.85	  domain(multiplication(one, domain(X)))
34.63/34.85	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.85	  domain(domain(X))
34.63/34.85	
34.63/34.85	Lemma 42: domain(domain(X)) = antidomain(c(X)).
34.63/34.85	Proof:
34.63/34.85	  domain(domain(X))
34.63/34.85	= { by axiom 23 (domain4) }
34.63/34.85	  antidomain(antidomain(domain(X)))
34.63/34.85	= { by axiom 29 (complement) }
34.63/34.85	  antidomain(c(X))
34.63/34.85	
34.63/34.85	Lemma 43: c(antidomain(X)) = antidomain(c(X)).
34.63/34.85	Proof:
34.63/34.86	  c(antidomain(X))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  domain(antidomain(antidomain(X)))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  domain(domain(X))
34.63/34.86	= { by lemma 42 }
34.63/34.86	  antidomain(c(X))
34.63/34.86	
34.63/34.86	Lemma 44: c(domain(X)) = domain(c(X)).
34.63/34.86	Proof:
34.63/34.86	  c(domain(X))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  domain(antidomain(domain(X)))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  domain(c(X))
34.63/34.86	
34.63/34.86	Lemma 45: addition(coantidomain(X), codomain(X)) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(coantidomain(X), codomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  addition(coantidomain(X), coantidomain(coantidomain(X)))
34.63/34.86	= { by lemma 31 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 46: coantidomain(zero) = one.
34.63/34.86	Proof:
34.63/34.86	  coantidomain(zero)
34.63/34.86	= { by lemma 36 }
34.63/34.86	  codomain(one)
34.63/34.86	= { by lemma 35 }
34.63/34.86	  addition(zero, codomain(one))
34.63/34.86	= { by lemma 33 }
34.63/34.86	  addition(coantidomain(one), codomain(one))
34.63/34.86	= { by lemma 45 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 47: multiplication(coantidomain(X), codomain(X)) = zero.
34.63/34.86	Proof:
34.63/34.86	  multiplication(coantidomain(X), codomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  multiplication(coantidomain(X), coantidomain(coantidomain(X)))
34.63/34.86	= { by axiom 19 (codomain1) }
34.63/34.86	  zero
34.63/34.86	
34.63/34.86	Lemma 48: addition(antidomain(X), domain(X)) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(antidomain(X), domain(X))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  addition(antidomain(X), antidomain(antidomain(X)))
34.63/34.86	= { by lemma 32 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 49: antidomain(zero) = one.
34.63/34.86	Proof:
34.63/34.86	  antidomain(zero)
34.63/34.86	= { by lemma 37 }
34.63/34.86	  domain(one)
34.63/34.86	= { by lemma 35 }
34.63/34.86	  addition(zero, domain(one))
34.63/34.86	= { by lemma 34 }
34.63/34.86	  addition(antidomain(one), domain(one))
34.63/34.86	= { by lemma 48 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 50: c(zero) = one.
34.63/34.86	Proof:
34.63/34.86	  c(zero)
34.63/34.86	= { by lemma 39 }
34.63/34.86	  domain(antidomain(zero))
34.63/34.86	= { by lemma 49 }
34.63/34.86	  domain(one)
34.63/34.86	= { by lemma 37 }
34.63/34.86	  antidomain(zero)
34.63/34.86	= { by lemma 49 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 51: addition(X, addition(X, Y)) = addition(X, Y).
34.63/34.86	Proof:
34.63/34.86	  addition(X, addition(X, Y))
34.63/34.86	= { by axiom 12 (additive_associativity) }
34.63/34.86	  addition(addition(X, X), Y)
34.63/34.86	= { by axiom 15 (additive_idempotence) }
34.63/34.86	  addition(X, Y)
34.63/34.86	
34.63/34.86	Lemma 52: addition(one, antidomain(X)) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(one, antidomain(X))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  addition(antidomain(X), one)
34.63/34.86	= { by lemma 32 }
34.63/34.86	  addition(antidomain(X), addition(antidomain(X), antidomain(antidomain(X))))
34.63/34.86	= { by lemma 51 }
34.63/34.86	  addition(antidomain(X), antidomain(antidomain(X)))
34.63/34.86	= { by lemma 32 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 53: leq(X, addition(X, Y)) = $$true.
34.63/34.86	Proof:
34.63/34.86	  leq(X, addition(X, Y))
34.63/34.86	= { by axiom 14 (order) }
34.63/34.86	  $$fresh2(addition(X, Y), addition(X, addition(X, Y)), X, addition(X, Y))
34.63/34.86	= { by lemma 51 }
34.63/34.86	  $$fresh2(addition(X, Y), addition(X, Y), X, addition(X, Y))
34.63/34.86	= { by axiom 1 (order) }
34.63/34.86	  $$true
34.63/34.86	
34.63/34.86	Lemma 54: addition(one, domain(X)) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(one, domain(X))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  addition(one, antidomain(antidomain(X)))
34.63/34.86	= { by lemma 52 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 55: addition(domain(X), c(X)) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(domain(X), c(X))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  addition(antidomain(antidomain(X)), c(X))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  addition(antidomain(antidomain(X)), domain(antidomain(X)))
34.63/34.86	= { by lemma 48 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 56: multiplication(antidomain(X), multiplication(X, Y)) = zero.
34.63/34.86	Proof:
34.63/34.86	  multiplication(antidomain(X), multiplication(X, Y))
34.63/34.86	= { by axiom 11 (multiplicative_associativity) }
34.63/34.86	  multiplication(multiplication(antidomain(X), X), Y)
34.63/34.86	= { by axiom 21 (domain1) }
34.63/34.86	  multiplication(zero, Y)
34.63/34.86	= { by axiom 8 (left_annihilation) }
34.63/34.86	  zero
34.63/34.86	
34.63/34.86	Lemma 57: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(domain(X), domain(Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(domain(X), antidomain(antidomain(Y)))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain_difference(X, antidomain(Y))
34.63/34.86	
34.63/34.86	Lemma 58: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  c(multiplication(X, domain(Y)))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  antidomain(domain(multiplication(X, domain(Y))))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  antidomain(forward_diamond(X, Y))
34.63/34.86	
34.63/34.86	Lemma 59: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(domain(X), c(Y))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  multiplication(domain(X), antidomain(domain(Y)))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain_difference(X, domain(Y))
34.63/34.86	
34.63/34.86	Lemma 60: addition(codomain(X), coantidomain(codomain(X))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(codomain(X), coantidomain(codomain(X)))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  addition(coantidomain(coantidomain(X)), coantidomain(codomain(X)))
34.63/34.86	= { by lemma 38 }
34.63/34.86	  addition(coantidomain(coantidomain(X)), codomain(coantidomain(X)))
34.63/34.86	= { by lemma 45 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 61: addition(c(X), antidomain(c(X))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(c(X), antidomain(c(X)))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  addition(antidomain(domain(X)), antidomain(c(X)))
34.63/34.86	= { by lemma 42 }
34.63/34.86	  addition(antidomain(domain(X)), domain(domain(X)))
34.63/34.86	= { by lemma 48 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 63: addition(X, addition(Y, Z)) = addition(Z, addition(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  addition(X, addition(Y, Z))
34.63/34.86	= { by axiom 12 (additive_associativity) }
34.63/34.86	  addition(addition(X, Y), Z)
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  addition(Z, addition(X, Y))
34.63/34.86	
34.63/34.86	Lemma 63: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
34.63/34.86	Proof:
34.63/34.86	  addition(Z, addition(X, Y))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  addition(addition(X, Y), Z)
34.63/34.86	= { by axiom 12 (additive_associativity) }
34.63/34.86	  addition(X, addition(Y, Z))
34.63/34.86	
34.63/34.86	Lemma 64: addition(X, addition(Y, Z)) = addition(Y, addition(X, Z)).
34.63/34.86	Proof:
34.63/34.86	  addition(X, addition(Y, Z))
34.63/34.86	= { by axiom 12 (additive_associativity) }
34.63/34.86	  addition(addition(X, Y), Z)
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  addition(addition(Y, X), Z)
34.63/34.86	= { by axiom 12 (additive_associativity) }
34.63/34.86	  addition(Y, addition(X, Z))
34.63/34.86	
34.63/34.86	Lemma 65: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y).
34.63/34.86	Proof:
34.63/34.86	  addition(Y, multiplication(X, Y))
34.63/34.86	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.86	  addition(multiplication(one, Y), multiplication(X, Y))
34.63/34.86	= { by axiom 4 (left_distributivity) }
34.63/34.86	  multiplication(addition(one, X), Y)
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(addition(X, one), Y)
34.63/34.86	
34.63/34.86	Lemma 66: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
34.63/34.86	Proof:
34.63/34.86	  addition(X, multiplication(X, Y))
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  addition(multiplication(X, one), multiplication(X, Y))
34.63/34.86	= { by axiom 3 (right_distributivity) }
34.63/34.86	  multiplication(X, addition(one, Y))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(X, addition(Y, one))
34.63/34.86	
34.63/34.86	Lemma 67: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(X, addition(Y, coantidomain(X)))
34.63/34.86	= { by axiom 3 (right_distributivity) }
34.63/34.86	  addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
34.63/34.86	= { by axiom 19 (codomain1) }
34.63/34.86	  addition(multiplication(X, Y), zero)
34.63/34.86	= { by axiom 9 (additive_identity) }
34.63/34.86	  multiplication(X, Y)
34.63/34.86	
34.63/34.86	Lemma 68: multiplication(coantidomain(X), coantidomain(X)) = coantidomain(X).
34.63/34.86	Proof:
34.63/34.86	  multiplication(coantidomain(X), coantidomain(X))
34.63/34.86	= { by lemma 67 }
34.63/34.86	  multiplication(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
34.63/34.86	= { by lemma 31 }
34.63/34.86	  multiplication(coantidomain(X), one)
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  coantidomain(X)
34.63/34.86	
34.63/34.86	Lemma 69: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(addition(X, antidomain(Y)), Y)
34.63/34.86	= { by axiom 4 (left_distributivity) }
34.63/34.86	  addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
34.63/34.86	= { by axiom 21 (domain1) }
34.63/34.86	  addition(multiplication(X, Y), zero)
34.63/34.86	= { by axiom 9 (additive_identity) }
34.63/34.86	  multiplication(X, Y)
34.63/34.86	
34.63/34.86	Lemma 70: c(codomain(codomain(c(X)))) = backward_box(one, X).
34.63/34.86	Proof:
34.63/34.86	  c(codomain(codomain(c(X))))
34.63/34.86	= { by lemma 40 }
34.63/34.86	  c(backward_diamond(one, c(X)))
34.63/34.86	= { by axiom 26 (backward_box) }
34.63/34.86	  backward_box(one, X)
34.63/34.86	
34.63/34.86	Lemma 71: multiplication(c(X), domain_difference(X, Y)) = zero.
34.63/34.86	Proof:
34.63/34.86	  multiplication(c(X), domain_difference(X, Y))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  multiplication(antidomain(domain(X)), domain_difference(X, Y))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  multiplication(antidomain(domain(X)), multiplication(domain(X), antidomain(Y)))
34.63/34.86	= { by lemma 56 }
34.63/34.86	  zero
34.63/34.86	
34.63/34.86	Lemma 72: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
34.63/34.86	Proof:
34.63/34.86	  domain(multiplication(X, c(Y)))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  domain(multiplication(X, domain(antidomain(Y))))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  forward_diamond(X, antidomain(Y))
34.63/34.86	
34.63/34.86	Lemma 73: forward_diamond(X, one) = domain(X).
34.63/34.86	Proof:
34.63/34.86	  forward_diamond(X, one)
34.63/34.86	= { by lemma 49 }
34.63/34.86	  forward_diamond(X, antidomain(zero))
34.63/34.86	= { by lemma 72 }
34.63/34.86	  domain(multiplication(X, c(zero)))
34.63/34.86	= { by lemma 50 }
34.63/34.86	  domain(multiplication(X, one))
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  domain(X)
34.63/34.86	
34.63/34.86	Lemma 74: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(c(X), antidomain(Y))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  multiplication(domain(antidomain(X)), antidomain(Y))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain_difference(antidomain(X), Y)
34.63/34.86	
34.63/34.86	Lemma 75: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y).
34.63/34.86	Proof:
34.63/34.86	  domain(domain_difference(X, antidomain(Y)))
34.63/34.86	= { by lemma 57 }
34.63/34.86	  domain(multiplication(domain(X), domain(Y)))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  forward_diamond(domain(X), Y)
34.63/34.86	
34.63/34.86	Lemma 76: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(addition(X, Y), coantidomain(X))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(addition(Y, X), coantidomain(X))
34.63/34.86	= { by axiom 4 (left_distributivity) }
34.63/34.86	  addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
34.63/34.86	= { by axiom 19 (codomain1) }
34.63/34.86	  addition(multiplication(Y, coantidomain(X)), zero)
34.63/34.86	= { by axiom 9 (additive_identity) }
34.63/34.86	  multiplication(Y, coantidomain(X))
34.63/34.86	
34.63/34.86	Lemma 77: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(antidomain(X), addition(X, Y))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(antidomain(X), addition(Y, X))
34.63/34.86	= { by axiom 3 (right_distributivity) }
34.63/34.86	  addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
34.63/34.86	= { by axiom 21 (domain1) }
34.63/34.86	  addition(multiplication(antidomain(X), Y), zero)
34.63/34.86	= { by axiom 9 (additive_identity) }
34.63/34.86	  multiplication(antidomain(X), Y)
34.63/34.86	
34.63/34.86	Lemma 78: multiplication(addition(X, one), X) = multiplication(X, addition(X, one)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(addition(X, one), X)
34.63/34.86	= { by lemma 65 }
34.63/34.86	  addition(X, multiplication(X, X))
34.63/34.86	= { by lemma 66 }
34.63/34.86	  multiplication(X, addition(X, one))
34.63/34.86	
34.63/34.86	Lemma 79: multiplication(X, codomain(X)) = X.
34.63/34.86	Proof:
34.63/34.86	  multiplication(X, codomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  multiplication(X, coantidomain(coantidomain(X)))
34.63/34.86	= { by lemma 67 }
34.63/34.86	  multiplication(X, addition(coantidomain(coantidomain(X)), coantidomain(X)))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(X, addition(coantidomain(X), coantidomain(coantidomain(X))))
34.63/34.86	= { by lemma 31 }
34.63/34.86	  multiplication(X, one)
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  X
34.63/34.86	
34.63/34.86	Lemma 80: multiplication(addition(antidomain(Y), X), Y) = multiplication(X, Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(addition(antidomain(Y), X), Y)
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(addition(X, antidomain(Y)), Y)
34.63/34.86	= { by lemma 69 }
34.63/34.86	  multiplication(X, Y)
34.63/34.86	
34.63/34.86	Lemma 81: multiplication(domain(X), X) = X.
34.63/34.86	Proof:
34.63/34.86	  multiplication(domain(X), X)
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(antidomain(antidomain(X)), X)
34.63/34.86	= { by lemma 80 }
34.63/34.86	  multiplication(addition(antidomain(X), antidomain(antidomain(X))), X)
34.63/34.86	= { by lemma 32 }
34.63/34.86	  multiplication(one, X)
34.63/34.86	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.86	  X
34.63/34.86	
34.63/34.86	Lemma 82: domain_difference(antidomain(X), X) = antidomain(X).
34.63/34.86	Proof:
34.63/34.86	  domain_difference(antidomain(X), X)
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  multiplication(domain(antidomain(X)), antidomain(X))
34.63/34.86	= { by lemma 81 }
34.63/34.86	  antidomain(X)
34.63/34.86	
34.63/34.86	Lemma 83: backward_diamond(X, one) = codomain(X).
34.63/34.86	Proof:
34.63/34.86	  backward_diamond(X, one)
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  codomain(multiplication(codomain(one), X))
34.63/34.86	= { by lemma 36 }
34.63/34.86	  codomain(multiplication(coantidomain(zero), X))
34.63/34.86	= { by lemma 46 }
34.63/34.86	  codomain(multiplication(one, X))
34.63/34.86	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.86	  codomain(X)
34.63/34.86	
34.63/34.86	Lemma 84: forward_box(X, backward_diamond(Y, c(Z))) = c(forward_diamond(X, backward_box(Y, Z))).
34.63/34.86	Proof:
34.63/34.86	  forward_box(X, backward_diamond(Y, c(Z)))
34.63/34.86	= { by axiom 25 (forward_box) }
34.63/34.86	  c(forward_diamond(X, c(backward_diamond(Y, c(Z)))))
34.63/34.86	= { by axiom 26 (backward_box) }
34.63/34.86	  c(forward_diamond(X, backward_box(Y, Z)))
34.63/34.86	
34.63/34.86	Lemma 85: domain_difference(multiplication(X, domain(Y)), Z) = multiplication(forward_diamond(X, Y), antidomain(Z)).
34.63/34.86	Proof:
34.63/34.86	  domain_difference(multiplication(X, domain(Y)), Z)
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  multiplication(forward_diamond(X, Y), antidomain(Z))
34.63/34.86	
34.63/34.86	Lemma 86: multiplication(antidomain(coantidomain(X)), codomain(X)) = antidomain(coantidomain(X)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(antidomain(coantidomain(X)), codomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  multiplication(antidomain(coantidomain(X)), coantidomain(coantidomain(X)))
34.63/34.86	= { by lemma 77 }
34.63/34.86	  multiplication(antidomain(coantidomain(X)), addition(coantidomain(X), coantidomain(coantidomain(X))))
34.63/34.86	= { by lemma 31 }
34.63/34.86	  multiplication(antidomain(coantidomain(X)), one)
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  antidomain(coantidomain(X))
34.63/34.86	
34.63/34.86	Lemma 87: addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y)))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  addition(domain(multiplication(X, domain(Y))), antidomain(forward_diamond(X, Y)))
34.63/34.86	= { by lemma 58 }
34.63/34.86	  addition(domain(multiplication(X, domain(Y))), c(multiplication(X, domain(Y))))
34.63/34.86	= { by lemma 55 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 88: addition(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y)))
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  addition(codomain(multiplication(codomain(Y), X)), coantidomain(backward_diamond(X, Y)))
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  addition(codomain(multiplication(codomain(Y), X)), coantidomain(codomain(multiplication(codomain(Y), X))))
34.63/34.86	= { by lemma 60 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 89: addition(forward_box(X, Y), antidomain(forward_box(X, Y))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(forward_box(X, Y), antidomain(forward_box(X, Y)))
34.63/34.86	= { by axiom 25 (forward_box) }
34.63/34.86	  addition(c(forward_diamond(X, c(Y))), antidomain(forward_box(X, Y)))
34.63/34.86	= { by axiom 25 (forward_box) }
34.63/34.86	  addition(c(forward_diamond(X, c(Y))), antidomain(c(forward_diamond(X, c(Y)))))
34.63/34.86	= { by lemma 61 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 90: addition(backward_box(X, Y), antidomain(backward_box(X, Y))) = one.
34.63/34.86	Proof:
34.63/34.86	  addition(backward_box(X, Y), antidomain(backward_box(X, Y)))
34.63/34.86	= { by axiom 26 (backward_box) }
34.63/34.86	  addition(c(backward_diamond(X, c(Y))), antidomain(backward_box(X, Y)))
34.63/34.86	= { by axiom 26 (backward_box) }
34.63/34.86	  addition(c(backward_diamond(X, c(Y))), antidomain(c(backward_diamond(X, c(Y)))))
34.63/34.86	= { by lemma 61 }
34.63/34.86	  one
34.63/34.86	
34.63/34.86	Lemma 91: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
34.63/34.86	Proof:
34.63/34.86	  multiplication(addition(X, Y), coantidomain(Y))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(addition(Y, X), coantidomain(Y))
34.63/34.86	= { by lemma 76 }
34.63/34.86	  multiplication(X, coantidomain(Y))
34.63/34.86	
34.63/34.86	Lemma 92: coantidomain(codomain(X)) = coantidomain(X).
34.63/34.86	Proof:
34.63/34.86	  coantidomain(codomain(X))
34.63/34.86	= { by lemma 38 }
34.63/34.86	  codomain(coantidomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  coantidomain(coantidomain(coantidomain(X)))
34.63/34.86	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.86	  multiplication(one, coantidomain(coantidomain(coantidomain(X))))
34.63/34.86	= { by lemma 31 }
34.63/34.86	  multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(X))))
34.63/34.86	= { by lemma 91 }
34.63/34.86	  multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X))))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  multiplication(coantidomain(X), codomain(coantidomain(X)))
34.63/34.86	= { by lemma 79 }
34.63/34.86	  coantidomain(X)
34.63/34.86	
34.63/34.86	Lemma 93: codomain(codomain(X)) = codomain(X).
34.63/34.86	Proof:
34.63/34.86	  codomain(codomain(X))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  coantidomain(coantidomain(codomain(X)))
34.63/34.86	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.86	  multiplication(one, coantidomain(coantidomain(codomain(X))))
34.63/34.86	= { by lemma 60 }
34.63/34.86	  multiplication(addition(codomain(X), coantidomain(codomain(X))), coantidomain(coantidomain(codomain(X))))
34.63/34.86	= { by lemma 91 }
34.63/34.86	  multiplication(codomain(X), coantidomain(coantidomain(codomain(X))))
34.63/34.86	= { by axiom 20 (codomain4) }
34.63/34.86	  multiplication(codomain(X), codomain(codomain(X)))
34.63/34.86	= { by lemma 79 }
34.63/34.86	  codomain(X)
34.63/34.86	
34.63/34.86	Lemma 94: backward_diamond(X, codomain(Y)) = backward_diamond(X, Y).
