0.10/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.11	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.11/0.31	% Computer   : n005.cluster.edu
0.11/0.31	% Model      : x86_64 x86_64
0.11/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.31	% Memory     : 8042.1875MB
0.11/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.31	% CPULimit   : 1200
0.11/0.31	% WCLimit    : 120
0.11/0.31	% DateTime   : Tue Jul 13 16:00:18 EDT 2021
0.11/0.31	% CPUTime    : 
724.57/91.92	% SZS status Theorem
724.57/91.92	
725.43/92.08	% SZS output start Proof
725.43/92.08	Take the following subset of the input axioms:
725.43/92.08	  fof(additive_associativity, axiom, ![A, C, B]: addition(addition(A, B), C)=addition(A, addition(B, C))).
725.43/92.08	  fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)).
725.43/92.08	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
725.43/92.08	  fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
725.43/92.08	  fof(backward_diamond, axiom, ![X0, X1]: codomain(multiplication(codomain(X1), X0))=backward_diamond(X0, X1)).
725.43/92.08	  fof(codomain1, axiom, ![X0]: zero=multiplication(X0, coantidomain(X0))).
725.43/92.08	  fof(codomain2, axiom, ![X0, X1]: addition(coantidomain(multiplication(X0, X1)), coantidomain(multiplication(coantidomain(coantidomain(X0)), X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0)), X1))).
725.43/92.08	  fof(codomain3, axiom, ![X0]: addition(coantidomain(coantidomain(X0)), coantidomain(X0))=one).
725.43/92.08	  fof(codomain4, axiom, ![X0]: coantidomain(coantidomain(X0))=codomain(X0)).
725.43/92.08	  fof(complement, axiom, ![X0]: c(X0)=antidomain(domain(X0))).
725.43/92.08	  fof(domain1, axiom, ![X0]: zero=multiplication(antidomain(X0), X0)).
725.43/92.08	  fof(domain3, axiom, ![X0]: addition(antidomain(antidomain(X0)), antidomain(X0))=one).
725.43/92.08	  fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))).
725.43/92.08	  fof(forward_box, axiom, ![X0, X1]: forward_box(X0, X1)=c(forward_diamond(X0, c(X1)))).
725.43/92.08	  fof(forward_diamond, axiom, ![X0, X1]: forward_diamond(X0, X1)=domain(multiplication(X0, domain(X1)))).
725.43/92.08	  fof(goals, conjecture, ![X0, X1]: addition(backward_diamond(X0, forward_box(X0, domain(X1))), domain(X1))=domain(X1)).
725.43/92.08	  fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)).
725.43/92.08	  fof(left_distributivity, axiom, ![A, C, B]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
725.43/92.08	  fof(multiplicative_associativity, axiom, ![A, C, B]: multiplication(A, multiplication(B, C))=multiplication(multiplication(A, B), C)).
725.43/92.08	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
725.43/92.08	  fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A).
725.43/92.08	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
725.43/92.08	  fof(right_distributivity, axiom, ![A, C, B]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
725.43/92.08	
725.43/92.08	Now clausify the problem and encode Horn clauses using encoding 3 of
725.43/92.08	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
725.43/92.08	We repeatedly replace C & s=t => u=v by the two clauses:
725.43/92.08	  fresh(y, y, x1...xn) = u
725.43/92.08	  C => fresh(s, t, x1...xn) = v
725.43/92.08	where fresh is a fresh function symbol and x1..xn are the free
725.43/92.08	variables of u and v.
725.43/92.08	A predicate p(X) is encoded as p(X)=true (this is sound, because the
725.43/92.08	input problem has no model of domain size 1).
725.43/92.08	
725.43/92.08	The encoding turns the above axioms into the following unit equations and goals:
725.43/92.08	
725.43/92.08	Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
725.43/92.08	Axiom 2 (left_annihilation): zero = multiplication(zero, X).
725.43/92.08	Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
725.43/92.08	Axiom 4 (additive_idempotence): X = addition(X, X).
725.43/92.08	Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
725.43/92.08	Axiom 6 (additive_identity): addition(X, zero) = X.
725.43/92.08	Axiom 7 (domain4): domain(X) = antidomain(antidomain(X)).
725.43/92.08	Axiom 8 (complement): c(X) = antidomain(domain(X)).
725.43/92.08	Axiom 9 (codomain4): coantidomain(coantidomain(X)) = codomain(X).
725.43/92.08	Axiom 10 (codomain1): zero = multiplication(X, coantidomain(X)).
725.43/92.08	Axiom 11 (domain1): zero = multiplication(antidomain(X), X).
725.43/92.08	Axiom 12 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
725.43/92.08	Axiom 13 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)).
725.43/92.08	Axiom 14 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
725.43/92.08	Axiom 15 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
725.43/92.08	Axiom 16 (backward_diamond): codomain(multiplication(codomain(X), Y)) = backward_diamond(Y, X).
725.43/92.08	Axiom 17 (order): fresh(X, X, Y, Z) = true.
725.43/92.08	Axiom 18 (order_1): fresh2(X, X, Y, Z) = Z.
725.43/92.08	Axiom 19 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
725.43/92.08	Axiom 20 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
725.43/92.08	Axiom 21 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
725.43/92.08	Axiom 22 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
725.43/92.08	Axiom 23 (order): fresh(X, addition(Y, X), Y, X) = leq(Y, X).
725.43/92.08	Axiom 24 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
725.43/92.08	Axiom 25 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
725.43/92.08	
725.43/92.08	Lemma 26: antidomain(one) = zero.
725.43/92.08	Proof:
725