34.63/34.86	Proof:
34.63/34.86	  backward_diamond(X, codomain(Y))
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  codomain(multiplication(codomain(codomain(Y)), X))
34.63/34.86	= { by lemma 93 }
34.63/34.86	  codomain(multiplication(codomain(Y), X))
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  backward_diamond(X, Y)
34.63/34.86	
34.63/34.86	Lemma 95: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
34.63/34.86	Proof:
34.63/34.86	  multiplication(antidomain(X), addition(Y, X))
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  multiplication(antidomain(X), addition(X, Y))
34.63/34.86	= { by lemma 77 }
34.63/34.86	  multiplication(antidomain(X), Y)
34.63/34.86	
34.63/34.86	Lemma 96: c(X) = antidomain(X).
34.63/34.86	Proof:
34.63/34.86	  c(X)
34.63/34.86	= { by lemma 39 }
34.63/34.86	  domain(antidomain(X))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(antidomain(antidomain(X)))
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  multiplication(antidomain(antidomain(antidomain(X))), one)
34.63/34.86	= { by lemma 32 }
34.63/34.86	  multiplication(antidomain(antidomain(antidomain(X))), addition(antidomain(X), antidomain(antidomain(X))))
34.63/34.86	= { by lemma 95 }
34.63/34.86	  multiplication(antidomain(antidomain(antidomain(X))), antidomain(X))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(domain(antidomain(X)), antidomain(X))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain_difference(antidomain(X), X)
34.63/34.86	= { by lemma 82 }
34.63/34.86	  antidomain(X)
34.63/34.86	
34.63/34.86	Lemma 97: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
34.63/34.86	Proof:
34.63/34.86	  domain(forward_diamond(X, Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(antidomain(forward_diamond(X, Y)))
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), one)
34.63/34.86	= { by lemma 87 }
34.63/34.86	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))))
34.63/34.86	= { by lemma 95 }
34.63/34.86	  multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(domain(forward_diamond(X, Y)), forward_diamond(X, Y))
34.63/34.86	= { by lemma 81 }
34.63/34.86	  forward_diamond(X, Y)
34.63/34.86	
34.63/34.86	Lemma 98: domain(backward_box(X, Y)) = backward_box(X, Y).
34.63/34.86	Proof:
34.63/34.86	  domain(backward_box(X, Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(antidomain(backward_box(X, Y)))
34.63/34.86	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.86	  multiplication(antidomain(antidomain(backward_box(X, Y))), one)
34.63/34.86	= { by lemma 90 }
34.63/34.86	  multiplication(antidomain(antidomain(backward_box(X, Y))), addition(backward_box(X, Y), antidomain(backward_box(X, Y))))
34.63/34.86	= { by lemma 95 }
34.63/34.86	  multiplication(antidomain(antidomain(backward_box(X, Y))), backward_box(X, Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(domain(backward_box(X, Y)), backward_box(X, Y))
34.63/34.86	= { by lemma 81 }
34.63/34.86	  backward_box(X, Y)
34.63/34.86	
34.63/34.86	Lemma 99: codomain(multiplication(coantidomain(Y), X)) = backward_diamond(X, coantidomain(Y)).
34.63/34.86	Proof:
34.63/34.86	  codomain(multiplication(coantidomain(Y), X))
34.63/34.86	= { by lemma 92 }
34.63/34.86	  codomain(multiplication(coantidomain(codomain(Y)), X))
34.63/34.86	= { by lemma 38 }
34.63/34.86	  codomain(multiplication(codomain(coantidomain(Y)), X))
34.63/34.86	= { by axiom 24 (backward_diamond) }
34.63/34.86	  backward_diamond(X, coantidomain(Y))
34.63/34.86	
34.63/34.86	Lemma 100: backward_diamond(coantidomain(X), coantidomain(X)) = coantidomain(X).
34.63/34.86	Proof:
34.63/34.86	  backward_diamond(coantidomain(X), coantidomain(X))
34.63/34.86	= { by lemma 99 }
34.63/34.86	  codomain(multiplication(coantidomain(X), coantidomain(X)))
34.63/34.86	= { by lemma 68 }
34.63/34.86	  codomain(coantidomain(X))
34.63/34.86	= { by lemma 38 }
34.63/34.86	  coantidomain(codomain(X))
34.63/34.86	= { by lemma 92 }
34.63/34.86	  coantidomain(X)
34.63/34.86	
34.63/34.86	Lemma 101: forward_diamond(X, domain(Y)) = forward_diamond(X, Y).
34.63/34.86	Proof:
34.63/34.86	  forward_diamond(X, domain(Y))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  domain(multiplication(X, domain(domain(Y))))
34.63/34.86	= { by lemma 42 }
34.63/34.86	  domain(multiplication(X, antidomain(c(Y))))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  domain(multiplication(X, antidomain(antidomain(Y))))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  domain(multiplication(X, domain(Y)))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  forward_diamond(X, Y)
34.63/34.86	
34.63/34.86	Lemma 102: domain_difference(domain(X), Y) = domain_difference(X, Y).
34.63/34.86	Proof:
34.63/34.86	  domain_difference(domain(X), Y)
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  multiplication(domain(domain(X)), antidomain(Y))
34.63/34.86	= { by lemma 42 }
34.63/34.86	  multiplication(antidomain(c(X)), antidomain(Y))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  multiplication(antidomain(antidomain(X)), antidomain(Y))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  multiplication(domain(X), antidomain(Y))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain_difference(X, Y)
34.63/34.86	
34.63/34.86	Lemma 103: forward_diamond(domain(X), antidomain(Y)) = domain(domain_difference(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  forward_diamond(domain(X), antidomain(Y))
34.63/34.86	= { by lemma 72 }
34.63/34.86	  domain(multiplication(domain(X), c(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  domain(multiplication(domain(X), antidomain(Y)))
34.63/34.86	= { by axiom 28 (domain_difference) }
34.63/34.86	  domain(domain_difference(X, Y))
34.63/34.86	
34.63/34.86	Lemma 104: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y).
34.63/34.86	Proof:
34.63/34.86	  antidomain(forward_diamond(X, antidomain(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  antidomain(forward_diamond(X, c(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  c(forward_diamond(X, c(Y)))
34.63/34.86	= { by axiom 25 (forward_box) }
34.63/34.86	  forward_box(X, Y)
34.63/34.86	
34.63/34.86	Lemma 105: antidomain(backward_diamond(X, antidomain(Y))) = backward_box(X, Y).
34.63/34.86	Proof:
34.63/34.86	  antidomain(backward_diamond(X, antidomain(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  antidomain(backward_diamond(X, c(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  c(backward_diamond(X, c(Y)))
34.63/34.86	= { by axiom 26 (backward_box) }
34.63/34.86	  backward_box(X, Y)
34.63/34.86	
34.63/34.86	Lemma 106: backward_box(X, domain(Y)) = backward_box(X, Y).
34.63/34.86	Proof:
34.63/34.86	  backward_box(X, domain(Y))
34.63/34.86	= { by lemma 105 }
34.63/34.86	  antidomain(backward_diamond(X, antidomain(domain(Y))))
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  antidomain(backward_diamond(X, c(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  antidomain(backward_diamond(X, antidomain(Y)))
34.63/34.86	= { by lemma 105 }
34.63/34.86	  backward_box(X, Y)
34.63/34.86	
34.63/34.86	Lemma 107: forward_diamond(addition(X, antidomain(Y)), Y) = forward_diamond(X, Y).
34.63/34.86	Proof:
34.63/34.86	  forward_diamond(addition(X, antidomain(Y)), Y)
34.63/34.86	= { by axiom 5 (additive_commutativity) }
34.63/34.86	  forward_diamond(addition(antidomain(Y), X), Y)
34.63/34.86	= { by lemma 96 }
34.63/34.86	  forward_diamond(addition(c(Y), X), Y)
34.63/34.86	= { by axiom 29 (complement) }
34.63/34.86	  forward_diamond(addition(antidomain(domain(Y)), X), Y)
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  domain(multiplication(addition(antidomain(domain(Y)), X), domain(Y)))
34.63/34.86	= { by lemma 80 }
34.63/34.86	  domain(multiplication(X, domain(Y)))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  forward_diamond(X, Y)
34.63/34.86	
34.63/34.86	Lemma 108: forward_diamond(X, antidomain(Y)) = antidomain(forward_box(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  forward_diamond(X, antidomain(Y))
34.63/34.86	= { by lemma 97 }
34.63/34.86	  domain(forward_diamond(X, antidomain(Y)))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(antidomain(forward_diamond(X, antidomain(Y))))
34.63/34.86	= { by lemma 104 }
34.63/34.86	  antidomain(forward_box(X, Y))
34.63/34.86	
34.63/34.86	Lemma 109: forward_box(X, antidomain(Y)) = antidomain(forward_diamond(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  forward_box(X, antidomain(Y))
34.63/34.86	= { by lemma 104 }
34.63/34.86	  antidomain(forward_diamond(X, antidomain(antidomain(Y))))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(forward_diamond(X, domain(Y)))
34.63/34.86	= { by lemma 101 }
34.63/34.86	  antidomain(forward_diamond(X, Y))
34.63/34.86	
34.63/34.86	Lemma 110: antidomain(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
34.63/34.86	Proof:
34.63/34.86	  antidomain(multiplication(X, domain(Y)))
34.63/34.86	= { by axiom 23 (domain4) }
34.63/34.86	  antidomain(multiplication(X, antidomain(antidomain(Y))))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  antidomain(multiplication(X, antidomain(c(Y))))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  c(multiplication(X, antidomain(c(Y))))
34.63/34.86	= { by lemma 42 }
34.63/34.86	  c(multiplication(X, domain(domain(Y))))
34.63/34.86	= { by lemma 58 }
34.63/34.86	  antidomain(forward_diamond(X, domain(Y)))
34.63/34.86	= { by lemma 101 }
34.63/34.86	  antidomain(forward_diamond(X, Y))
34.63/34.86	
34.63/34.86	Lemma 111: antidomain(multiplication(X, antidomain(Y))) = forward_box(X, Y).
34.63/34.86	Proof:
34.63/34.86	  antidomain(multiplication(X, antidomain(Y)))
34.63/34.86	= { by lemma 96 }
34.63/34.86	  antidomain(multiplication(X, c(Y)))
34.63/34.86	= { by lemma 39 }
34.63/34.86	  antidomain(multiplication(X, domain(antidomain(Y))))
34.63/34.86	= { by lemma 110 }
34.63/34.86	  antidomain(forward_diamond(X, antidomain(Y)))
34.63/34.86	= { by lemma 104 }
34.63/34.86	  forward_box(X, Y)
34.63/34.86	
34.63/34.86	Lemma 112: domain(multiplication(X, multiplication(Y, domain(Z)))) = forward_diamond(multiplication(X, Y), Z).
34.63/34.86	Proof:
34.63/34.86	  domain(multiplication(X, multiplication(Y, domain(Z))))
34.63/34.86	= { by axiom 11 (multiplicative_associativity) }
34.63/34.86	  domain(multiplication(multiplication(X, Y), domain(Z)))
34.63/34.86	= { by axiom 27 (forward_diamond) }
34.63/34.86	  forward_diamond(multiplication(X, Y), Z)
34.63/34.86	
34.63/34.86	Lemma 113: multiplication(domain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(X, Y), Z).
34.63/34.87	Proof:
34.63/34.87	  multiplication(domain(X), multiplication(antidomain(Y), Z))
34.63/34.87	= { by axiom 11 (multiplicative_associativity) }
34.63/34.87	  multiplication(multiplication(domain(X), antidomain(Y)), Z)
34.63/34.87	= { by axiom 28 (domain_difference) }
34.63/34.87	  multiplication(domain_difference(X, Y), Z)
34.63/34.87	
34.63/34.87	Lemma 114: multiplication(antidomain(X), coantidomain(domain(X))) = coantidomain(domain(X)).
34.63/34.87	Proof:
34.63/34.87	  multiplication(antidomain(X), coantidomain(domain(X)))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  multiplication(c(X), coantidomain(domain(X)))
34.63/34.87	= { by lemma 76 }
34.63/34.87	  multiplication(addition(domain(X), c(X)), coantidomain(domain(X)))
34.63/34.87	= { by lemma 55 }
34.63/34.87	  multiplication(one, coantidomain(domain(X)))
34.63/34.87	= { by axiom 6 (multiplicative_left_identity) }
34.63/34.87	  coantidomain(domain(X))
34.63/34.87	
34.63/34.87	Lemma 115: backward_diamond(addition(X, coantidomain(Y)), Y) = backward_diamond(X, Y).
34.63/34.87	Proof:
34.63/34.87	  backward_diamond(addition(X, coantidomain(Y)), Y)
34.63/34.87	= { by lemma 92 }
34.63/34.87	  backward_diamond(addition(X, coantidomain(codomain(Y))), Y)
34.63/34.87	= { by axiom 24 (backward_diamond) }
34.63/34.87	  codomain(multiplication(codomain(Y), addition(X, coantidomain(codomain(Y)))))
34.63/34.87	= { by lemma 67 }
34.63/34.87	  codomain(multiplication(codomain(Y), X))
34.63/34.87	= { by axiom 24 (backward_diamond) }
34.63/34.87	  backward_diamond(X, Y)
34.63/34.87	
34.63/34.87	Lemma 116: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
34.63/34.87	Proof:
34.63/34.87	  coantidomain(multiplication(codomain(X), Y))
34.63/34.87	= { by lemma 92 }
34.63/34.87	  coantidomain(codomain(multiplication(codomain(X), Y)))
34.63/34.87	= { by axiom 24 (backward_diamond) }
34.63/34.87	  coantidomain(backward_diamond(Y, X))
34.63/34.87	
34.63/34.87	Lemma 117: multiplication(antidomain(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
34.63/34.87	Proof:
34.63/34.87	  multiplication(antidomain(X), antidomain(Y))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  multiplication(c(X), antidomain(Y))
34.63/34.87	= { by lemma 74 }
34.63/34.87	  domain_difference(antidomain(X), Y)
34.63/34.87	
34.63/34.87	Lemma 118: antidomain(backward_diamond(X, domain(Y))) = backward_box(X, antidomain(Y)).
34.63/34.87	Proof:
34.63/34.87	  antidomain(backward_diamond(X, domain(Y)))
34.63/34.87	= { by axiom 23 (domain4) }
34.63/34.87	  antidomain(backward_diamond(X, antidomain(antidomain(Y))))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  antidomain(backward_diamond(X, antidomain(c(Y))))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  antidomain(backward_diamond(X, c(c(Y))))
34.63/34.87	= { by lemma 39 }
34.63/34.87	  antidomain(backward_diamond(X, domain(antidomain(c(Y)))))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  antidomain(backward_diamond(X, domain(c(c(Y)))))
34.63/34.87	= { by axiom 23 (domain4) }
34.63/34.87	  antidomain(backward_diamond(X, antidomain(antidomain(c(c(Y))))))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  antidomain(backward_diamond(X, antidomain(c(c(c(Y))))))
34.63/34.87	= { by lemma 42 }
34.63/34.87	  antidomain(backward_diamond(X, domain(domain(c(c(Y))))))
34.63/34.87	= { by lemma 44 }
34.63/34.87	  antidomain(backward_diamond(X, domain(c(domain(c(Y))))))
34.63/34.87	= { by lemma 44 }
34.63/34.87	  antidomain(backward_diamond(X, c(domain(domain(c(Y))))))
34.63/34.87	= { by lemma 41 }
34.63/34.87	  antidomain(backward_diamond(X, c(forward_diamond(one, c(Y)))))
34.63/34.87	= { by axiom 25 (forward_box) }
34.63/34.87	  antidomain(backward_diamond(X, forward_box(one, Y)))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  c(backward_diamond(X, forward_box(one, Y)))
34.63/34.87	= { by axiom 25 (forward_box) }
34.63/34.87	  c(backward_diamond(X, c(forward_diamond(one, c(Y)))))
34.63/34.87	= { by axiom 26 (backward_box) }
34.63/34.87	  backward_box(X, forward_diamond(one, c(Y)))
34.63/34.87	= { by lemma 41 }
34.63/34.87	  backward_box(X, domain(domain(c(Y))))
34.63/34.87	= { by lemma 106 }
34.63/34.87	  backward_box(X, domain(c(Y)))
34.63/34.87	= { by lemma 106 }
34.63/34.87	  backward_box(X, c(Y))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  backward_box(X, antidomain(Y))
34.63/34.87	
34.63/34.87	Lemma 119: antidomain(backward_diamond(X, backward_box(one, Y))) = backward_box(X, codomain(antidomain(Y))).
34.63/34.87	Proof:
34.63/34.87	  antidomain(backward_diamond(X, backward_box(one, Y)))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  c(backward_diamond(X, backward_box(one, Y)))
34.63/34.87	= { by axiom 26 (backward_box) }
34.63/34.87	  c(backward_diamond(X, c(backward_diamond(one, c(Y)))))
34.63/34.87	= { by axiom 26 (backward_box) }
34.63/34.87	  backward_box(X, backward_diamond(one, c(Y)))
34.63/34.87	= { by lemma 40 }
34.63/34.87	  backward_box(X, codomain(codomain(c(Y))))
34.63/34.87	= { by lemma 93 }
34.63/34.87	  backward_box(X, codomain(c(Y)))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  backward_box(X, codomain(antidomain(Y)))
34.63/34.87	
34.63/34.87	Lemma 120: coantidomain(backward_diamond(Y, coantidomain(X))) = coantidomain(multiplication(coantidomain(X), Y)).
34.63/34.87	Proof:
34.63/34.87	  coantidomain(backward_diamond(Y, coantidomain(X)))
34.63/34.87	= { by lemma 99 }
34.63/34.87	  coantidomain(codomain(multiplication(coantidomain(X), Y)))
34.63/34.87	= { by lemma 92 }
34.63/34.87	  coantidomain(multiplication(coantidomain(X), Y))
34.63/34.87	
34.63/34.87	Lemma 121: domain(multiplication(X, forward_diamond(Y, Z))) = forward_diamond(X, forward_diamond(Y, Z)).
34.63/34.87	Proof:
34.63/34.87	  domain(multiplication(X, forward_diamond(Y, Z)))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  domain(multiplication(X, domain(multiplication(Y, domain(Z)))))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  forward_diamond(X, multiplication(Y, domain(Z)))
34.63/34.87	= { by lemma 101 }
34.63/34.87	  forward_diamond(X, domain(multiplication(Y, domain(Z))))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  forward_diamond(X, forward_diamond(Y, Z))
34.63/34.87	
34.63/34.87	Lemma 122: multiplication(forward_diamond(X, Y), antidomain(Z)) = domain_difference(forward_diamond(X, Y), Z).
34.63/34.87	Proof:
34.63/34.87	  multiplication(forward_diamond(X, Y), antidomain(Z))
34.63/34.87	= { by lemma 85 }
34.63/34.87	  domain_difference(multiplication(X, domain(Y)), Z)
34.63/34.87	= { by lemma 102 }
34.63/34.87	  domain_difference(domain(multiplication(X, domain(Y))), Z)
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  domain_difference(forward_diamond(X, Y), Z)
34.63/34.87	
34.63/34.87	Lemma 123: multiplication(X, multiplication(codomain(X), Y)) = multiplication(X, Y).
34.63/34.87	Proof:
34.63/34.87	  multiplication(X, multiplication(codomain(X), Y))
34.63/34.87	= { by axiom 11 (multiplicative_associativity) }
34.63/34.87	  multiplication(multiplication(X, codomain(X)), Y)
34.63/34.87	= { by lemma 79 }
34.63/34.87	  multiplication(X, Y)
34.63/34.87	
34.63/34.87	Lemma 124: backward_box(X, antidomain(forward_diamond(Y, Z))) = antidomain(backward_diamond(X, forward_diamond(Y, Z))).
34.63/34.87	Proof:
34.63/34.87	  backward_box(X, antidomain(forward_diamond(Y, Z)))
34.63/34.87	= { by lemma 110 }
34.63/34.87	  backward_box(X, antidomain(multiplication(Y, domain(Z))))
34.63/34.87	= { by lemma 118 }
34.63/34.87	  antidomain(backward_diamond(X, domain(multiplication(Y, domain(Z)))))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  antidomain(backward_diamond(X, forward_diamond(Y, Z)))
34.63/34.87	
34.63/34.87	Lemma 125: forward_diamond(X, addition(Y, one)) = domain(X).
34.63/34.87	Proof:
34.63/34.87	  forward_diamond(X, addition(Y, one))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  domain(multiplication(X, domain(addition(Y, one))))
34.63/34.87	= { by axiom 23 (domain4) }
34.63/34.87	  domain(multiplication(X, antidomain(antidomain(addition(Y, one)))))
34.63/34.87	= { by lemma 35 }
34.63/34.87	  domain(multiplication(X, addition(zero, antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by axiom 21 (domain1) }
34.63/34.87	  domain(multiplication(X, addition(multiplication(antidomain(addition(one, Y)), addition(one, Y)), antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by lemma 51 }
34.63/34.87	  domain(multiplication(X, addition(multiplication(antidomain(addition(one, Y)), addition(one, addition(one, Y))), antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by lemma 95 }
34.63/34.87	  domain(multiplication(X, addition(multiplication(antidomain(addition(one, Y)), one), antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.87	  domain(multiplication(X, addition(antidomain(addition(one, Y)), antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by axiom 5 (additive_commutativity) }
34.63/34.87	  domain(multiplication(X, addition(antidomain(addition(Y, one)), antidomain(antidomain(addition(Y, one))))))
34.63/34.87	= { by lemma 32 }
34.63/34.87	  domain(multiplication(X, one))
34.63/34.87	= { by axiom 7 (multiplicative_right_identity) }
34.63/34.87	  domain(X)
34.63/34.87	
34.63/34.87	Lemma 126: multiplication(domain_difference(X, antidomain(Y)), Y) = multiplication(domain(X), Y).
34.63/34.87	Proof:
34.63/34.87	  multiplication(domain_difference(X, antidomain(Y)), Y)
34.63/34.87	= { by lemma 57 }
34.63/34.87	  multiplication(multiplication(domain(X), domain(Y)), Y)
34.63/34.87	= { by axiom 11 (multiplicative_associativity) }
34.63/34.87	  multiplication(domain(X), multiplication(domain(Y), Y))
34.63/34.87	= { by lemma 81 }
34.63/34.87	  multiplication(domain(X), Y)
34.63/34.87	
34.63/34.87	Lemma 127: domain_difference(X, antidomain(forward_diamond(Y, Z))) = multiplication(domain(X), forward_diamond(Y, Z)).
34.63/34.87	Proof:
34.63/34.87	  domain_difference(X, antidomain(forward_diamond(Y, Z)))
34.63/34.87	= { by lemma 110 }
34.63/34.87	  domain_difference(X, antidomain(multiplication(Y, domain(Z))))
34.63/34.87	= { by lemma 57 }
34.63/34.87	  multiplication(domain(X), domain(multiplication(Y, domain(Z))))
34.63/34.87	= { by axiom 27 (forward_diamond) }
34.63/34.87	  multiplication(domain(X), forward_diamond(Y, Z))
34.63/34.87	
34.63/34.87	Lemma 128: multiplication(antidomain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(antidomain(X), Y), Z).
34.63/34.87	Proof:
34.63/34.87	  multiplication(antidomain(X), multiplication(antidomain(Y), Z))
34.63/34.87	= { by lemma 96 }
34.63/34.87	  multiplication(c(X), multiplication(antidomain(Y), Z))
34.63/34.87	= { by lemma 39 }
34.63/34.87	  multiplication(domain(antidomain(X)), multiplication(antidomain(Y), Z))
34.63/34.87	= { by lemma 113 }
34.69/34.88	  multiplication(domain_difference(antidomain(X), Y), Z)
34.69/34.88	
34.69/34.88	Lemma 129: domain_difference(forward_diamond(X, Y), multiplication(X, Y)) = zero.
34.69/34.88	Proof:
34.69/34.88	  domain_difference(forward_diamond(X, Y), multiplication(X, Y))
34.69/34.88	= { by lemma 122 }
34.69/34.88	  multiplication(forward_diamond(X, Y), antidomain(multiplication(X, Y)))
34.69/34.88	= { by lemma 85 }
34.69/34.88	  domain_difference(multiplication(X, domain(Y)), multiplication(X, Y))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain_difference(multiplication(X, antidomain(antidomain(Y))), multiplication(X, Y))
34.69/34.88	= { by lemma 102 }
34.69/34.88	  domain_difference(domain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, Y))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain_difference(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, Y))
34.69/34.88	= { by lemma 117 }
34.69/34.88	  multiplication(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(multiplication(X, Y)))
34.69/34.88	= { by lemma 95 }
34.69/34.88	  multiplication(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))))
34.69/34.88	= { by axiom 18 (domain2) }
34.69/34.88	  multiplication(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), antidomain(multiplication(X, antidomain(antidomain(Y)))))
34.69/34.88	= { by lemma 117 }
34.69/34.88	  domain_difference(antidomain(antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, antidomain(antidomain(Y))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain_difference(domain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, antidomain(antidomain(Y))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain_difference(domain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, domain(Y)))
34.69/34.88	= { by lemma 102 }
34.69/34.88	  domain_difference(multiplication(X, antidomain(antidomain(Y))), multiplication(X, domain(Y)))
34.69/34.88	= { by axiom 28 (domain_difference) }
34.69/34.88	  multiplication(domain(multiplication(X, antidomain(antidomain(Y)))), antidomain(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 110 }
34.69/34.88	  multiplication(domain(multiplication(X, antidomain(antidomain(Y)))), antidomain(forward_diamond(X, Y)))
34.69/34.88	= { by axiom 28 (domain_difference) }
34.69/34.88	  domain_difference(multiplication(X, antidomain(antidomain(Y))), forward_diamond(X, Y))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain_difference(multiplication(X, domain(Y)), forward_diamond(X, Y))
34.69/34.88	= { by axiom 27 (forward_diamond) }
34.69/34.88	  domain_difference(multiplication(X, domain(Y)), domain(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 59 }
34.69/34.88	  multiplication(domain(multiplication(X, domain(Y))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 69 }
34.69/34.88	  multiplication(addition(domain(multiplication(X, domain(Y))), antidomain(c(multiplication(X, domain(Y))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  multiplication(addition(antidomain(antidomain(multiplication(X, domain(Y)))), antidomain(c(multiplication(X, domain(Y))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.88	  multiplication(addition(antidomain(multiplication(one, antidomain(multiplication(X, domain(Y))))), antidomain(c(multiplication(X, domain(Y))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 43 }
34.69/34.88	  multiplication(addition(antidomain(multiplication(one, antidomain(multiplication(X, domain(Y))))), c(antidomain(multiplication(X, domain(Y))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 39 }
34.69/34.88	  multiplication(addition(antidomain(multiplication(one, antidomain(multiplication(X, domain(Y))))), domain(antidomain(antidomain(multiplication(X, domain(Y)))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  multiplication(addition(antidomain(multiplication(one, antidomain(multiplication(X, domain(Y))))), antidomain(antidomain(antidomain(antidomain(multiplication(X, domain(Y))))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.88	  multiplication(addition(antidomain(multiplication(one, antidomain(multiplication(X, domain(Y))))), antidomain(multiplication(one, antidomain(antidomain(antidomain(multiplication(X, domain(Y)))))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 18 (domain2) }
34.69/34.88	  multiplication(antidomain(multiplication(one, antidomain(antidomain(antidomain(multiplication(X, domain(Y))))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.88	  multiplication(antidomain(antidomain(antidomain(antidomain(multiplication(X, domain(Y)))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  multiplication(domain(antidomain(antidomain(multiplication(X, domain(Y))))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 39 }
34.69/34.88	  multiplication(c(antidomain(multiplication(X, domain(Y)))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by lemma 43 }
34.69/34.88	  multiplication(antidomain(c(multiplication(X, domain(Y)))), c(multiplication(X, domain(Y))))
34.69/34.88	= { by axiom 21 (domain1) }
34.69/34.88	  zero
34.69/34.88	
34.69/34.88	Lemma 130: domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), coantidomain(X)) = zero.
34.69/34.88	Proof:
34.69/34.88	  domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), coantidomain(X))
34.69/34.88	= { by lemma 68 }
34.69/34.88	  domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), multiplication(coantidomain(X), coantidomain(X)))
34.69/34.88	= { by lemma 129 }
34.69/34.88	  zero
34.69/34.88	
34.69/34.88	Lemma 131: addition(antidomain(X), domain_difference(Y, X)) = antidomain(X).
34.69/34.88	Proof:
34.69/34.88	  addition(antidomain(X), domain_difference(Y, X))
34.69/34.88	= { by axiom 28 (domain_difference) }
34.69/34.88	  addition(antidomain(X), multiplication(domain(Y), antidomain(X)))
34.69/34.88	= { by lemma 65 }
34.69/34.88	  multiplication(addition(domain(Y), one), antidomain(X))
34.69/34.88	= { by axiom 5 (additive_commutativity) }
34.69/34.88	  multiplication(addition(one, domain(Y)), antidomain(X))
34.69/34.88	= { by lemma 54 }
34.69/34.88	  multiplication(one, antidomain(X))
34.69/34.88	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.88	  antidomain(X)
34.69/34.88	
34.69/34.88	Lemma 132: multiplication(domain(X), domain_difference(Y, X)) = zero.
34.69/34.88	Proof:
34.69/34.88	  multiplication(domain(X), domain_difference(Y, X))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  multiplication(antidomain(antidomain(X)), domain_difference(Y, X))
34.69/34.88	= { by lemma 77 }
34.69/34.88	  multiplication(antidomain(antidomain(X)), addition(antidomain(X), domain_difference(Y, X)))
34.69/34.88	= { by lemma 131 }
34.69/34.88	  multiplication(antidomain(antidomain(X)), antidomain(X))
34.69/34.88	= { by axiom 21 (domain1) }
34.69/34.88	  zero
34.69/34.88	
34.69/34.88	Lemma 133: multiplication(antidomain(X), addition(Z, multiplication(X, Y))) = multiplication(antidomain(X), Z).
34.69/34.88	Proof:
34.69/34.88	  multiplication(antidomain(X), addition(Z, multiplication(X, Y)))
34.69/34.88	= { by axiom 5 (additive_commutativity) }
34.69/34.88	  multiplication(antidomain(X), addition(multiplication(X, Y), Z))
34.69/34.88	= { by axiom 3 (right_distributivity) }
34.69/34.88	  addition(multiplication(antidomain(X), multiplication(X, Y)), multiplication(antidomain(X), Z))
34.69/34.88	= { by lemma 56 }
34.69/34.88	  addition(zero, multiplication(antidomain(X), Z))
34.69/34.88	= { by lemma 35 }
34.69/34.88	  multiplication(antidomain(X), Z)
34.69/34.88	
34.69/34.88	Lemma 134: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
34.69/34.88	Proof:
34.69/34.88	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  multiplication(antidomain(multiplication(X, Y)), multiplication(X, antidomain(antidomain(Y))))
34.69/34.88	= { by lemma 69 }
34.69/34.88	  multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, antidomain(antidomain(Y))))
34.69/34.88	= { by axiom 18 (domain2) }
34.69/34.88	  multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, antidomain(antidomain(Y))))
34.69/34.88	= { by axiom 21 (domain1) }
34.69/34.88	  zero
34.69/34.88	
34.69/34.88	Lemma 135: addition(domain_difference(X, Y), multiplication(domain(X), Z)) = multiplication(domain(X), addition(Z, antidomain(Y))).
34.69/34.88	Proof:
34.69/34.88	  addition(domain_difference(X, Y), multiplication(domain(X), Z))
34.69/34.88	= { by axiom 28 (domain_difference) }
34.69/34.88	  addition(multiplication(domain(X), antidomain(Y)), multiplication(domain(X), Z))
34.69/34.88	= { by axiom 3 (right_distributivity) }
34.69/34.88	  multiplication(domain(X), addition(antidomain(Y), Z))
34.69/34.88	= { by axiom 5 (additive_commutativity) }
34.69/34.88	  multiplication(domain(X), addition(Z, antidomain(Y)))
34.69/34.88	
34.69/34.88	Lemma 136: domain_difference(X, antidomain(backward_box(Y, Z))) = multiplication(domain(X), backward_box(Y, Z)).
34.69/34.88	Proof:
34.69/34.88	  domain_difference(X, antidomain(backward_box(Y, Z)))
34.69/34.88	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.88	  multiplication(domain_difference(X, antidomain(backward_box(Y, Z))), one)
34.69/34.88	= { by lemma 90 }
34.69/34.88	  multiplication(domain_difference(X, antidomain(backward_box(Y, Z))), addition(backward_box(Y, Z), antidomain(backward_box(Y, Z))))
34.69/34.88	= { by axiom 5 (additive_commutativity) }
34.69/34.88	  multiplication(domain_difference(X, antidomain(backward_box(Y, Z))), addition(antidomain(backward_box(Y, Z)), backward_box(Y, Z)))
34.69/34.88	= { by lemma 113 }
34.69/34.88	  multiplication(domain(X), multiplication(antidomain(antidomain(backward_box(Y, Z))), addition(antidomain(backward_box(Y, Z)), backward_box(Y, Z))))
34.69/34.88	= { by lemma 77 }
34.69/34.88	  multiplication(domain(X), multiplication(antidomain(antidomain(backward_box(Y, Z))), backward_box(Y, Z)))
34.69/34.88	= { by lemma 113 }
34.69/34.88	  multiplication(domain_difference(X, antidomain(backward_box(Y, Z))), backward_box(Y, Z))
34.69/34.88	= { by lemma 126 }
34.69/34.88	  multiplication(domain(X), backward_box(Y, Z))
34.69/34.88	
34.69/34.88	Lemma 137: domain(multiplication(X, backward_box(Y, Z))) = forward_diamond(X, backward_box(Y, Z)).
34.69/34.88	Proof:
34.69/34.88	  domain(multiplication(X, backward_box(Y, Z)))
34.69/34.88	= { by axiom 26 (backward_box) }
34.69/34.88	  domain(multiplication(X, c(backward_diamond(Y, c(Z)))))
34.69/34.88	= { by lemma 72 }
34.69/34.88	  forward_diamond(X, antidomain(backward_diamond(Y, c(Z))))
34.69/34.88	= { by lemma 108 }
34.69/34.88	  antidomain(forward_box(X, backward_diamond(Y, c(Z))))
34.69/34.88	= { by lemma 84 }
34.69/34.88	  antidomain(c(forward_diamond(X, backward_box(Y, Z))))
34.69/34.88	= { by lemma 96 }
34.69/34.88	  antidomain(antidomain(forward_diamond(X, backward_box(Y, Z))))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain(forward_diamond(X, backward_box(Y, Z)))
34.69/34.88	= { by lemma 97 }
34.69/34.88	  forward_diamond(X, backward_box(Y, Z))
34.69/34.88	
34.69/34.88	Lemma 138: leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) = $$true.
34.69/34.88	Proof:
34.69/34.88	  leq(multiplication(X, Y), multiplication(addition(X, Z), Y))
34.69/34.88	= { by axiom 4 (left_distributivity) }
34.69/34.88	  leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
34.69/34.88	= { by lemma 53 }
34.69/34.88	  $$true
34.69/34.88	
34.69/34.88	Lemma 139: addition(coantidomain(multiplication(X, Y)), coantidomain(backward_diamond(Y, X))) = coantidomain(backward_diamond(Y, X)).
34.69/34.88	Proof:
34.69/34.88	  addition(coantidomain(multiplication(X, Y)), coantidomain(backward_diamond(Y, X)))
34.69/34.88	= { by lemma 116 }
34.69/34.88	  addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y)))
34.69/34.88	= { by axiom 20 (codomain4) }
34.69/34.88	  addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y)))
34.69/34.88	= { by axiom 16 (codomain2) }
34.69/34.88	  coantidomain(multiplication(coantidomain(coantidomain(X)), Y))
34.69/34.88	= { by axiom 20 (codomain4) }
34.69/34.88	  coantidomain(multiplication(codomain(X), Y))
34.69/34.88	= { by lemma 116 }
34.69/34.88	  coantidomain(backward_diamond(Y, X))
34.69/34.88	
34.69/34.88	Lemma 140: codomain(coantidomain(X)) = coantidomain(X).
34.69/34.88	Proof:
34.69/34.88	  codomain(coantidomain(X))
34.69/34.88	= { by axiom 20 (codomain4) }
34.69/34.88	  coantidomain(coantidomain(coantidomain(X)))
34.69/34.88	= { by axiom 20 (codomain4) }
34.69/34.88	  coantidomain(codomain(X))
34.69/34.88	= { by lemma 92 }
34.69/34.88	  coantidomain(X)
34.69/34.88	
34.69/34.88	Lemma 141: domain(antidomain(X)) = antidomain(X).
34.69/34.88	Proof:
34.69/34.88	  domain(antidomain(X))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  antidomain(antidomain(antidomain(X)))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  antidomain(domain(X))
34.69/34.88	= { by axiom 29 (complement) }
34.69/34.88	  c(X)
34.69/34.88	= { by lemma 96 }
34.69/34.88	  antidomain(X)
34.69/34.88	
34.69/34.88	Lemma 142: forward_diamond(one, X) = domain(X).
34.69/34.88	Proof:
34.69/34.88	  forward_diamond(one, X)
34.69/34.88	= { by axiom 27 (forward_diamond) }
34.69/34.88	  domain(multiplication(one, domain(X)))
34.69/34.88	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.88	  domain(domain(X))
34.69/34.88	= { by lemma 42 }
34.69/34.88	  antidomain(c(X))
34.69/34.88	= { by lemma 96 }
34.69/34.88	  antidomain(antidomain(X))
34.69/34.88	= { by axiom 23 (domain4) }
34.69/34.88	  domain(X)
34.69/34.88	
34.69/34.88	Lemma 143: domain_difference(antidomain(X), antidomain(Y)) = multiplication(antidomain(X), domain(Y)).
34.69/34.88	Proof:
34.69/34.88	  domain_difference(antidomain(X), antidomain(Y))
34.69/34.88	= { by lemma 57 }
34.69/34.88	  multiplication(domain(antidomain(X)), domain(Y))
34.69/34.88	= { by lemma 39 }
34.69/34.88	  multiplication(c(X), domain(Y))
34.69/34.88	= { by lemma 96 }
34.69/34.88	  multiplication(antidomain(X), domain(Y))
34.69/34.88	
34.69/34.88	Lemma 144: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z).
34.69/34.88	Proof:
34.69/34.88	  domain_difference(multiplication(X, domain(Y)), Z)
34.69/34.88	= { by axiom 28 (domain_difference) }
34.69/34.88	  multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
34.69/34.88	= { by axiom 27 (forward_diamond) }
34.69/34.88	  multiplication(forward_diamond(X, Y), antidomain(Z))
34.69/34.88	= { by lemma 122 }
34.69/34.88	  domain_difference(forward_diamond(X, Y), Z)
34.69/34.88	
34.69/34.89	Lemma 145: addition(multiplication(X, Y), addition(W, multiplication(X, Z))) = addition(W, multiplication(X, addition(Y, Z))).
34.69/34.89	Proof:
34.69/34.89	  addition(multiplication(X, Y), addition(W, multiplication(X, Z)))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  addition(multiplication(X, Y), addition(multiplication(X, Z), W))
34.69/34.89	= { by axiom 12 (additive_associativity) }
34.69/34.89	  addition(addition(multiplication(X, Y), multiplication(X, Z)), W)
34.69/34.89	= { by axiom 3 (right_distributivity) }
34.69/34.89	  addition(multiplication(X, addition(Y, Z)), W)
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  addition(W, multiplication(X, addition(Y, Z)))
34.69/34.89	
34.69/34.89	Lemma 146: multiplication(X, multiplication(Y, backward_diamond(Y, X))) = multiplication(X, Y).
34.69/34.89	Proof:
34.69/34.89	  multiplication(X, multiplication(Y, backward_diamond(Y, X)))
34.69/34.89	= { by axiom 24 (backward_diamond) }
34.69/34.89	  multiplication(X, multiplication(Y, codomain(multiplication(codomain(X), Y))))
34.69/34.89	= { by lemma 123 }
34.69/34.89	  multiplication(X, multiplication(codomain(X), multiplication(Y, codomain(multiplication(codomain(X), Y)))))
34.69/34.89	= { by axiom 11 (multiplicative_associativity) }
34.69/34.89	  multiplication(X, multiplication(multiplication(codomain(X), Y), codomain(multiplication(codomain(X), Y))))
34.69/34.89	= { by lemma 79 }
34.69/34.89	  multiplication(X, multiplication(codomain(X), Y))
34.69/34.89	= { by lemma 123 }
34.69/34.89	  multiplication(X, Y)
34.69/34.89	
34.69/34.89	Lemma 147: multiplication(antidomain(addition(X, Y)), addition(X, Z)) = multiplication(antidomain(addition(X, Y)), Z).
34.69/34.89	Proof:
34.69/34.89	  multiplication(antidomain(addition(X, Y)), addition(X, Z))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  multiplication(antidomain(addition(X, Y)), addition(Z, X))
34.69/34.89	= { by lemma 77 }
34.69/34.89	  multiplication(antidomain(addition(X, Y)), addition(addition(X, Y), addition(Z, X)))
34.69/34.89	= { by lemma 63 }
34.69/34.89	  multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, addition(X, Y))))
34.69/34.89	= { by lemma 51 }
34.69/34.89	  multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, Y)))
34.69/34.89	= { by lemma 95 }
34.69/34.89	  multiplication(antidomain(addition(X, Y)), Z)
34.69/34.89	
34.69/34.89	Lemma 148: multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X))) = multiplication(codomain(X), Y).
34.69/34.89	Proof:
34.69/34.89	  multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X)))
34.69/34.89	= { by axiom 11 (multiplicative_associativity) }
34.69/34.89	  multiplication(multiplication(codomain(X), Y), backward_diamond(Y, X))
34.69/34.89	= { by axiom 24 (backward_diamond) }
34.69/34.89	  multiplication(multiplication(codomain(X), Y), codomain(multiplication(codomain(X), Y)))
34.69/34.89	= { by lemma 79 }
34.69/34.89	  multiplication(codomain(X), Y)
34.69/34.89	
34.69/34.89	Lemma 149: multiplication(coantidomain(X), antidomain(codomain(X))) = coantidomain(X).
34.69/34.89	Proof:
34.69/34.89	  multiplication(coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by lemma 35 }
34.69/34.89	  addition(zero, multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by lemma 134 }
34.69/34.89	  addition(multiplication(antidomain(multiplication(coantidomain(X), codomain(X))), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by lemma 47 }
34.69/34.89	  addition(multiplication(antidomain(zero), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by lemma 49 }
34.69/34.89	  addition(multiplication(one, multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.89	  addition(multiplication(coantidomain(X), domain(codomain(X))), multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by axiom 3 (right_distributivity) }
34.69/34.89	  multiplication(coantidomain(X), addition(domain(codomain(X)), antidomain(codomain(X))))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  multiplication(coantidomain(X), addition(antidomain(codomain(X)), domain(codomain(X))))
34.69/34.89	= { by lemma 48 }
34.69/34.89	  multiplication(coantidomain(X), one)
34.69/34.89	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.89	  coantidomain(X)
34.69/34.89	
34.69/34.89	Lemma 150: forward_box(coantidomain(X), codomain(X)) = antidomain(coantidomain(X)).
34.69/34.89	Proof:
34.69/34.89	  forward_box(coantidomain(X), codomain(X))
34.69/34.89	= { by lemma 111 }
34.69/34.89	  antidomain(multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.89	= { by lemma 149 }
34.69/34.89	  antidomain(coantidomain(X))
34.69/34.89	
34.69/34.89	Lemma 151: antidomain(codomain(X)) = coantidomain(X).
34.69/34.89	Proof:
34.69/34.89	  antidomain(codomain(X))
34.69/34.89	= { by axiom 2 (order_1) }
34.69/34.89	  $$fresh($$true, $$true, coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by lemma 138 }
34.69/34.89	  $$fresh(leq(multiplication(coantidomain(X), antidomain(codomain(X))), multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by lemma 149 }
34.69/34.89	  $$fresh(leq(coantidomain(X), multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by lemma 31 }
34.69/34.89	  $$fresh(leq(coantidomain(X), multiplication(one, antidomain(codomain(X)))), $$true, coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.89	  $$fresh(leq(coantidomain(X), antidomain(codomain(X))), $$true, coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by axiom 13 (order_1) }
34.69/34.89	  addition(coantidomain(X), antidomain(codomain(X)))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  addition(antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 13 (order_1) }
34.69/34.89	  $$fresh(leq(antidomain(codomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 20 (codomain4) }
34.69/34.89	  $$fresh(leq(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), one), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 31 }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), addition(coantidomain(X), coantidomain(coantidomain(X)))), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 95 }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(coantidomain(coantidomain(X))), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 20 (codomain4) }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), multiplication(one, coantidomain(X))), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 32 }
34.69/34.89	  $$fresh(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), multiplication(addition(antidomain(codomain(X)), antidomain(antidomain(codomain(X)))), coantidomain(X))), $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 138 }
34.69/34.89	  $$fresh($$true, $$true, antidomain(codomain(X)), coantidomain(X))
34.69/34.89	= { by axiom 2 (order_1) }
34.69/34.89	  coantidomain(X)
34.69/34.89	
34.69/34.89	Lemma 152: antidomain(coantidomain(X)) = codomain(X).
34.69/34.89	Proof:
34.69/34.89	  antidomain(coantidomain(X))
34.69/34.89	= { by lemma 140 }
34.69/34.89	  antidomain(codomain(coantidomain(X)))
34.69/34.89	= { by lemma 151 }
34.69/34.89	  coantidomain(coantidomain(X))
34.69/34.89	= { by axiom 20 (codomain4) }
34.69/34.89	  codomain(X)
34.69/34.89	
34.69/34.89	Lemma 153: backward_box(one, X) = coantidomain(antidomain(X)).
34.69/34.89	Proof:
34.69/34.89	  backward_box(one, X)
34.69/34.89	= { by lemma 70 }
34.69/34.89	  c(codomain(codomain(c(X))))
34.69/34.89	= { by lemma 96 }
34.69/34.89	  antidomain(codomain(codomain(c(X))))
34.69/34.89	= { by lemma 93 }
34.69/34.89	  antidomain(codomain(c(X)))
34.69/34.89	= { by lemma 96 }
34.69/34.89	  antidomain(codomain(antidomain(X)))
34.69/34.89	= { by lemma 151 }
34.69/34.89	  coantidomain(antidomain(X))
34.69/34.89	
34.69/34.89	Lemma 154: forward_box(codomain(X), coantidomain(X)) = coantidomain(X).
34.69/34.89	Proof:
34.69/34.89	  forward_box(codomain(X), coantidomain(X))
34.69/34.89	= { by axiom 20 (codomain4) }
34.69/34.89	  forward_box(coantidomain(coantidomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 140 }
34.69/34.89	  forward_box(coantidomain(coantidomain(X)), codomain(coantidomain(X)))
34.69/34.89	= { by lemma 150 }
34.69/34.89	  antidomain(coantidomain(coantidomain(X)))
34.69/34.89	= { by lemma 152 }
34.69/34.89	  codomain(coantidomain(X))
34.69/34.89	= { by lemma 140 }
34.69/34.89	  coantidomain(X)
34.69/34.89	
34.69/34.89	Lemma 155: forward_diamond(coantidomain(X), coantidomain(X)) = coantidomain(X).
34.69/34.89	Proof:
34.69/34.89	  forward_diamond(coantidomain(X), coantidomain(X))
34.69/34.89	= { by lemma 154 }
34.69/34.89	  forward_diamond(forward_box(codomain(X), coantidomain(X)), coantidomain(X))
34.69/34.89	= { by lemma 154 }
34.69/34.89	  forward_diamond(forward_box(codomain(X), coantidomain(X)), forward_box(codomain(X), coantidomain(X)))
34.69/34.89	= { by lemma 107 }
34.69/34.89	  forward_diamond(addition(forward_box(codomain(X), coantidomain(X)), antidomain(forward_box(codomain(X), coantidomain(X)))), forward_box(codomain(X), coantidomain(X)))
34.69/34.89	= { by lemma 89 }
34.69/34.89	  forward_diamond(one, forward_box(codomain(X), coantidomain(X)))
34.69/34.89	= { by lemma 41 }
34.69/34.89	  domain(domain(forward_box(codomain(X), coantidomain(X))))
34.69/34.89	= { by lemma 42 }
34.69/34.89	  antidomain(c(forward_box(codomain(X), coantidomain(X))))
34.69/34.89	= { by lemma 96 }
34.69/34.89	  antidomain(antidomain(forward_box(codomain(X), coantidomain(X))))
34.69/34.89	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.89	  multiplication(antidomain(antidomain(forward_box(codomain(X), coantidomain(X)))), one)
34.69/34.89	= { by lemma 89 }
34.69/34.89	  multiplication(antidomain(antidomain(forward_box(codomain(X), coantidomain(X)))), addition(forward_box(codomain(X), coantidomain(X)), antidomain(forward_box(codomain(X), coantidomain(X)))))
34.69/34.89	= { by lemma 95 }
34.69/34.89	  multiplication(antidomain(antidomain(forward_box(codomain(X), coantidomain(X)))), forward_box(codomain(X), coantidomain(X)))
34.69/34.89	= { by axiom 23 (domain4) }
34.69/34.89	  multiplication(domain(forward_box(codomain(X), coantidomain(X))), forward_box(codomain(X), coantidomain(X)))
34.69/34.89	= { by lemma 81 }
34.69/34.89	  forward_box(codomain(X), coantidomain(X))
34.69/34.89	= { by lemma 154 }
34.69/34.89	  coantidomain(X)
34.69/34.89	
34.69/34.89	Lemma 156: coantidomain(backward_diamond(X, Y)) = antidomain(backward_diamond(X, Y)).
34.69/34.89	Proof:
34.69/34.89	  coantidomain(backward_diamond(X, Y))
34.69/34.89	= { by lemma 116 }
34.69/34.89	  coantidomain(multiplication(codomain(Y), X))
34.69/34.89	= { by lemma 151 }
34.69/34.89	  antidomain(codomain(multiplication(codomain(Y), X)))
34.69/34.89	= { by axiom 24 (backward_diamond) }
34.69/34.89	  antidomain(backward_diamond(X, Y))
34.69/34.89	
34.69/34.89	Lemma 157: multiplication(coantidomain(X), antidomain(Y)) = domain_difference(coantidomain(X), Y).
34.69/34.89	Proof:
34.69/34.89	  multiplication(coantidomain(X), antidomain(Y))
34.69/34.89	= { by lemma 35 }
34.69/34.89	  addition(zero, multiplication(coantidomain(X), antidomain(Y)))
34.69/34.89	= { by lemma 47 }
34.69/34.89	  addition(multiplication(coantidomain(X), codomain(X)), multiplication(coantidomain(X), antidomain(Y)))
34.69/34.89	= { by axiom 3 (right_distributivity) }
34.69/34.89	  multiplication(coantidomain(X), addition(codomain(X), antidomain(Y)))
34.69/34.89	= { by lemma 155 }
34.69/34.89	  multiplication(forward_diamond(coantidomain(X), coantidomain(X)), addition(codomain(X), antidomain(Y)))
34.69/34.89	= { by lemma 97 }
34.69/34.89	  multiplication(domain(forward_diamond(coantidomain(X), coantidomain(X))), addition(codomain(X), antidomain(Y)))
34.69/34.89	= { by lemma 152 }
34.69/34.89	  multiplication(domain(forward_diamond(coantidomain(X), coantidomain(X))), addition(antidomain(coantidomain(X)), antidomain(Y)))
34.69/34.89	= { by lemma 135 }
34.69/34.89	  addition(domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), Y), multiplication(domain(forward_diamond(coantidomain(X), coantidomain(X))), antidomain(coantidomain(X))))
34.69/34.89	= { by axiom 28 (domain_difference) }
34.69/34.89	  addition(domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), Y), domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), coantidomain(X)))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  addition(domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), coantidomain(X)), domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), Y))
34.69/34.89	= { by lemma 130 }
34.69/34.89	  addition(zero, domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), Y))
34.69/34.89	= { by lemma 35 }
34.69/34.89	  domain_difference(forward_diamond(coantidomain(X), coantidomain(X)), Y)
34.69/34.89	= { by lemma 155 }
34.69/34.89	  domain_difference(coantidomain(X), Y)
34.69/34.89	
34.69/34.89	Lemma 158: multiplication(domain(X), codomain(Y)) = domain_difference(X, coantidomain(Y)).
34.69/34.89	Proof:
34.69/34.89	  multiplication(domain(X), codomain(Y))
34.69/34.89	= { by lemma 35 }
34.69/34.89	  addition(zero, multiplication(domain(X), codomain(Y)))
34.69/34.89	= { by lemma 47 }
34.69/34.89	  addition(multiplication(coantidomain(Y), codomain(Y)), multiplication(domain(X), codomain(Y)))
34.69/34.89	= { by axiom 4 (left_distributivity) }
34.69/34.89	  multiplication(addition(coantidomain(Y), domain(X)), codomain(Y))
34.69/34.89	= { by lemma 155 }
34.69/34.89	  multiplication(addition(forward_diamond(coantidomain(Y), coantidomain(Y)), domain(X)), codomain(Y))
34.69/34.89	= { by lemma 97 }
34.69/34.89	  multiplication(addition(domain(forward_diamond(coantidomain(Y), coantidomain(Y))), domain(X)), codomain(Y))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  multiplication(addition(domain(X), domain(forward_diamond(coantidomain(Y), coantidomain(Y)))), codomain(Y))
34.69/34.89	= { by lemma 152 }
34.69/34.89	  multiplication(addition(domain(X), domain(forward_diamond(coantidomain(Y), coantidomain(Y)))), antidomain(coantidomain(Y)))
34.69/34.89	= { by axiom 4 (left_distributivity) }
34.69/34.89	  addition(multiplication(domain(X), antidomain(coantidomain(Y))), multiplication(domain(forward_diamond(coantidomain(Y), coantidomain(Y))), antidomain(coantidomain(Y))))
34.69/34.89	= { by axiom 28 (domain_difference) }
34.69/34.89	  addition(domain_difference(X, coantidomain(Y)), multiplication(domain(forward_diamond(coantidomain(Y), coantidomain(Y))), antidomain(coantidomain(Y))))
34.69/34.89	= { by axiom 28 (domain_difference) }
34.69/34.89	  addition(domain_difference(X, coantidomain(Y)), domain_difference(forward_diamond(coantidomain(Y), coantidomain(Y)), coantidomain(Y)))
34.69/34.89	= { by axiom 5 (additive_commutativity) }
34.69/34.89	  addition(domain_difference(forward_diamond(coantidomain(Y), coantidomain(Y)), coantidomain(Y)), domain_difference(X, coantidomain(Y)))
34.69/34.89	= { by lemma 130 }
34.69/34.89	  addition(zero, domain_difference(X, coantidomain(Y)))
34.69/34.89	= { by lemma 35 }
34.69/34.89	  domain_difference(X, coantidomain(Y))
34.69/34.89	
34.69/34.89	Lemma 159: domain_difference(antidomain(addition(X, domain(Y))), Y) = antidomain(addition(X, domain(Y))).
34.69/34.89	Proof:
34.69/34.89	  domain_difference(antidomain(addition(X, domain(Y))), Y)
34.69/34.89	= { by lemma 117 }
34.69/34.89	  multiplication(antidomain(addition(X, domain(Y))), antidomain(Y))
34.69/34.89	= { by lemma 95 }
34.69/34.90	  multiplication(antidomain(addition(X, domain(Y))), addition(antidomain(Y), addition(X, domain(Y))))
34.69/34.90	= { by lemma 64 }
34.69/34.90	  multiplication(antidomain(addition(X, domain(Y))), addition(X, addition(antidomain(Y), domain(Y))))
34.69/34.90	= { by lemma 48 }
34.69/34.90	  multiplication(antidomain(addition(X, domain(Y))), addition(X, one))
34.69/34.90	= { by lemma 147 }
34.69/34.90	  multiplication(antidomain(addition(X, domain(Y))), one)
34.69/34.90	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.90	  antidomain(addition(X, domain(Y)))
34.69/34.90	
34.69/34.90	Lemma 160: multiplication(domain(X), coantidomain(Y)) = domain_difference(X, codomain(Y)).
34.69/34.90	Proof:
34.69/34.90	  multiplication(domain(X), coantidomain(Y))
34.69/34.90	= { by lemma 100 }
34.69/34.90	  multiplication(domain(X), backward_diamond(coantidomain(Y), coantidomain(Y)))
34.69/34.90	= { by lemma 126 }
34.69/34.90	  multiplication(domain_difference(X, antidomain(backward_diamond(coantidomain(Y), coantidomain(Y)))), backward_diamond(coantidomain(Y), coantidomain(Y)))
34.69/34.90	= { by lemma 156 }
34.69/34.90	  multiplication(domain_difference(X, coantidomain(backward_diamond(coantidomain(Y), coantidomain(Y)))), backward_diamond(coantidomain(Y), coantidomain(Y)))
34.69/34.90	= { by lemma 116 }
34.69/34.90	  multiplication(domain_difference(X, coantidomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y)))), backward_diamond(coantidomain(Y), coantidomain(Y)))
34.69/34.90	= { by axiom 24 (backward_diamond) }
34.69/34.90	  multiplication(domain_difference(X, coantidomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y)))), codomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y))))
34.69/34.90	= { by lemma 113 }
34.69/34.90	  multiplication(domain(X), multiplication(antidomain(coantidomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y)))), codomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y)))))
34.69/34.90	= { by lemma 86 }
34.69/34.90	  multiplication(domain(X), antidomain(coantidomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y)))))
34.69/34.90	= { by axiom 28 (domain_difference) }
34.69/34.90	  domain_difference(X, coantidomain(multiplication(codomain(coantidomain(Y)), coantidomain(Y))))
34.69/34.90	= { by lemma 116 }
34.69/34.90	  domain_difference(X, coantidomain(backward_diamond(coantidomain(Y), coantidomain(Y))))
34.69/34.90	= { by lemma 156 }
34.69/34.90	  domain_difference(X, antidomain(backward_diamond(coantidomain(Y), coantidomain(Y))))
34.69/34.90	= { by lemma 100 }
34.69/34.90	  domain_difference(X, antidomain(coantidomain(Y)))
34.69/34.90	= { by lemma 152 }
34.69/34.90	  domain_difference(X, codomain(Y))
34.69/34.90	
34.69/34.90	Lemma 161: multiplication(codomain(X), antidomain(Y)) = domain_difference(codomain(X), Y).
34.69/34.90	Proof:
34.69/34.90	  multiplication(codomain(X), antidomain(Y))
34.69/34.90	= { by axiom 20 (codomain4) }
34.69/34.90	  multiplication(coantidomain(coantidomain(X)), antidomain(Y))
34.69/34.90	= { by lemma 157 }
34.69/34.90	  domain_difference(coantidomain(coantidomain(X)), Y)
34.69/34.90	= { by axiom 20 (codomain4) }
34.69/34.90	  domain_difference(codomain(X), Y)
34.69/34.90	
34.69/34.90	Lemma 162: backward_box(coantidomain(antidomain(X)), codomain(antidomain(X))) = codomain(antidomain(X)).
34.69/34.90	Proof:
34.69/34.90	  backward_box(coantidomain(antidomain(X)), codomain(antidomain(X)))
34.69/34.90	= { by lemma 119 }
34.69/34.90	  antidomain(backward_diamond(coantidomain(antidomain(X)), backward_box(one, X)))
34.69/34.90	= { by lemma 115 }
34.69/34.90	  antidomain(backward_diamond(addition(coantidomain(antidomain(X)), coantidomain(backward_box(one, X))), backward_box(one, X)))
34.69/34.90	= { by lemma 119 }
34.69/34.90	  backward_box(addition(coantidomain(antidomain(X)), coantidomain(backward_box(one, X))), codomain(antidomain(X)))
34.69/34.90	= { by lemma 153 }
34.69/34.90	  backward_box(addition(coantidomain(antidomain(X)), coantidomain(coantidomain(antidomain(X)))), codomain(antidomain(X)))
34.69/34.90	= { by axiom 20 (codomain4) }
34.69/34.90	  backward_box(addition(coantidomain(antidomain(X)), codomain(antidomain(X))), codomain(antidomain(X)))
34.69/34.90	= { by lemma 45 }
34.69/34.90	  backward_box(one, codomain(antidomain(X)))
34.69/34.90	= { by lemma 119 }
34.69/34.90	  antidomain(backward_diamond(one, backward_box(one, X)))
34.69/34.90	= { by lemma 40 }
34.69/34.90	  antidomain(codomain(codomain(backward_box(one, X))))
34.69/34.90	= { by lemma 93 }
34.69/34.90	  antidomain(codomain(backward_box(one, X)))
34.69/34.90	= { by lemma 151 }
34.69/34.90	  coantidomain(backward_box(one, X))
34.69/34.90	= { by lemma 153 }
34.69/34.90	  coantidomain(coantidomain(antidomain(X)))
34.69/34.90	= { by axiom 20 (codomain4) }
34.69/34.90	  codomain(antidomain(X))
34.69/34.90	
34.69/34.90	Lemma 163: addition(antidomain(X), antidomain(addition(Y, domain(X)))) = antidomain(X).
34.69/34.90	Proof:
34.69/34.90	  addition(antidomain(X), antidomain(addition(Y, domain(X))))
34.69/34.90	= { by lemma 159 }
34.69/34.90	  addition(antidomain(X), domain_difference(antidomain(addition(Y, domain(X))), X))
34.69/34.90	= { by lemma 131 }
34.69/34.90	  antidomain(X)
34.69/34.90	
34.69/34.90	Lemma 164: domain_difference(codomain(X), antidomain(Y)) = multiplication(codomain(X), domain(Y)).
34.69/34.90	Proof:
34.69/34.90	  domain_difference(codomain(X), antidomain(Y))
34.69/34.90	= { by lemma 161 }
34.69/34.90	  multiplication(codomain(X), antidomain(antidomain(Y)))
34.69/34.90	= { by axiom 23 (domain4) }
34.69/34.90	  multiplication(codomain(X), domain(Y))
34.69/34.90	
34.69/34.90	Lemma 165: backward_diamond(backward_diamond(X, coantidomain(Y)), multiplication(coantidomain(Y), X)) = backward_diamond(X, coantidomain(Y)).
34.69/34.90	Proof:
34.69/34.90	  backward_diamond(backward_diamond(X, coantidomain(Y)), multiplication(coantidomain(Y), X))
34.69/34.90	= { by lemma 115 }
34.69/34.90	  backward_diamond(addition(backward_diamond(X, coantidomain(Y)), coantidomain(multiplication(coantidomain(Y), X))), multiplication(coantidomain(Y), X))
34.69/34.90	= { by axiom 5 (additive_commutativity) }
34.69/34.90	  backward_diamond(addition(coantidomain(multiplication(coantidomain(Y), X)), backward_diamond(X, coantidomain(Y))), multiplication(coantidomain(Y), X))
34.69/34.90	= { by lemma 99 }
34.69/34.90	  backward_diamond(addition(coantidomain(multiplication(coantidomain(Y), X)), codomain(multiplication(coantidomain(Y), X))), multiplication(coantidomain(Y), X))
34.69/34.90	= { by lemma 45 }
34.69/34.90	  backward_diamond(one, multiplication(coantidomain(Y), X))
34.69/34.90	= { by axiom 24 (backward_diamond) }
34.69/34.90	  codomain(multiplication(codomain(multiplication(coantidomain(Y), X)), one))
34.69/34.90	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.90	  codomain(codomain(multiplication(coantidomain(Y), X)))
34.69/34.90	= { by lemma 93 }
34.69/34.90	  codomain(multiplication(coantidomain(Y), X))
34.69/34.90	= { by lemma 99 }
34.69/34.90	  backward_diamond(X, coantidomain(Y))
34.69/34.90	
34.69/34.90	Lemma 166: coantidomain(multiplication(codomain(X), Y)) = antidomain(backward_diamond(Y, X)).
34.69/34.90	Proof:
34.69/34.90	  coantidomain(multiplication(codomain(X), Y))
34.69/34.90	= { by lemma 92 }
34.69/34.90	  coantidomain(codomain(multiplication(codomain(X), Y)))
34.69/34.90	= { by axiom 24 (backward_diamond) }
34.69/34.90	  coantidomain(backward_diamond(Y, X))
34.69/34.90	= { by lemma 156 }
34.69/34.90	  antidomain(backward_diamond(Y, X))
34.69/34.90	
34.69/34.90	Lemma 167: antidomain(backward_box(X, antidomain(Y))) = backward_diamond(X, domain(Y)).
34.69/34.90	Proof:
34.69/34.90	  antidomain(backward_box(X, antidomain(Y)))
34.69/34.90	= { by lemma 118 }
34.69/34.90	  antidomain(antidomain(backward_diamond(X, domain(Y))))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  antidomain(c(backward_diamond(X, domain(Y))))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  c(c(backward_diamond(X, domain(Y))))
34.69/34.90	= { by lemma 39 }
34.69/34.90	  domain(antidomain(c(backward_diamond(X, domain(Y)))))
34.69/34.90	= { by lemma 43 }
34.69/34.90	  domain(c(antidomain(backward_diamond(X, domain(Y)))))
34.69/34.90	= { by axiom 23 (domain4) }
34.69/34.90	  antidomain(antidomain(c(antidomain(backward_diamond(X, domain(Y))))))
34.69/34.90	= { by lemma 42 }
34.69/34.90	  antidomain(domain(domain(antidomain(backward_diamond(X, domain(Y))))))
34.69/34.90	= { by axiom 23 (domain4) }
34.69/34.90	  antidomain(domain(antidomain(antidomain(antidomain(backward_diamond(X, domain(Y)))))))
34.69/34.90	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.90	  antidomain(domain(multiplication(one, antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))))))
34.69/34.90	= { by lemma 32 }
34.69/34.90	  antidomain(domain(multiplication(addition(antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))), antidomain(antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))))), antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))))))
34.69/34.90	= { by lemma 69 }
34.69/34.90	  antidomain(domain(multiplication(antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))), antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))))))
34.69/34.90	= { by axiom 23 (domain4) }
34.69/34.90	  antidomain(domain(multiplication(domain(antidomain(backward_diamond(X, domain(Y)))), antidomain(antidomain(antidomain(backward_diamond(X, domain(Y))))))))
34.69/34.90	= { by axiom 28 (domain_difference) }
34.69/34.90	  antidomain(domain(domain_difference(antidomain(backward_diamond(X, domain(Y))), antidomain(antidomain(backward_diamond(X, domain(Y)))))))
34.69/34.90	= { by lemma 75 }
34.69/34.90	  antidomain(forward_diamond(domain(antidomain(backward_diamond(X, domain(Y)))), antidomain(backward_diamond(X, domain(Y)))))
34.69/34.90	= { by lemma 104 }
34.69/34.90	  forward_box(domain(antidomain(backward_diamond(X, domain(Y)))), backward_diamond(X, domain(Y)))
34.69/34.90	= { by lemma 39 }
34.69/34.90	  forward_box(c(backward_diamond(X, domain(Y))), backward_diamond(X, domain(Y)))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  forward_box(antidomain(backward_diamond(X, domain(Y))), backward_diamond(X, domain(Y)))
34.69/34.90	= { by lemma 156 }
34.69/34.90	  forward_box(coantidomain(backward_diamond(X, domain(Y))), backward_diamond(X, domain(Y)))
34.69/34.90	= { by lemma 116 }
34.69/34.90	  forward_box(coantidomain(multiplication(codomain(domain(Y)), X)), backward_diamond(X, domain(Y)))
34.69/34.90	= { by axiom 24 (backward_diamond) }
34.69/34.90	  forward_box(coantidomain(multiplication(codomain(domain(Y)), X)), codomain(multiplication(codomain(domain(Y)), X)))
34.69/34.90	= { by lemma 150 }
34.69/34.90	  antidomain(coantidomain(multiplication(codomain(domain(Y)), X)))
34.69/34.90	= { by lemma 152 }
34.69/34.90	  codomain(multiplication(codomain(domain(Y)), X))
34.69/34.90	= { by axiom 24 (backward_diamond) }
34.69/34.90	  backward_diamond(X, domain(Y))
34.69/34.90	
34.69/34.90	Lemma 168: backward_box(X, zero) = coantidomain(X).
34.69/34.90	Proof:
34.69/34.90	  backward_box(X, zero)
34.69/34.90	= { by axiom 26 (backward_box) }
34.69/34.90	  c(backward_diamond(X, c(zero)))
34.69/34.90	= { by lemma 50 }
34.69/34.90	  c(backward_diamond(X, one))
34.69/34.90	= { by lemma 83 }
34.69/34.90	  c(codomain(X))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  antidomain(codomain(X))
34.69/34.90	= { by lemma 151 }
34.69/34.90	  coantidomain(X)
34.69/34.90	
34.69/34.90	Lemma 169: antidomain(multiplication(X, coantidomain(antidomain(Y)))) = forward_box(X, codomain(antidomain(Y))).
34.69/34.90	Proof:
34.69/34.90	  antidomain(multiplication(X, coantidomain(antidomain(Y))))
34.69/34.90	= { by lemma 153 }
34.69/34.90	  antidomain(multiplication(X, backward_box(one, Y)))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  c(multiplication(X, backward_box(one, Y)))
34.69/34.90	= { by axiom 29 (complement) }
34.69/34.90	  antidomain(domain(multiplication(X, backward_box(one, Y))))
34.69/34.90	= { by lemma 137 }
34.69/34.90	  antidomain(forward_diamond(X, backward_box(one, Y)))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  c(forward_diamond(X, backward_box(one, Y)))
34.69/34.90	= { by lemma 84 }
34.69/34.90	  forward_box(X, backward_diamond(one, c(Y)))
34.69/34.90	= { by lemma 40 }
34.69/34.90	  forward_box(X, codomain(codomain(c(Y))))
34.69/34.90	= { by lemma 93 }
34.69/34.90	  forward_box(X, codomain(c(Y)))
34.69/34.90	= { by lemma 96 }
34.69/34.90	  forward_box(X, codomain(antidomain(Y)))
34.69/34.90	
34.69/34.90	Lemma 170: forward_box(coantidomain(X), codomain(X)) = codomain(X).
34.69/34.90	Proof:
34.69/34.90	  forward_box(coantidomain(X), codomain(X))
34.69/34.90	= { by lemma 111 }
34.69/34.90	  antidomain(multiplication(coantidomain(X), antidomain(codomain(X))))
34.69/34.90	= { by lemma 149 }
34.69/34.90	  antidomain(coantidomain(X))
34.69/34.90	= { by lemma 152 }
34.69/34.90	  codomain(X)
34.69/34.90	
34.69/34.90	Lemma 171: codomain(antidomain(backward_diamond(X, Y))) = antidomain(backward_diamond(X, Y)).
34.69/34.90	Proof:
34.69/34.90	  codomain(antidomain(backward_diamond(X, Y)))
34.69/34.90	= { by lemma 82 }
34.69/34.90	  codomain(domain_difference(antidomain(backward_diamond(X, Y)), backward_diamond(X, Y)))
34.69/34.90	= { by lemma 166 }
34.69/34.90	  codomain(domain_difference(coantidomain(multiplication(codomain(Y), X)), backward_diamond(X, Y)))
34.69/34.90	= { by lemma 157 }
34.69/34.90	  codomain(multiplication(coantidomain(multiplication(codomain(Y), X)), antidomain(backward_diamond(X, Y))))
34.69/34.90	= { by lemma 99 }
34.69/34.90	  backward_diamond(antidomain(backward_diamond(X, Y)), coantidomain(multiplication(codomain(Y), X)))
34.69/34.90	= { by lemma 166 }
34.69/34.90	  backward_diamond(coantidomain(multiplication(codomain(Y), X)), coantidomain(multiplication(codomain(Y), X)))
34.69/34.90	= { by lemma 100 }
34.69/34.90	  coantidomain(multiplication(codomain(Y), X))
34.69/34.90	= { by lemma 166 }
34.69/34.91	  antidomain(backward_diamond(X, Y))
34.69/34.91	
34.69/34.91	Lemma 172: backward_diamond(backward_box(X, Y), backward_box(X, Y)) = backward_box(X, Y).
34.69/34.91	Proof:
34.69/34.91	  backward_diamond(backward_box(X, Y), backward_box(X, Y))
34.69/34.91	= { by lemma 106 }
34.69/34.91	  backward_diamond(backward_box(X, domain(Y)), backward_box(X, Y))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  backward_diamond(backward_box(X, antidomain(antidomain(Y))), backward_box(X, Y))
34.69/34.91	= { by lemma 118 }
34.69/34.91	  backward_diamond(antidomain(backward_diamond(X, domain(antidomain(Y)))), backward_box(X, Y))
34.69/34.91	= { by lemma 171 }
34.69/34.91	  backward_diamond(codomain(antidomain(backward_diamond(X, domain(antidomain(Y))))), backward_box(X, Y))
34.69/34.91	= { by lemma 106 }
34.69/34.91	  backward_diamond(codomain(antidomain(backward_diamond(X, domain(antidomain(Y))))), backward_box(X, domain(Y)))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  backward_diamond(codomain(antidomain(backward_diamond(X, domain(antidomain(Y))))), backward_box(X, antidomain(antidomain(Y))))
34.69/34.91	= { by lemma 98 }
34.69/34.91	  backward_diamond(codomain(antidomain(backward_diamond(X, domain(antidomain(Y))))), domain(backward_box(X, antidomain(antidomain(Y)))))
34.69/34.91	= { by lemma 167 }
34.69/34.91	  antidomain(backward_box(codomain(antidomain(backward_diamond(X, domain(antidomain(Y))))), antidomain(backward_box(X, antidomain(antidomain(Y))))))
34.69/34.91	= { by lemma 96 }
34.69/34.91	  antidomain(backward_box(codomain(c(backward_diamond(X, domain(antidomain(Y))))), antidomain(backward_box(X, antidomain(antidomain(Y))))))
34.69/34.91	= { by lemma 167 }
34.69/34.91	  antidomain(backward_box(codomain(c(backward_diamond(X, domain(antidomain(Y))))), backward_diamond(X, domain(antidomain(Y)))))
34.69/34.91	= { by axiom 26 (backward_box) }
34.69/34.91	  antidomain(c(backward_diamond(codomain(c(backward_diamond(X, domain(antidomain(Y))))), c(backward_diamond(X, domain(antidomain(Y)))))))
34.69/34.91	= { by axiom 24 (backward_diamond) }
34.69/34.91	  antidomain(c(codomain(multiplication(codomain(c(backward_diamond(X, domain(antidomain(Y))))), codomain(c(backward_diamond(X, domain(antidomain(Y)))))))))
34.69/34.91	= { by lemma 67 }
34.69/34.91	  antidomain(c(codomain(multiplication(codomain(c(backward_diamond(X, domain(antidomain(Y))))), addition(codomain(c(backward_diamond(X, domain(antidomain(Y))))), coantidomain(codomain(c(backward_diamond(X, domain(antidomain(Y)))))))))))
34.69/34.91	= { by lemma 60 }
34.69/34.91	  antidomain(c(codomain(multiplication(codomain(c(backward_diamond(X, domain(antidomain(Y))))), one))))
34.69/34.91	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.91	  antidomain(c(codomain(codomain(c(backward_diamond(X, domain(antidomain(Y))))))))
34.69/34.91	= { by lemma 70 }
34.69/34.91	  antidomain(backward_box(one, backward_diamond(X, domain(antidomain(Y)))))
34.69/34.91	= { by lemma 153 }
34.69/34.91	  antidomain(coantidomain(antidomain(backward_diamond(X, domain(antidomain(Y))))))
34.69/34.91	= { by lemma 152 }
34.69/34.91	  codomain(antidomain(backward_diamond(X, domain(antidomain(Y)))))
34.69/34.91	= { by lemma 171 }
34.69/34.91	  antidomain(backward_diamond(X, domain(antidomain(Y))))
34.69/34.91	= { by lemma 118 }
34.69/34.91	  backward_box(X, antidomain(antidomain(Y)))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  backward_box(X, domain(Y))
34.69/34.91	= { by lemma 106 }
34.69/34.91	  backward_box(X, Y)
34.69/34.91	
34.69/34.91	Lemma 173: coantidomain(antidomain(backward_box(X, Y))) = backward_box(X, Y).
34.69/34.91	Proof:
34.69/34.91	  coantidomain(antidomain(backward_box(X, Y)))
34.69/34.91	= { by lemma 172 }
34.69/34.91	  coantidomain(antidomain(backward_diamond(backward_box(X, Y), backward_box(X, Y))))
34.69/34.91	= { by lemma 94 }
34.69/34.91	  coantidomain(antidomain(backward_diamond(backward_box(X, Y), codomain(backward_box(X, Y)))))
34.69/34.91	= { by axiom 20 (codomain4) }
34.69/34.91	  coantidomain(antidomain(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y))))))
34.69/34.91	= { by lemma 165 }
34.69/34.91	  coantidomain(antidomain(backward_diamond(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))), multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y)))))
34.69/34.91	= { by lemma 156 }
34.69/34.91	  coantidomain(coantidomain(backward_diamond(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))), multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y)))))
34.69/34.91	= { by lemma 139 }
34.69/34.91	  coantidomain(addition(coantidomain(multiplication(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y)), backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))))), coantidomain(backward_diamond(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))), multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))))))
34.69/34.91	= { by axiom 11 (multiplicative_associativity) }
34.69/34.91	  coantidomain(addition(coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), multiplication(backward_box(X, Y), backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y))))))), coantidomain(backward_diamond(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))), multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))))))
34.69/34.91	= { by lemma 146 }
34.69/34.91	  coantidomain(addition(coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))), coantidomain(backward_diamond(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))), multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))))))
34.69/34.91	= { by lemma 165 }
34.69/34.91	  coantidomain(addition(coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))), coantidomain(backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y)))))))
34.69/34.91	= { by lemma 120 }
34.69/34.91	  coantidomain(addition(coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))), coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y)))))
34.69/34.91	= { by axiom 15 (additive_idempotence) }
34.69/34.91	  coantidomain(coantidomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y))))
34.69/34.91	= { by axiom 20 (codomain4) }
34.69/34.91	  codomain(multiplication(coantidomain(coantidomain(backward_box(X, Y))), backward_box(X, Y)))
34.69/34.91	= { by lemma 99 }
34.69/34.91	  backward_diamond(backward_box(X, Y), coantidomain(coantidomain(backward_box(X, Y))))
34.69/34.91	= { by axiom 20 (codomain4) }
34.69/34.91	  backward_diamond(backward_box(X, Y), codomain(backward_box(X, Y)))
34.69/34.91	= { by lemma 94 }
34.69/34.91	  backward_diamond(backward_box(X, Y), backward_box(X, Y))
34.69/34.91	= { by lemma 172 }
34.69/34.91	  backward_box(X, Y)
34.69/34.91	
34.69/34.91	Lemma 174: multiplication(antidomain(X), domain_difference(Y, X)) = domain_difference(Y, X).
34.69/34.91	Proof:
34.69/34.91	  multiplication(antidomain(X), domain_difference(Y, X))
34.69/34.91	= { by lemma 35 }
34.69/34.91	  addition(zero, multiplication(antidomain(X), domain_difference(Y, X)))
34.69/34.91	= { by lemma 132 }
34.69/34.91	  addition(multiplication(domain(X), domain_difference(Y, X)), multiplication(antidomain(X), domain_difference(Y, X)))
34.69/34.91	= { by axiom 4 (left_distributivity) }
34.69/34.91	  multiplication(addition(domain(X), antidomain(X)), domain_difference(Y, X))
34.69/34.91	= { by axiom 5 (additive_commutativity) }
34.69/34.91	  multiplication(addition(antidomain(X), domain(X)), domain_difference(Y, X))
34.69/34.91	= { by lemma 48 }
34.69/34.91	  multiplication(one, domain_difference(Y, X))
34.69/34.91	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.91	  domain_difference(Y, X)
34.69/34.91	
34.69/34.91	Lemma 175: multiplication(antidomain(Y), domain(X)) = domain_difference(X, Y).
34.69/34.91	Proof:
34.69/34.91	  multiplication(antidomain(Y), domain(X))
34.69/34.91	= { by lemma 143 }
34.69/34.91	  domain_difference(antidomain(Y), antidomain(X))
34.69/34.91	= { by lemma 174 }
34.69/34.91	  multiplication(antidomain(antidomain(X)), domain_difference(antidomain(Y), antidomain(X)))
34.69/34.91	= { by lemma 143 }
34.69/34.91	  multiplication(antidomain(antidomain(X)), multiplication(antidomain(Y), domain(X)))
34.69/34.91	= { by lemma 128 }
34.69/34.91	  multiplication(domain_difference(antidomain(antidomain(X)), Y), domain(X))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  multiplication(domain_difference(domain(X), Y), domain(X))
34.69/34.91	= { by lemma 73 }
34.69/34.91	  multiplication(domain_difference(forward_diamond(X, one), Y), domain(X))
34.69/34.91	= { by lemma 73 }
34.69/34.91	  multiplication(domain_difference(forward_diamond(X, one), Y), forward_diamond(X, one))
34.69/34.91	= { by lemma 113 }
34.69/34.91	  multiplication(domain(forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one)))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  multiplication(antidomain(antidomain(forward_diamond(X, one))), multiplication(antidomain(Y), forward_diamond(X, one)))
34.69/34.91	= { by lemma 95 }
34.69/34.91	  multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(multiplication(antidomain(Y), forward_diamond(X, one)), antidomain(forward_diamond(X, one))))
34.69/34.91	= { by lemma 131 }
34.69/34.91	  multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(multiplication(antidomain(Y), forward_diamond(X, one)), addition(antidomain(forward_diamond(X, one)), domain_difference(antidomain(Y), forward_diamond(X, one)))))
34.69/34.91	= { by lemma 63 }
34.69/34.91	  multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(antidomain(forward_diamond(X, one)), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one)))))
34.69/34.91	= { by lemma 77 }
34.69/34.91	  multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one))))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  multiplication(domain(forward_diamond(X, one)), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one))))
34.69/34.91	= { by lemma 141 }
34.69/34.91	  multiplication(domain(forward_diamond(X, one)), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(domain(antidomain(Y)), forward_diamond(X, one))))
34.69/34.91	= { by lemma 135 }
34.69/34.91	  multiplication(domain(forward_diamond(X, one)), multiplication(domain(antidomain(Y)), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one)))))
34.69/34.91	= { by lemma 141 }
34.69/34.91	  multiplication(domain(forward_diamond(X, one)), multiplication(antidomain(Y), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one)))))
34.69/34.91	= { by lemma 113 }
34.69/34.91	  multiplication(domain_difference(forward_diamond(X, one), Y), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one))))
34.69/34.91	= { by lemma 87 }
34.69/34.91	  multiplication(domain_difference(forward_diamond(X, one), Y), one)
34.69/34.91	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.91	  domain_difference(forward_diamond(X, one), Y)
34.69/34.91	= { by lemma 73 }
34.69/34.91	  domain_difference(domain(X), Y)
34.69/34.91	= { by lemma 102 }
34.69/34.91	  domain_difference(X, Y)
34.69/34.91	
34.69/34.91	Lemma 176: domain_difference(X, antidomain(Y)) = domain_difference(Y, antidomain(X)).
34.69/34.91	Proof:
34.69/34.91	  domain_difference(X, antidomain(Y))
34.69/34.91	= { by lemma 175 }
34.69/34.91	  multiplication(antidomain(antidomain(Y)), domain(X))
34.69/34.91	= { by axiom 23 (domain4) }
34.69/34.91	  multiplication(domain(Y), domain(X))
34.69/34.91	= { by lemma 57 }
34.69/34.91	  domain_difference(Y, antidomain(X))
34.69/34.91	
34.69/34.91	Lemma 177: forward_diamond(antidomain(X), Y) = domain(domain_difference(Y, X)).
34.69/34.91	Proof:
34.69/34.91	  forward_diamond(antidomain(X), Y)
34.69/34.91	= { by axiom 27 (forward_diamond) }
34.69/34.91	  domain(multiplication(antidomain(X), domain(Y)))
34.69/34.91	= { by lemma 175 }
34.69/34.91	  domain(domain_difference(Y, X))
34.69/34.91	
34.69/34.91	Lemma 178: domain_difference(antidomain(X), Y) = domain_difference(antidomain(Y), X).
34.69/34.91	Proof:
34.69/34.91	  domain_difference(antidomain(X), Y)
34.69/34.91	= { by lemma 175 }
34.69/34.91	  multiplication(antidomain(Y), domain(antidomain(X)))
34.69/34.91	= { by lemma 141 }
34.69/34.91	  multiplication(antidomain(Y), antidomain(X))
34.69/34.91	= { by lemma 117 }
34.69/34.91	  domain_difference(antidomain(Y), X)
34.69/34.91	
34.69/34.91	Lemma 179: multiplication(codomain(X), domain(Y)) = domain_difference(Y, coantidomain(X)).
34.69/34.91	Proof:
34.69/34.91	  multiplication(codomain(X), domain(Y))
34.69/34.91	= { by lemma 164 }
34.69/34.91	  domain_difference(codomain(X), antidomain(Y))
34.69/34.91	= { by lemma 176 }
34.69/34.91	  domain_difference(Y, antidomain(codomain(X)))
34.69/34.91	= { by lemma 151 }
34.69/34.91	  domain_difference(Y, coantidomain(X))
34.69/34.91	
34.69/34.91	Lemma 180: multiplication(antidomain(Y), codomain(X)) = domain_difference(codomain(X), Y).
34.69/34.91	Proof:
34.69/34.91	  multiplication(antidomain(Y), codomain(X))
34.69/34.91	= { by lemma 141 }
34.69/34.91	  multiplication(domain(antidomain(Y)), codomain(X))
34.69/34.91	= { by lemma 158 }
34.69/34.91	  domain_difference(antidomain(Y), coantidomain(X))
34.69/34.91	= { by lemma 178 }
34.69/34.91	  domain_difference(antidomain(coantidomain(X)), Y)
34.69/34.91	= { by lemma 152 }
34.69/34.92	  domain_difference(codomain(X), Y)
34.69/34.92	
34.69/34.92	Lemma 181: addition(antidomain(X), coantidomain(addition(Y, domain(X)))) = antidomain(X).
34.69/34.92	Proof:
34.69/34.92	  addition(antidomain(X), coantidomain(addition(Y, domain(X))))
34.69/34.92	= { by axiom 5 (additive_commutativity) }
34.69/34.92	  addition(antidomain(X), coantidomain(addition(domain(X), Y)))
34.69/34.92	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.92	  addition(antidomain(X), multiplication(one, coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 48 }
34.69/34.92	  addition(antidomain(X), multiplication(addition(antidomain(X), domain(X)), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 76 }
34.69/34.92	  addition(antidomain(X), multiplication(addition(addition(domain(X), Y), addition(antidomain(X), domain(X))), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 63 }
34.69/34.92	  addition(antidomain(X), multiplication(addition(antidomain(X), addition(domain(X), addition(domain(X), Y))), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 51 }
34.69/34.92	  addition(antidomain(X), multiplication(addition(antidomain(X), addition(domain(X), Y)), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 91 }
34.69/34.92	  addition(antidomain(X), multiplication(antidomain(X), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 141 }
34.69/34.92	  addition(antidomain(X), multiplication(domain(antidomain(X)), coantidomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 160 }
34.69/34.92	  addition(antidomain(X), domain_difference(antidomain(X), codomain(addition(domain(X), Y))))
34.69/34.92	= { by lemma 178 }
34.69/34.92	  addition(antidomain(X), domain_difference(antidomain(codomain(addition(domain(X), Y))), X))
34.69/34.92	= { by lemma 151 }
34.69/34.92	  addition(antidomain(X), domain_difference(coantidomain(addition(domain(X), Y)), X))
34.69/34.92	= { by axiom 5 (additive_commutativity) }
34.69/34.92	  addition(antidomain(X), domain_difference(coantidomain(addition(Y, domain(X))), X))
34.69/34.92	= { by lemma 131 }
34.69/34.93	  antidomain(X)
34.69/34.93	
34.69/34.93	Lemma 182: forward_diamond(domain(Z), backward_box(X, Y)) = forward_diamond(backward_box(X, Y), Z).
34.69/34.93	Proof:
34.69/34.93	  forward_diamond(domain(Z), backward_box(X, Y))
34.69/34.93	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.93	  forward_diamond(multiplication(domain(Z), one), backward_box(X, Y))
34.69/34.93	= { by lemma 90 }
34.69/34.93	  forward_diamond(multiplication(domain(Z), addition(backward_box(X, Y), antidomain(backward_box(X, Y)))), backward_box(X, Y))
34.69/34.93	= { by lemma 96 }
34.69/34.93	  forward_diamond(multiplication(domain(Z), addition(backward_box(X, Y), c(backward_box(X, Y)))), backward_box(X, Y))
34.69/34.93	= { by axiom 29 (complement) }
34.69/34.93	  forward_diamond(multiplication(domain(Z), addition(backward_box(X, Y), antidomain(domain(backward_box(X, Y))))), backward_box(X, Y))
34.69/34.93	= { by lemma 112 }
34.69/34.93	  domain(multiplication(domain(Z), multiplication(addition(backward_box(X, Y), antidomain(domain(backward_box(X, Y)))), domain(backward_box(X, Y)))))
34.69/34.93	= { by lemma 69 }
34.69/34.93	  domain(multiplication(domain(Z), multiplication(backward_box(X, Y), domain(backward_box(X, Y)))))
34.69/34.93	= { by lemma 112 }
34.69/34.93	  forward_diamond(multiplication(domain(Z), backward_box(X, Y)), backward_box(X, Y))
34.69/34.93	= { by lemma 136 }
34.69/34.93	  forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), backward_box(X, Y))
34.69/34.93	= { by lemma 173 }
34.69/34.93	  forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(antidomain(backward_box(X, Y))))
34.69/34.93	= { by lemma 141 }
34.69/34.93	  forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(domain(antidomain(backward_box(X, Y)))))
34.69/34.93	= { by axiom 23 (domain4) }
34.69/34.93	  forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(antidomain(antidomain(antidomain(backward_box(X, Y))))))
34.69/34.93	= { by lemma 153 }
34.69/34.93	  forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), backward_box(one, antidomain(antidomain(backward_box(X, Y)))))
34.69/34.93	= { by lemma 97 }
34.69/34.93	  domain(forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), backward_box(one, antidomain(antidomain(backward_box(X, Y))))))
34.69/34.93	= { by axiom 23 (domain4) }
34.69/34.93	  antidomain(antidomain(forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), backward_box(one, antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 96 }
34.69/34.93	  antidomain(c(forward_diamond(domain_difference(Z, antidomain(backward_box(X, Y))), backward_box(one, antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 84 }
34.69/34.93	  antidomain(forward_box(domain_difference(Z, antidomain(backward_box(X, Y))), backward_diamond(one, c(antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 40 }
34.69/34.93	  antidomain(forward_box(domain_difference(Z, antidomain(backward_box(X, Y))), codomain(codomain(c(antidomain(antidomain(backward_box(X, Y))))))))
34.69/34.93	= { by lemma 93 }
34.69/34.93	  antidomain(forward_box(domain_difference(Z, antidomain(backward_box(X, Y))), codomain(c(antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 96 }
34.69/34.93	  antidomain(forward_box(domain_difference(Z, antidomain(backward_box(X, Y))), codomain(antidomain(antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 169 }
34.69/34.93	  antidomain(antidomain(multiplication(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(antidomain(antidomain(antidomain(backward_box(X, Y))))))))
34.69/34.93	= { by axiom 23 (domain4) }
34.69/34.93	  domain(multiplication(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(antidomain(antidomain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by axiom 23 (domain4) }
34.69/34.93	  domain(multiplication(domain_difference(Z, antidomain(backward_box(X, Y))), coantidomain(domain(antidomain(backward_box(X, Y))))))
34.69/34.93	= { by lemma 113 }
34.69/34.93	  domain(multiplication(domain(Z), multiplication(antidomain(antidomain(backward_box(X, Y))), coantidomain(domain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 114 }
34.69/34.93	  domain(multiplication(domain(Z), coantidomain(domain(antidomain(backward_box(X, Y))))))
34.69/34.93	= { by lemma 168 }
34.69/34.93	  domain(multiplication(domain(Z), backward_box(domain(antidomain(backward_box(X, Y))), zero)))
34.69/34.93	= { by lemma 137 }
34.69/34.93	  forward_diamond(domain(Z), backward_box(domain(antidomain(backward_box(X, Y))), zero))
34.69/34.93	= { by lemma 168 }
34.69/34.93	  forward_diamond(domain(Z), coantidomain(domain(antidomain(backward_box(X, Y)))))
34.69/34.93	= { by lemma 75 }
34.69/34.93	  domain(domain_difference(Z, antidomain(coantidomain(domain(antidomain(backward_box(X, Y)))))))
34.69/34.93	= { by lemma 176 }
34.69/34.93	  domain(domain_difference(coantidomain(domain(antidomain(backward_box(X, Y)))), antidomain(Z)))
34.69/34.93	= { by lemma 157 }
34.69/34.93	  domain(multiplication(coantidomain(domain(antidomain(backward_box(X, Y)))), antidomain(antidomain(Z))))
34.69/34.93	= { by lemma 96 }
34.69/34.93	  domain(multiplication(coantidomain(domain(antidomain(backward_box(X, Y)))), c(antidomain(Z))))
34.69/34.93	= { by lemma 72 }
34.69/34.93	  forward_diamond(coantidomain(domain(antidomain(backward_box(X, Y)))), antidomain(antidomain(Z)))
34.69/34.93	= { by lemma 108 }
34.69/34.93	  antidomain(forward_box(coantidomain(domain(antidomain(backward_box(X, Y)))), antidomain(Z)))
34.69/34.93	= { by lemma 109 }
34.69/34.93	  antidomain(antidomain(forward_diamond(coantidomain(domain(antidomain(backward_box(X, Y)))), Z)))
34.69/34.93	= { by axiom 23 (domain4) }
34.69/34.93	  domain(forward_diamond(coantidomain(domain(antidomain(backward_box(X, Y)))), Z))
34.69/34.93	= { by lemma 97 }
34.69/34.93	  forward_diamond(coantidomain(domain(antidomain(backward_box(X, Y)))), Z)
34.69/34.93	= { by lemma 141 }
34.69/34.93	  forward_diamond(coantidomain(antidomain(backward_box(X, Y))), Z)
34.69/34.93	= { by lemma 173 }
34.69/34.93	  forward_diamond(backward_box(X, Y), Z)
34.69/34.93	
34.69/34.93	Lemma 183: multiplication(antidomain(Z), forward_diamond(X, Y)) = domain_difference(forward_diamond(X, Y), Z).
34.69/34.93	Proof:
34.69/34.93	  multiplication(antidomain(Z), forward_diamond(X, Y))
34.69/34.93	= { by axiom 27 (forward_diamond) }
34.69/34.93	  multiplication(antidomain(Z), domain(multiplication(X, domain(Y))))
34.69/34.93	= { by lemma 175 }
34.69/34.93	  domain_difference(multiplication(X, domain(Y)), Z)
34.69/34.93	= { by lemma 144 }
34.69/34.93	  domain_difference(forward_diamond(X, Y), Z)
34.69/34.93	
34.69/34.93	Lemma 185: multiplication(domain(X), forward_diamond(Z, Y)) = multiplication(forward_diamond(Z, Y), domain(X)).
34.69/34.93	Proof:
34.69/34.93	  multiplication(domain(X), forward_diamond(Z, Y))
34.69/34.93	= { by lemma 127 }
34.69/34.93	  domain_difference(X, antidomain(forward_diamond(Z, Y)))
34.69/34.93	= { by lemma 176 }
34.69/34.93	  domain_difference(forward_diamond(Z, Y), antidomain(X))
34.69/34.93	= { by lemma 122 }
34.69/34.93	  multiplication(forward_diamond(Z, Y), antidomain(antidomain(X)))
34.69/34.93	= { by lemma 85 }
34.69/34.93	  domain_difference(multiplication(Z, domain(Y)), antidomain(X))
34.69/34.93	= { by lemma 57 }
34.69/34.93	  multiplication(domain(multiplication(Z, domain(Y))), domain(X))
34.69/34.93	= { by axiom 27 (forward_diamond) }
34.69/34.93	  multiplication(forward_diamond(Z, Y), domain(X))
34.69/34.93	
34.69/34.93	Lemma 185: multiplication(forward_diamond(Z, Y), domain(X)) = multiplication(domain(X), forward_diamond(Z, Y)).
34.69/34.93	Proof:
34.69/34.93	  multiplication(forward_diamond(Z, Y), domain(X))
34.69/34.93	= { by axiom 27 (forward_diamond) }
34.69/34.93	  multiplication(domain(multiplication(Z, domain(Y))), domain(X))
34.69/34.93	= { by lemma 57 }
34.69/34.93	  domain_difference(multiplication(Z, domain(Y)), antidomain(X))
34.69/34.93	= { by lemma 85 }
34.69/34.93	  multiplication(forward_diamond(Z, Y), antidomain(antidomain(X)))
34.69/34.93	= { by lemma 122 }
34.69/34.93	  domain_difference(forward_diamond(Z, Y), antidomain(X))
34.69/34.93	= { by lemma 176 }
34.69/34.93	  domain_difference(X, antidomain(forward_diamond(Z, Y)))
34.69/34.93	= { by lemma 127 }
34.69/34.93	  multiplication(domain(X), forward_diamond(Z, Y))
34.69/34.93	
34.69/34.93	Lemma 186: forward_diamond(domain(Z), forward_diamond(X, Y)) = forward_diamond(forward_diamond(X, Y), Z).
34.69/34.93	Proof:
34.69/34.93	  forward_diamond(domain(Z), forward_diamond(X, Y))
34.69/34.93	= { by lemma 121 }
34.69/34.93	  domain(multiplication(domain(Z), forward_diamond(X, Y)))
34.69/34.93	= { by lemma 185 }
34.69/34.93	  domain(multiplication(forward_diamond(X, Y), domain(Z)))
34.69/34.93	= { by axiom 27 (forward_diamond) }
34.69/34.93	  forward_diamond(forward_diamond(X, Y), Z)
34.69/34.93	
34.69/34.93	Lemma 187: addition(multiplication(Z, W), multiplication(addition(X, Z), Y)) = addition(multiplication(X, Y), multiplication(Z, addition(Y, W))).
34.69/34.93	Proof:
34.69/34.93	  addition(multiplication(Z, W), multiplication(addition(X, Z), Y))
34.69/34.93	= { by axiom 4 (left_distributivity) }
34.69/34.93	  addition(multiplication(Z, W), addition(multiplication(X, Y), multiplication(Z, Y)))
34.69/34.93	= { by lemma 145 }
34.69/34.93	  addition(multiplication(X, Y), multiplication(Z, addition(W, Y)))
34.69/34.93	= { by axiom 5 (additive_commutativity) }
34.69/34.93	  addition(multiplication(X, Y), multiplication(Z, addition(Y, W)))
34.69/34.93	
34.69/34.93	Lemma 188: addition(X, multiplication(antidomain(Y), addition(X, Z))) = addition(X, multiplication(antidomain(Y), Z)).
34.69/34.93	Proof:
34.69/34.93	  addition(X, multiplication(antidomain(Y), addition(X, Z)))
34.69/34.93	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.93	  addition(multiplication(one, X), multiplication(antidomain(Y), addition(X, Z)))
34.69/34.93	= { by lemma 187 }
34.69/34.93	  addition(multiplication(antidomain(Y), Z), multiplication(addition(one, antidomain(Y)), X))
34.69/34.93	= { by lemma 52 }
34.69/34.93	  addition(multiplication(antidomain(Y), Z), multiplication(one, X))
34.69/34.93	= { by axiom 6 (multiplicative_left_identity) }
34.69/34.93	  addition(multiplication(antidomain(Y), Z), X)
34.69/34.93	= { by axiom 5 (additive_commutativity) }
34.69/34.93	  addition(X, multiplication(antidomain(Y), Z))
34.69/34.93	
34.69/34.93	Lemma 189: addition(antidomain(X), domain_difference(X, Y)) = addition(antidomain(X), antidomain(Y)).
34.69/34.93	Proof:
34.69/34.93	  addition(antidomain(X), domain_difference(X, Y))
34.69/34.93	= { by lemma 175 }
34.69/34.93	  addition(antidomain(X), multiplication(antidomain(Y), domain(X)))
34.69/34.93	= { by lemma 188 }
34.69/34.93	  addition(antidomain(X), multiplication(antidomain(Y), addition(antidomain(X), domain(X))))
34.69/34.93	= { by lemma 48 }
34.69/34.93	  addition(antidomain(X), multiplication(antidomain(Y), one))
34.69/34.93	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.94	  addition(antidomain(X), antidomain(Y))
34.69/34.94	
34.69/34.94	Lemma 190: antidomain(forward_diamond(X, Y)) = antidomain(multiplication(X, Y)).
34.69/34.94	Proof:
34.69/34.94	  antidomain(forward_diamond(X, Y))
34.69/34.94	= { by lemma 109 }
34.69/34.94	  forward_box(X, antidomain(Y))
34.69/34.94	= { by lemma 111 }
34.69/34.94	  antidomain(multiplication(X, antidomain(antidomain(Y))))
34.69/34.94	= { by axiom 18 (domain2) }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
34.69/34.94	= { by lemma 111 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), forward_box(X, antidomain(Y)))
34.69/34.94	= { by lemma 109 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), antidomain(forward_diamond(X, Y)))
34.69/34.94	= { by lemma 189 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), domain_difference(multiplication(X, Y), forward_diamond(X, Y)))
34.69/34.94	= { by lemma 175 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 143 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), domain_difference(antidomain(forward_diamond(X, Y)), antidomain(multiplication(X, Y))))
34.69/34.94	= { by lemma 57 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain(antidomain(forward_diamond(X, Y))), domain(multiplication(X, Y))))
34.69/34.94	= { by axiom 7 (multiplicative_right_identity) }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(multiplication(domain(antidomain(forward_diamond(X, Y))), one), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 49 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(multiplication(domain(antidomain(forward_diamond(X, Y))), antidomain(zero)), domain(multiplication(X, Y))))
34.69/34.94	= { by axiom 28 (domain_difference) }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), zero), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 56 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 110 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y))))
34.69/34.94	= { by axiom 11 (multiplicative_associativity) }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, multiplication(domain(Y), Y)))), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 81 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 178 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), forward_diamond(X, Y)), domain(multiplication(X, Y))))
34.69/34.94	= { by lemma 128 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), multiplication(antidomain(forward_diamond(X, Y)), domain(multiplication(X, Y)))))
34.69/34.94	= { by lemma 134 }
34.69/34.94	  addition(antidomain(multiplication(X, Y)), zero)
34.69/34.94	= { by axiom 9 (additive_identity) }
34.69/34.94	  antidomain(multiplication(X, Y))
34.69/34.94	
34.69/34.94	Lemma 191: domain(multiplication(X, Y)) = forward_diamond(X, Y).
34.69/34.94	Proof:
34.69/34.94	  domain(multiplication(X, Y))
34.69/34.94	= { by axiom 23 (domain4) }
34.69/34.94	  antidomain(antidomain(multiplication(X, Y)))
34.69/34.94	= { by lemma 190 }
34.69/34.94	  antidomain(antidomain(forward_diamond(X, Y)))
34.69/34.94	= { by axiom 23 (domain4) }
34.69/34.94	  domain(forward_diamond(X, Y))
34.69/34.94	= { by lemma 97 }
35.45/35.70	  forward_diamond(X, Y)
35.45/35.70	
35.45/35.70	Lemma 192: codomain(antidomain(X)) = antidomain(X).
35.45/35.70	Proof:
35.45/35.70	  codomain(antidomain(X))
35.45/35.70	= { by axiom 6 (multiplicative_left_identity) }
35.45/35.70	  multiplication(one, codomain(antidomain(X)))
35.45/35.70	= { by lemma 52 }
35.45/35.70	  multiplication(addition(one, antidomain(X)), codomain(antidomain(X)))
35.45/35.70	= { by axiom 5 (additive_commutativity) }
35.45/35.70	  multiplication(addition(antidomain(X), one), codomain(antidomain(X)))
35.45/35.70	= { by lemma 65 }
35.45/35.70	  addition(codomain(antidomain(X)), multiplication(antidomain(X), codomain(antidomain(X))))
35.45/35.70	= { by lemma 79 }
35.45/35.70	  addition(codomain(antidomain(X)), antidomain(X))
35.45/35.70	= { by axiom 5 (additive_commutativity) }
35.45/35.70	  addition(antidomain(X), codomain(antidomain(X)))
35.45/35.70	= { by lemma 152 }
35.45/35.70	  addition(antidomain(X), antidomain(coantidomain(antidomain(X))))
35.45/35.70	= { by lemma 189 }
35.45/35.70	  addition(antidomain(X), domain_difference(X, coantidomain(antidomain(X))))
35.45/35.70	= { by lemma 81 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(antidomain(X)))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 141 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(domain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 79 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(domain(multiplication(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 162 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(domain(multiplication(antidomain(X), backward_box(coantidomain(antidomain(X)), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 137 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(forward_diamond(antidomain(X), backward_box(coantidomain(antidomain(X)), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 162 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(domain_difference(X, coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 96 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(c(domain_difference(X, coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 29 (complement) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(domain(domain_difference(X, coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 103 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(forward_diamond(domain(X), antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 104 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(forward_box(domain(X), coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 168 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(forward_box(domain(X), backward_box(forward_diamond(antidomain(X), codomain(antidomain(X))), zero))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 34 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(forward_box(domain(X), backward_box(forward_diamond(antidomain(X), codomain(antidomain(X))), antidomain(one)))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 111 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), antidomain(backward_box(forward_diamond(antidomain(X), codomain(antidomain(X))), antidomain(one)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 167 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), backward_diamond(forward_diamond(antidomain(X), codomain(antidomain(X))), domain(one))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), backward_diamond(forward_diamond(antidomain(X), codomain(antidomain(X))), antidomain(antidomain(one)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 34 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), backward_diamond(forward_diamond(antidomain(X), codomain(antidomain(X))), antidomain(zero))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 49 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), backward_diamond(forward_diamond(antidomain(X), codomain(antidomain(X))), one)))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 83 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(multiplication(domain(X), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(domain(X), codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 170 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(domain(X), forward_box(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 25 (forward_box) }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(domain(X), c(forward_diamond(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), c(codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 72 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(domain(X), antidomain(forward_diamond(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), c(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 108 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(forward_box(domain(X), forward_diamond(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), c(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 25 (forward_box) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(c(forward_diamond(domain(X), c(forward_diamond(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), c(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 25 (forward_box) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(c(forward_diamond(domain(X), forward_box(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 96 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(forward_diamond(domain(X), forward_box(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(domain(forward_diamond(domain(X), forward_box(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 97 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(domain(X), forward_box(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 170 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(domain(X), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 75 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(domain_difference(X, antidomain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 174 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(antidomain(antidomain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain_difference(X, antidomain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 57 }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(antidomain(antidomain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), multiplication(domain(X), domain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 112 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(antidomain(antidomain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))), domain(X)), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(domain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain(X)), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 57 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(domain_difference(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(X)), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 161 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 152 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), antidomain(antidomain(X))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 86 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), antidomain(antidomain(X))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 7 (multiplicative_right_identity) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), one)), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 52 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(one, antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 46 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(zero), antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 129 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(domain_difference(forward_diamond(antidomain(X), codomain(antidomain(X))), multiplication(antidomain(X), codomain(antidomain(X))))), antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 79 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(domain_difference(forward_diamond(antidomain(X), codomain(antidomain(X))), antidomain(X))), antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 28 (domain_difference) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(multiplication(domain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))), antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 124 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(multiplication(domain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))), backward_box(antidomain(antidomain(X)), antidomain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 118 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(multiplication(domain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))), antidomain(backward_diamond(antidomain(antidomain(X)), domain(forward_diamond(antidomain(X), codomain(antidomain(X))))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 156 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), addition(coantidomain(multiplication(domain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))), coantidomain(backward_diamond(antidomain(antidomain(X)), domain(forward_diamond(antidomain(X), codomain(antidomain(X))))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 139 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), coantidomain(backward_diamond(antidomain(antidomain(X)), domain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 156 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), antidomain(backward_diamond(antidomain(antidomain(X)), domain(forward_diamond(antidomain(X), codomain(antidomain(X)))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 118 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), backward_box(antidomain(antidomain(X)), antidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 124 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), antidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 156 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), coantidomain(backward_diamond(antidomain(antidomain(X)), forward_diamond(antidomain(X), codomain(antidomain(X))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 116 }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(antidomain(antidomain(X)), coantidomain(multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 11 (multiplicative_associativity) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), multiplication(antidomain(antidomain(X)), coantidomain(multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X))))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 11 (multiplicative_associativity) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), multiplication(multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X))), coantidomain(multiplication(codomain(forward_diamond(antidomain(X), codomain(antidomain(X)))), antidomain(antidomain(X)))))), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 19 (codomain1) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(multiplication(antidomain(coantidomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), zero), codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 10 (right_annihilation) }
35.45/35.70	  addition(antidomain(X), multiplication(forward_diamond(zero, codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 27 (forward_diamond) }
35.45/35.70	  addition(antidomain(X), multiplication(domain(multiplication(zero, domain(codomain(forward_diamond(antidomain(X), codomain(antidomain(X))))))), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 8 (left_annihilation) }
35.45/35.70	  addition(antidomain(X), multiplication(domain(zero), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 23 (domain4) }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(antidomain(zero)), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 49 }
35.45/35.70	  addition(antidomain(X), multiplication(antidomain(one), domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by lemma 34 }
35.45/35.70	  addition(antidomain(X), multiplication(zero, domain_difference(X, coantidomain(antidomain(X)))))
35.45/35.70	= { by axiom 8 (left_annihilation) }
35.45/35.70	  addition(antidomain(X), zero)
35.45/35.70	= { by axiom 9 (additive_identity) }
35.45/35.70	  antidomain(X)
35.45/35.70	
35.45/35.70	Lemma 193: coantidomain(antidomain(X)) = domain(X).
35.45/35.70	Proof:
35.45/35.70	  coantidomain(antidomain(X))
35.45/35.70	= { by lemma 151 }
35.45/35.70	  antidomain(codomain(antidomain(X)))
35.45/35.70	= { by lemma 192 }
35.45/35.70	  antidomain(antidomain(X))
35.45/35.70	= { by axiom 23 (domain4) }
35.53/35.74	  domain(X)
35.53/35.74	
35.53/35.74	Lemma 194: coantidomain(domain(X)) = antidomain(X).
35.53/35.74	Proof:
35.53/35.74	  coantidomain(domain(X))
35.53/35.74	= { by axiom 9 (additive_identity) }
35.53/35.74	  addition(coantidomain(domain(X)), zero)
35.53/35.74	= { by axiom 10 (right_annihilation) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), zero))
35.53/35.74	= { by axiom 10 (right_annihilation) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), zero)))
35.53/35.74	= { by axiom 19 (codomain1) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), coantidomain(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))))))))
35.53/35.74	= { by lemma 116 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X)))))))
35.53/35.74	= { by axiom 20 (codomain4) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X)))))))
35.53/35.74	= { by axiom 16 (codomain2) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(domain_difference(one, antidomain(X)), antidomain(X))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X))))))))
35.53/35.74	= { by lemma 113 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(domain(one), multiplication(antidomain(antidomain(X)), antidomain(X)))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X))))))))
35.53/35.74	= { by axiom 21 (domain1) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(domain(one), zero)), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X))))))))
35.53/35.74	= { by axiom 10 (right_annihilation) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(zero), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X))))))))
35.53/35.74	= { by lemma 46 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(one, coantidomain(multiplication(coantidomain(coantidomain(domain_difference(one, antidomain(X)))), antidomain(X))))))))
35.53/35.74	= { by axiom 20 (codomain4) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(one, coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))))))))
35.53/35.74	= { by axiom 5 (additive_commutativity) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))), one)))))
35.53/35.74	= { by lemma 31 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))), addition(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))), coantidomain(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))))))))))
35.53/35.74	= { by lemma 51 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), addition(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X))), coantidomain(coantidomain(multiplication(codomain(domain_difference(one, antidomain(X))), antidomain(X)))))))))
35.53/35.74	= { by lemma 31 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), multiplication(backward_diamond(antidomain(X), domain_difference(one, antidomain(X))), one))))
35.53/35.74	= { by axiom 7 (multiplicative_right_identity) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), backward_diamond(antidomain(X), domain_difference(one, antidomain(X))))))
35.53/35.74	= { by lemma 49 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), backward_diamond(antidomain(X), domain_difference(antidomain(zero), antidomain(X))))))
35.53/35.74	= { by lemma 74 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), backward_diamond(antidomain(X), multiplication(c(zero), antidomain(antidomain(X)))))))
35.53/35.74	= { by lemma 50 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), backward_diamond(antidomain(X), multiplication(one, antidomain(antidomain(X)))))))
35.53/35.74	= { by axiom 6 (multiplicative_left_identity) }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), multiplication(antidomain(X), backward_diamond(antidomain(X), antidomain(antidomain(X))))))
35.53/35.74	= { by lemma 148 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(codomain(antidomain(antidomain(X))), antidomain(X)))
35.53/35.74	= { by lemma 161 }
35.53/35.74	  addition(coantidomain(domain(X)), domain_difference(codomain(antidomain(antidomain(X))), X))
35.53/35.74	= { by axiom 23 (domain4) }
35.53/35.74	  addition(coantidomain(domain(X)), domain_difference(codomain(domain(X)), X))
35.53/35.74	= { by lemma 180 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(antidomain(X), codomain(domain(X))))
35.53/35.74	= { by lemma 188 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(antidomain(X), addition(coantidomain(domain(X)), codomain(domain(X)))))
35.53/35.74	= { by lemma 45 }
35.53/35.74	  addition(coantidomain(domain(X)), multiplication(antidomain(X), one))
35.53/35.74	= { by axiom 7 (multiplicative_right_identity) }
35.53/35.74	  addition(coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by axiom 13 (order_1) }
35.53/35.74	  $$fresh(leq(coantidomain(domain(X)), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by lemma 114 }
35.53/35.74	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), antidomain(X)), $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by axiom 7 (multiplicative_right_identity) }
35.53/35.74	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), multiplication(antidomain(X), one)), $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by lemma 31 }
35.53/35.74	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), multiplication(antidomain(X), addition(coantidomain(domain(X)), coantidomain(coantidomain(domain(X)))))), $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by axiom 3 (right_distributivity) }
35.53/35.74	  $$fresh(leq(multiplication(antidomain(X), coantidomain(domain(X))), addition(multiplication(antidomain(X), coantidomain(domain(X))), multiplication(antidomain(X), coantidomain(coantidomain(domain(X)))))), $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by lemma 53 }
35.53/35.74	  $$fresh($$true, $$true, coantidomain(domain(X)), antidomain(X))
35.53/35.74	= { by axiom 2 (order_1) }
35.53/35.74	  antidomain(X)
35.53/35.74	
35.53/35.74	Lemma 195: domain_difference(forward_diamond(X, Y), Z) = domain_difference(multiplication(X, Y), Z).
35.53/35.74	Proof:
35.53/35.74	  domain_difference(forward_diamond(X, Y), Z)
35.53/35.74	= { by lemma 122 }
35.53/35.74	  multiplication(forward_diamond(X, Y), antidomain(Z))
35.53/35.74	= { by lemma 191 }
35.53/35.74	  multiplication(domain(multiplication(X, Y)), antidomain(Z))
35.53/35.74	= { by axiom 28 (domain_difference) }
35.71/35.92	  domain_difference(multiplication(X, Y), Z)
35.71/35.92	
35.71/35.92	Lemma 196: domain(domain_difference(X, Y)) = domain_difference(X, Y).
35.71/35.92	Proof:
35.71/35.92	  domain(domain_difference(X, Y))
35.71/35.92	= { by axiom 23 (domain4) }
35.71/35.92	  antidomain(antidomain(domain_difference(X, Y)))
35.71/35.92	= { by lemma 181 }
35.71/35.92	  antidomain(addition(antidomain(domain_difference(X, Y)), coantidomain(addition(domain(X), domain(domain_difference(X, Y))))))
35.71/35.92	= { by axiom 5 (additive_commutativity) }
35.71/35.92	  antidomain(addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y))))
35.71/35.92	= { by axiom 7 (multiplicative_right_identity) }
35.71/35.92	  multiplication(antidomain(addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), one)
35.71/35.92	= { by lemma 45 }
35.71/35.92	  multiplication(antidomain(addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), codomain(addition(domain(X), domain(domain_difference(X, Y))))))
35.71/35.92	= { by lemma 147 }
35.71/35.92	  multiplication(antidomain(addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), codomain(addition(domain(X), domain(domain_difference(X, Y)))))
35.71/35.92	= { by lemma 180 }
35.71/35.92	  domain_difference(codomain(addition(domain(X), domain(domain_difference(X, Y)))), addition(coantidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y))))
35.71/35.92	= { by axiom 5 (additive_commutativity) }
35.71/35.92	  domain_difference(codomain(addition(domain(X), domain(domain_difference(X, Y)))), addition(antidomain(domain_difference(X, Y)), coantidomain(addition(domain(X), domain(domain_difference(X, Y))))))
35.71/35.92	= { by lemma 181 }
35.71/35.92	  domain_difference(codomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))
35.71/35.92	= { by lemma 164 }
35.71/35.92	  multiplication(codomain(addition(domain(X), domain(domain_difference(X, Y)))), domain(domain_difference(X, Y)))
35.71/35.92	= { by lemma 179 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(domain(X), domain(domain_difference(X, Y)))))
35.71/35.92	= { by lemma 193 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), domain(domain_difference(X, Y)))))
35.71/35.92	= { by axiom 6 (multiplicative_left_identity) }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(one, coantidomain(antidomain(X))), domain(domain_difference(X, Y)))))
35.71/35.92	= { by axiom 7 (multiplicative_right_identity) }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(one, coantidomain(antidomain(X))), multiplication(domain(domain_difference(X, Y)), one))))
35.71/35.92	= { by lemma 45 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(one, coantidomain(antidomain(X))), multiplication(domain(domain_difference(X, Y)), addition(coantidomain(antidomain(X)), codomain(antidomain(X)))))))
35.71/35.92	= { by lemma 187 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(domain(domain_difference(X, Y)), codomain(antidomain(X))), multiplication(addition(one, domain(domain_difference(X, Y))), coantidomain(antidomain(X))))))
35.71/35.92	= { by lemma 54 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(domain(domain_difference(X, Y)), codomain(antidomain(X))), multiplication(one, coantidomain(antidomain(X))))))
35.71/35.92	= { by axiom 6 (multiplicative_left_identity) }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(multiplication(domain(domain_difference(X, Y)), codomain(antidomain(X))), coantidomain(antidomain(X)))))
35.71/35.92	= { by axiom 5 (additive_commutativity) }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), codomain(antidomain(X))))))
35.71/35.92	= { by lemma 158 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), domain_difference(domain_difference(X, Y), coantidomain(antidomain(X))))))
35.71/35.92	= { by lemma 179 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), domain(domain_difference(X, Y))))))
35.71/35.92	= { by lemma 148 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), backward_diamond(domain(domain_difference(X, Y)), antidomain(X)))))))
35.71/35.92	= { by axiom 6 (multiplicative_left_identity) }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(one, backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.71/35.92	= { by lemma 46 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(zero), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.71/35.92	= { by lemma 134 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(multiplication(c(X), domain_difference(X, Y))), multiplication(c(X), domain(domain_difference(X, Y))))), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.71/35.92	= { by lemma 71 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(zero), multiplication(c(X), domain(domain_difference(X, Y))))), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.71/35.92	= { by lemma 49 }
35.71/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(one, multiplication(c(X), domain(domain_difference(X, Y))))), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.71/35.92	= { by lemma 96 }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(one, multiplication(antidomain(X), domain(domain_difference(X, Y))))), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.75/35.92	= { by axiom 6 (multiplicative_left_identity) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), backward_diamond(domain(domain_difference(X, Y)), antidomain(X))))))))
35.75/35.92	= { by lemma 94 }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), backward_diamond(domain(domain_difference(X, Y)), codomain(antidomain(X)))))))))
35.75/35.92	= { by axiom 20 (codomain4) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), backward_diamond(domain(domain_difference(X, Y)), coantidomain(coantidomain(antidomain(X))))))))))
35.75/35.92	= { by lemma 99 }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), codomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y))))))))))
35.75/35.92	= { by axiom 20 (codomain4) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y)))))))))))
35.75/35.92	= { by lemma 91 }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(addition(coantidomain(multiplication(antidomain(X), domain(domain_difference(X, Y)))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y))))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y)))))))))))
35.75/35.92	= { by axiom 16 (codomain2) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), multiplication(coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y)))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(domain_difference(X, Y)))))))))))
35.75/35.92	= { by axiom 19 (codomain1) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(domain_difference(X, Y)), zero)))))
35.75/35.92	= { by axiom 10 (right_annihilation) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), zero))))
35.75/35.92	= { by axiom 10 (right_annihilation) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(addition(coantidomain(antidomain(X)), zero)))
35.75/35.92	= { by axiom 9 (additive_identity) }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(coantidomain(antidomain(X))))
35.75/35.92	= { by lemma 193 }
35.75/35.92	  domain_difference(domain_difference(X, Y), coantidomain(domain(X)))
35.75/35.92	= { by lemma 194 }
35.75/35.92	  domain_difference(domain_difference(X, Y), antidomain(X))
35.75/35.92	= { by lemma 176 }
35.75/35.92	  domain_difference(X, antidomain(domain_difference(X, Y)))
35.75/35.92	= { by lemma 57 }
35.75/35.92	  multiplication(domain(X), domain(domain_difference(X, Y)))
35.75/35.92	= { by axiom 7 (multiplicative_right_identity) }
35.75/35.92	  multiplication(domain(X), multiplication(domain(domain_difference(X, Y)), one))
35.75/35.92	= { by lemma 52 }
35.75/35.92	  multiplication(domain(X), multiplication(domain(domain_difference(X, Y)), addition(one, antidomain(Y))))
35.75/35.92	= { by axiom 5 (additive_commutativity) }
35.75/35.92	  multiplication(domain(X), multiplication(domain(domain_difference(X, Y)), addition(antidomain(Y), one)))
35.75/35.92	= { by lemma 66 }
35.75/35.92	  multiplication(domain(X), addition(domain(domain_difference(X, Y)), multiplication(domain(domain_difference(X, Y)), antidomain(Y))))
35.75/35.92	= { by axiom 28 (domain_difference) }
35.75/35.92	  multiplication(domain(X), addition(domain(domain_difference(X, Y)), domain_difference(domain_difference(X, Y), Y)))
35.75/35.92	= { by lemma 102 }
35.75/35.92	  multiplication(domain(X), addition(domain(domain_difference(X, Y)), domain_difference(domain(domain_difference(X, Y)), Y)))
35.75/35.92	= { by axiom 5 (additive_commutativity) }
35.75/35.92	  multiplication(domain(X), addition(domain_difference(domain(domain_difference(X, Y)), Y), domain(domain_difference(X, Y))))
35.75/35.92	= { by lemma 81 }
35.75/35.92	  multiplication(domain(X), addition(domain_difference(domain(domain_difference(X, Y)), Y), multiplication(domain(domain(domain_difference(X, Y))), domain(domain_difference(X, Y)))))
35.75/35.92	= { by lemma 135 }
35.75/35.92	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(domain(domain_difference(X, Y)), antidomain(Y))))
35.75/35.92	= { by axiom 5 (additive_commutativity) }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), domain(domain_difference(X, Y)))))
35.75/35.93	= { by axiom 23 (domain4) }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), antidomain(antidomain(domain_difference(X, Y))))))
35.75/35.93	= { by lemma 189 }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), domain_difference(Y, antidomain(domain_difference(X, Y))))))
35.75/35.93	= { by axiom 6 (multiplicative_left_identity) }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(one, domain_difference(Y, antidomain(domain_difference(X, Y)))))))
35.75/35.93	= { by lemma 49 }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(antidomain(zero), domain_difference(Y, antidomain(domain_difference(X, Y)))))))
35.75/35.93	= { by lemma 132 }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), domain_difference(X, Y))), domain_difference(Y, antidomain(domain_difference(X, Y)))))))
35.75/35.93	= { by lemma 57 }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), domain_difference(X, Y))), multiplication(domain(Y), domain(domain_difference(X, Y)))))))
35.75/35.93	= { by lemma 134 }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), zero)))
35.75/35.93	= { by axiom 9 (additive_identity) }
35.75/35.93	  multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), antidomain(Y)))
35.75/35.93	= { by axiom 28 (domain_difference) }
35.75/35.93	  multiplication(domain(X), domain_difference(domain(domain_difference(X, Y)), Y))
35.75/35.93	= { by lemma 102 }
35.75/35.93	  multiplication(domain(X), domain_difference(domain_difference(X, Y), Y))
35.75/35.93	= { by axiom 6 (multiplicative_left_identity) }
35.75/35.93	  multiplication(domain(X), domain_difference(multiplication(one, domain_difference(X, Y)), Y))
35.75/35.93	= { by lemma 195 }
35.75/35.93	  multiplication(domain(X), domain_difference(forward_diamond(one, domain_difference(X, Y)), Y))
35.75/35.93	= { by lemma 122 }
35.75/35.93	  multiplication(domain(X), multiplication(forward_diamond(one, domain_difference(X, Y)), antidomain(Y)))
35.75/35.93	= { by axiom 11 (multiplicative_associativity) }
35.75/35.93	  multiplication(multiplication(domain(X), forward_diamond(one, domain_difference(X, Y))), antidomain(Y))
35.75/35.93	= { by lemma 185 }
35.75/35.93	  multiplication(multiplication(forward_diamond(one, domain_difference(X, Y)), domain(X)), antidomain(Y))
35.75/35.93	= { by axiom 11 (multiplicative_associativity) }
35.75/35.93	  multiplication(forward_diamond(one, domain_difference(X, Y)), multiplication(domain(X), antidomain(Y)))
35.75/35.93	= { by lemma 142 }
35.75/35.93	  multiplication(domain(domain_difference(X, Y)), multiplication(domain(X), antidomain(Y)))
35.75/35.93	= { by axiom 28 (domain_difference) }
35.75/35.93	  multiplication(domain(domain_difference(X, Y)), domain_difference(X, Y))
35.75/35.93	= { by lemma 81 }
35.75/35.95	  domain_difference(X, Y)
35.75/35.95	
35.75/35.95	Lemma 197: domain_difference(X, antidomain(Y)) = forward_diamond(domain(X), Y).
35.75/35.95	Proof:
35.75/35.95	  domain_difference(X, antidomain(Y))
35.75/35.95	= { by lemma 57 }
35.75/35.95	  multiplication(domain(X), domain(Y))
35.75/35.95	= { by lemma 142 }
35.75/35.95	  multiplication(domain(X), forward_diamond(one, Y))
35.75/35.95	= { by axiom 15 (additive_idempotence) }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(one, one), Y))
35.75/35.95	= { by axiom 27 (forward_diamond) }
35.75/35.95	  multiplication(domain(X), domain(multiplication(addition(one, one), domain(Y))))
35.75/35.95	= { by lemma 48 }
35.75/35.95	  multiplication(domain(X), domain(multiplication(addition(addition(antidomain(domain(Y)), domain(domain(Y))), one), domain(Y))))
35.75/35.95	= { by axiom 12 (additive_associativity) }
35.75/35.95	  multiplication(domain(X), domain(multiplication(addition(antidomain(domain(Y)), addition(domain(domain(Y)), one)), domain(Y))))
35.75/35.95	= { by lemma 80 }
35.75/35.95	  multiplication(domain(X), domain(multiplication(addition(domain(domain(Y)), one), domain(Y))))
35.75/35.95	= { by axiom 5 (additive_commutativity) }
35.75/35.95	  multiplication(domain(X), domain(multiplication(addition(one, domain(domain(Y))), domain(Y))))
35.75/35.95	= { by axiom 27 (forward_diamond) }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(one, domain(domain(Y))), Y))
35.75/35.95	= { by lemma 42 }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(one, antidomain(c(Y))), Y))
35.75/35.95	= { by lemma 96 }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(one, antidomain(antidomain(Y))), Y))
35.75/35.95	= { by axiom 23 (domain4) }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(one, domain(Y)), Y))
35.75/35.95	= { by axiom 5 (additive_commutativity) }
35.75/35.95	  multiplication(domain(X), forward_diamond(addition(domain(Y), one), Y))
35.75/35.95	= { by lemma 127 }
35.75/35.95	  domain_difference(X, antidomain(forward_diamond(addition(domain(Y), one), Y)))
35.75/35.95	= { by lemma 109 }
35.75/35.95	  domain_difference(X, forward_box(addition(domain(Y), one), antidomain(Y)))
35.75/35.95	= { by lemma 193 }
35.75/35.95	  domain_difference(X, forward_box(addition(coantidomain(antidomain(Y)), one), antidomain(Y)))
35.75/35.95	= { by lemma 192 }
35.75/35.95	  domain_difference(X, forward_box(addition(coantidomain(antidomain(Y)), one), codomain(antidomain(Y))))
35.75/35.95	= { by lemma 196 }
35.75/35.95	  domain(domain_difference(X, forward_box(addition(coantidomain(antidomain(Y)), one), codomain(antidomain(Y)))))
35.75/35.95	= { by lemma 169 }
35.75/35.95	  domain(domain_difference(X, antidomain(multiplication(addition(coantidomain(antidomain(Y)), one), coantidomain(antidomain(Y))))))
35.75/35.95	= { by lemma 75 }
35.75/35.95	  forward_diamond(domain(X), multiplication(addition(coantidomain(antidomain(Y)), one), coantidomain(antidomain(Y))))
35.75/35.95	= { by lemma 78 }
35.75/35.95	  forward_diamond(domain(X), multiplication(coantidomain(antidomain(Y)), addition(coantidomain(antidomain(Y)), one)))
35.75/35.95	= { by lemma 75 }
35.75/35.95	  domain(domain_difference(X, antidomain(multiplication(coantidomain(antidomain(Y)), addition(coantidomain(antidomain(Y)), one)))))
35.75/35.95	= { by lemma 190 }
35.75/35.95	  domain(domain_difference(X, antidomain(forward_diamond(coantidomain(antidomain(Y)), addition(coantidomain(antidomain(Y)), one)))))
35.75/35.95	= { by lemma 75 }
35.75/35.95	  forward_diamond(domain(X), forward_diamond(coantidomain(antidomain(Y)), addition(coantidomain(antidomain(Y)), one)))
35.75/35.95	= { by lemma 186 }
35.75/35.95	  forward_diamond(forward_diamond(coantidomain(antidomain(Y)), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by lemma 168 }
35.75/35.95	  forward_diamond(forward_diamond(backward_box(antidomain(Y), zero), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by lemma 98 }
35.75/35.95	  forward_diamond(forward_diamond(domain(backward_box(antidomain(Y), zero)), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by axiom 23 (domain4) }
35.75/35.95	  forward_diamond(forward_diamond(antidomain(antidomain(backward_box(antidomain(Y), zero))), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by lemma 163 }
35.75/35.95	  forward_diamond(forward_diamond(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by lemma 141 }
35.75/35.95	  forward_diamond(forward_diamond(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero))))))), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by lemma 141 }
35.75/35.95	  forward_diamond(forward_diamond(domain(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))))), addition(coantidomain(antidomain(Y)), one)), X)
35.75/35.95	= { by axiom 5 (additive_commutativity) }
35.75/35.95	  forward_diamond(forward_diamond(domain(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))))), addition(one, coantidomain(antidomain(Y)))), X)
35.75/35.95	= { by lemma 168 }
35.75/35.95	  forward_diamond(forward_diamond(domain(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))))), addition(one, backward_box(antidomain(Y), zero))), X)
35.75/35.95	= { by lemma 98 }
35.75/35.95	  forward_diamond(forward_diamond(domain(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))))), addition(one, domain(backward_box(antidomain(Y), zero)))), X)
35.75/35.95	= { by lemma 75 }
35.75/35.95	  forward_diamond(domain(domain_difference(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero))))))), antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))), X)
35.75/35.95	= { by lemma 159 }
35.75/35.95	  forward_diamond(domain(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))))), X)
35.75/35.95	= { by lemma 141 }
35.75/35.95	  forward_diamond(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), domain(antidomain(addition(one, domain(backward_box(antidomain(Y), zero))))))), X)
35.75/35.95	= { by lemma 141 }
35.75/35.95	  forward_diamond(antidomain(addition(antidomain(backward_box(antidomain(Y), zero)), antidomain(addition(one, domain(backward_box(antidomain(Y), zero)))))), X)
35.75/35.95	= { by lemma 163 }
35.75/35.95	  forward_diamond(antidomain(antidomain(backward_box(antidomain(Y), zero))), X)
35.75/35.95	= { by axiom 23 (domain4) }
35.75/35.95	  forward_diamond(domain(backward_box(antidomain(Y), zero)), X)
35.75/35.95	= { by lemma 98 }
35.75/35.95	  forward_diamond(backward_box(antidomain(Y), zero), X)
35.75/35.95	= { by lemma 168 }
35.75/35.95	  forward_diamond(coantidomain(antidomain(Y)), X)
35.75/35.95	= { by lemma 193 }
35.75/35.95	  forward_diamond(domain(Y), X)
35.75/35.95	= { by lemma 75 }
35.75/35.95	  domain(domain_difference(Y, antidomain(X)))
35.75/35.95	= { by lemma 176 }
35.75/35.95	  domain(domain_difference(X, antidomain(Y)))
35.75/35.95	= { by lemma 75 }
35.75/35.96	  forward_diamond(domain(X), Y)
35.75/35.96	
35.75/35.96	Lemma 198: multiplication(domain(X), forward_diamond(Y, Z)) = forward_diamond(forward_diamond(Y, Z), X).
35.75/35.96	Proof:
35.75/35.96	  multiplication(domain(X), forward_diamond(Y, Z))
35.75/35.96	= { by lemma 127 }
35.75/35.96	  domain_difference(X, antidomain(forward_diamond(Y, Z)))
35.75/35.96	= { by lemma 57 }
35.75/35.96	  multiplication(domain(X), domain(forward_diamond(Y, Z)))
35.75/35.96	= { by lemma 125 }
35.75/35.96	  multiplication(domain(X), forward_diamond(forward_diamond(Y, Z), addition(forward_diamond(Y, Z), one)))
35.75/35.96	= { by lemma 191 }
35.75/35.96	  multiplication(domain(X), domain(multiplication(forward_diamond(Y, Z), addition(forward_diamond(Y, Z), one))))
35.75/35.96	= { by lemma 78 }
35.75/35.96	  multiplication(domain(X), domain(multiplication(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z))))
35.75/35.96	= { by lemma 191 }
35.75/35.96	  multiplication(domain(X), forward_diamond(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z)))
35.75/35.96	= { by lemma 185 }
35.75/35.96	  multiplication(forward_diamond(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z)), domain(X))
35.75/35.96	= { by axiom 27 (forward_diamond) }
35.75/35.96	  multiplication(domain(multiplication(addition(forward_diamond(Y, Z), one), domain(forward_diamond(Y, Z)))), domain(X))
35.75/35.96	= { by lemma 57 }
35.75/35.96	  domain_difference(multiplication(addition(forward_diamond(Y, Z), one), domain(forward_diamond(Y, Z))), antidomain(X))
35.75/35.96	= { by lemma 85 }
35.75/35.96	  multiplication(forward_diamond(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z)), antidomain(antidomain(X)))
35.75/35.96	= { by lemma 122 }
35.75/35.96	  domain_difference(forward_diamond(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z)), antidomain(X))
35.75/35.96	= { by lemma 121 }
35.75/35.96	  domain_difference(domain(multiplication(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z))), antidomain(X))
35.75/35.96	= { by axiom 27 (forward_diamond) }
35.75/35.96	  domain_difference(domain(multiplication(addition(forward_diamond(Y, Z), one), domain(multiplication(Y, domain(Z))))), antidomain(X))
35.75/35.96	= { by axiom 27 (forward_diamond) }
35.75/35.96	  domain_difference(forward_diamond(addition(forward_diamond(Y, Z), one), multiplication(Y, domain(Z))), antidomain(X))
35.75/35.96	= { by lemma 144 }
35.75/35.96	  domain_difference(multiplication(addition(forward_diamond(Y, Z), one), domain(multiplication(Y, domain(Z)))), antidomain(X))
35.75/35.96	= { by axiom 27 (forward_diamond) }
35.75/35.96	  domain_difference(multiplication(addition(forward_diamond(Y, Z), one), forward_diamond(Y, Z)), antidomain(X))
35.75/35.96	= { by lemma 78 }
35.75/35.96	  domain_difference(multiplication(forward_diamond(Y, Z), addition(forward_diamond(Y, Z), one)), antidomain(X))
35.75/35.96	= { by lemma 197 }
35.75/35.96	  forward_diamond(domain(multiplication(forward_diamond(Y, Z), addition(forward_diamond(Y, Z), one))), X)
35.75/35.96	= { by lemma 191 }
35.75/35.96	  forward_diamond(forward_diamond(forward_diamond(Y, Z), addition(forward_diamond(Y, Z), one)), X)
35.75/35.96	= { by lemma 125 }
35.75/35.96	  forward_diamond(domain(forward_diamond(Y, Z)), X)
35.75/35.96	= { by lemma 97 }
36.34/36.53	  forward_diamond(forward_diamond(Y, Z), X)
36.34/36.53	
36.34/36.53	Goal 1 (goals_1): zero = multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)).
36.34/36.53	Proof:
36.34/36.53	  zero
36.34/36.53	= { by lemma 134 }
36.34/36.53	  multiplication(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  multiplication(antidomain(multiplication(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0)), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1)))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  multiplication(antidomain(multiplication(multiplication(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0)), coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0)))), domain(sK2_goals_X1))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 19 (codomain1) }
36.34/36.53	  multiplication(antidomain(multiplication(zero, domain(sK2_goals_X1))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 8 (left_annihilation) }
36.34/36.53	  multiplication(antidomain(zero), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by lemma 49 }
36.34/36.53	  multiplication(one, multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))))
36.34/36.53	= { by lemma 191 }
36.34/36.53	  multiplication(one, multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1)))))
36.34/36.53	= { by axiom 6 (multiplicative_left_identity) }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0))), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 133 }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), addition(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(antidomain(sK1_goals_X2), ?)))), domain(sK2_goals_X1))))
36.34/36.53	= { by axiom 5 (additive_commutativity) }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), addition(multiplication(antidomain(sK1_goals_X2), ?), multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0)))), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 65 }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), addition(multiplication(antidomain(sK1_goals_X2), ?), addition(sK3_goals_X0, multiplication(antidomain(sK1_goals_X2), sK3_goals_X0))))), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 145 }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), addition(sK3_goals_X0, multiplication(antidomain(sK1_goals_X2), addition(?, sK3_goals_X0))))), domain(sK2_goals_X1))))
36.34/36.53	= { by axiom 5 (additive_commutativity) }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), addition(sK3_goals_X0, multiplication(antidomain(sK1_goals_X2), addition(sK3_goals_X0, ?))))), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 133 }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by axiom 27 (forward_diamond) }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), domain(multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by lemma 175 }
36.34/36.53	  domain_difference(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), domain(multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 144 }
36.34/36.53	  domain_difference(forward_diamond(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 195 }
36.34/36.53	  domain_difference(multiplication(multiplication(addition(antidomain(sK1_goals_X2), one), sK3_goals_X0), multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  domain_difference(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 195 }
36.34/36.53	  domain_difference(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 183 }
36.34/36.53	  multiplication(antidomain(antidomain(sK1_goals_X2)), forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 141 }
36.34/36.53	  multiplication(domain(antidomain(antidomain(sK1_goals_X2))), forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 198 }
36.34/36.53	  forward_diamond(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(antidomain(sK1_goals_X2)))
36.34/36.53	= { by lemma 186 }
36.34/36.53	  forward_diamond(domain(antidomain(antidomain(sK1_goals_X2))), forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 121 }
36.34/36.53	  domain(multiplication(domain(antidomain(antidomain(sK1_goals_X2))), forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 23 (domain4) }
36.34/36.53	  domain(multiplication(antidomain(antidomain(antidomain(antidomain(sK1_goals_X2)))), forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by lemma 183 }
36.34/36.53	  domain(domain_difference(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(antidomain(antidomain(sK1_goals_X2)))))
36.34/36.53	= { by lemma 195 }
36.34/36.53	  domain(domain_difference(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))), antidomain(antidomain(antidomain(sK1_goals_X2)))))
36.34/36.53	= { by lemma 176 }
36.34/36.53	  domain(domain_difference(antidomain(antidomain(sK1_goals_X2)), antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 103 }
36.34/36.53	  forward_diamond(domain(antidomain(antidomain(sK1_goals_X2))), antidomain(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 39 }
36.34/36.53	  forward_diamond(c(antidomain(sK1_goals_X2)), antidomain(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 108 }
36.34/36.53	  antidomain(forward_box(c(antidomain(sK1_goals_X2)), antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 96 }
36.34/36.53	  antidomain(forward_box(antidomain(antidomain(sK1_goals_X2)), antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 109 }
36.34/36.53	  antidomain(antidomain(forward_diamond(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by axiom 23 (domain4) }
36.34/36.53	  domain(forward_diamond(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by lemma 97 }
36.34/36.53	  forward_diamond(antidomain(antidomain(sK1_goals_X2)), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 107 }
36.34/36.53	  forward_diamond(addition(antidomain(antidomain(sK1_goals_X2)), antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by axiom 5 (additive_commutativity) }
36.34/36.53	  forward_diamond(addition(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 190 }
36.34/36.53	  forward_diamond(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 191 }
36.34/36.53	  domain(multiplication(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), multiplication(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  domain(multiplication(multiplication(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), addition(antidomain(sK1_goals_X2), one)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))
36.34/36.53	= { by lemma 191 }
36.34/36.53	  forward_diamond(multiplication(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), addition(antidomain(sK1_goals_X2), one)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 66 }
36.34/36.53	  forward_diamond(addition(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), multiplication(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 69 }
36.34/36.53	  forward_diamond(addition(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))), multiplication(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by axiom 12 (additive_associativity) }
36.34/36.53	  forward_diamond(addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), addition(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(sK1_goals_X2)))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 64 }
36.34/36.53	  forward_diamond(addition(antidomain(antidomain(sK1_goals_X2)), addition(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), multiplication(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), antidomain(sK1_goals_X2)))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 66 }
36.34/36.53	  forward_diamond(addition(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(forward_diamond(addition(antidomain(sK1_goals_X2), one), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), addition(antidomain(sK1_goals_X2), one))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 110 }
36.34/36.53	  forward_diamond(addition(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), addition(antidomain(sK1_goals_X2), one))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by axiom 27 (forward_diamond) }
36.34/36.53	  domain(multiplication(addition(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), addition(antidomain(sK1_goals_X2), one))), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 5 (additive_commutativity) }
36.34/36.53	  domain(multiplication(addition(multiplication(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), addition(antidomain(sK1_goals_X2), one)), antidomain(antidomain(sK1_goals_X2))), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 4 (left_distributivity) }
36.34/36.53	  domain(addition(multiplication(multiplication(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), addition(antidomain(sK1_goals_X2), one)), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  domain(addition(multiplication(antidomain(multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), multiplication(addition(antidomain(sK1_goals_X2), one), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))), multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by axiom 21 (domain1) }
36.34/36.53	  domain(addition(zero, multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1)))))))
36.34/36.53	= { by lemma 35 }
36.34/36.53	  domain(multiplication(antidomain(antidomain(sK1_goals_X2)), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))))
36.34/36.53	= { by axiom 27 (forward_diamond) }
36.34/36.53	  forward_diamond(antidomain(antidomain(sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))))
36.34/36.53	= { by lemma 177 }
36.34/36.53	  domain(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2)))
36.34/36.53	= { by lemma 196 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 194 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 15 (additive_idempotence) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(addition(coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0)), coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 146 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(addition(coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2))))))), coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 11 (multiplicative_associativity) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(addition(coantidomain(multiplication(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0), backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))))), coantidomain(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 120 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(addition(coantidomain(multiplication(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0), backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))))), coantidomain(backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 165 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(addition(coantidomain(multiplication(multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0), backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))))), coantidomain(backward_diamond(backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))), multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0)))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 139 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(backward_diamond(backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))), multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 156 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2)))), multiplication(coantidomain(domain(antidomain(sK1_goals_X2))), sK3_goals_X0))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 165 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, coantidomain(domain(antidomain(sK1_goals_X2))))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 168 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, backward_box(domain(antidomain(sK1_goals_X2)), zero))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 98 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, domain(backward_box(domain(antidomain(sK1_goals_X2)), zero)))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 118 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, antidomain(backward_box(domain(antidomain(sK1_goals_X2)), zero))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 168 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, antidomain(coantidomain(domain(antidomain(sK1_goals_X2))))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 152 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, codomain(domain(antidomain(sK1_goals_X2)))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 23 (domain4) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, codomain(antidomain(antidomain(antidomain(sK1_goals_X2))))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 192 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, antidomain(antidomain(antidomain(sK1_goals_X2)))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 23 (domain4) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, domain(antidomain(sK1_goals_X2))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 106 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, antidomain(sK1_goals_X2)), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 26 (backward_box) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(c(backward_diamond(sK3_goals_X0, c(antidomain(sK1_goals_X2)))), domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 23 (domain4) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, multiplication(c(backward_diamond(sK3_goals_X0, c(antidomain(sK1_goals_X2)))), antidomain(antidomain(sK2_goals_X1)))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 74 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain_difference(antidomain(backward_diamond(sK3_goals_X0, c(antidomain(sK1_goals_X2)))), antidomain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 96 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain_difference(antidomain(backward_diamond(sK3_goals_X0, antidomain(antidomain(sK1_goals_X2)))), antidomain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 105 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain_difference(backward_box(sK3_goals_X0, antidomain(sK1_goals_X2)), antidomain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 176 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain_difference(sK2_goals_X1, antidomain(backward_box(sK3_goals_X0, antidomain(sK1_goals_X2))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 197 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, forward_diamond(domain(sK2_goals_X1), backward_box(sK3_goals_X0, antidomain(sK1_goals_X2)))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 137 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), backward_box(sK3_goals_X0, antidomain(sK1_goals_X2))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 118 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), antidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 28 (domain_difference) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 79 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, multiplication(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)), codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 20 (codomain4) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, multiplication(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 91 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, multiplication(addition(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 88 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, multiplication(one, coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 6 (multiplicative_left_identity) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, coantidomain(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 20 (codomain4) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(domain_difference(sK2_goals_X1, codomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 160 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 9 (additive_identity) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(addition(multiplication(domain(sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), zero))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 30 (goals) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(addition(multiplication(domain(sK2_goals_X1), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))), multiplication(domain(sK2_goals_X1), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 3 (right_distributivity) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), addition(coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))), backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 5 (additive_commutativity) }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), addition(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2)), coantidomain(backward_diamond(sK3_goals_X0, domain(sK1_goals_X2))))))), antidomain(sK1_goals_X2))
36.34/36.53	= { by lemma 88 }
36.34/36.53	  domain_difference(multiplication(sK3_goals_X0, domain(multiplication(domain(sK2_goals_X1), one))), antidomain(sK1_goals_X2))
36.34/36.53	= { by axiom 7 (multiplicative_right_identity) }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, domain(domain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.54	= { by axiom 23 (domain4) }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, antidomain(antidomain(domain(sK2_goals_X1)))), antidomain(sK1_goals_X2))
36.34/36.54	= { by axiom 29 (complement) }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, antidomain(c(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.54	= { by lemma 96 }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, antidomain(antidomain(sK2_goals_X1))), antidomain(sK1_goals_X2))
36.34/36.54	= { by axiom 23 (domain4) }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, domain(sK2_goals_X1)), antidomain(sK1_goals_X2))
36.34/36.54	= { by axiom 28 (domain_difference) }
36.34/36.54	  multiplication(domain(multiplication(sK3_goals_X0, domain(sK2_goals_X1))), antidomain(antidomain(sK1_goals_X2)))
36.34/36.54	= { by axiom 27 (forward_diamond) }
36.34/36.54	  multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(antidomain(sK1_goals_X2)))
36.34/36.54	= { by lemma 122 }
36.34/36.54	  domain_difference(forward_diamond(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))
36.34/36.54	= { by lemma 195 }
36.34/36.54	  domain_difference(multiplication(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))
36.34/36.54	= { by lemma 197 }
36.34/36.54	  forward_diamond(domain(multiplication(sK3_goals_X0, sK2_goals_X1)), sK1_goals_X2)
36.34/36.54	= { by lemma 191 }
36.34/36.54	  forward_diamond(forward_diamond(sK3_goals_X0, sK2_goals_X1), sK1_goals_X2)
36.34/36.54	= { by lemma 198 }
36.34/36.54	  multiplication(domain(sK1_goals_X2), forward_diamond(sK3_goals_X0, sK2_goals_X1))
36.34/36.54	= { by lemma 185 }
36.34/36.54	  multiplication(forward_diamond(sK3_goals_X0, sK2_goals_X1), domain(sK1_goals_X2))
36.34/36.54	= { by lemma 101 }
36.34/36.54	  multiplication(forward_diamond(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))
36.34/36.54	% SZS output end Proof
36.34/36.54	
36.34/36.54	RESULT: Theorem (the conjecture is true).
36.34/36.55	EOF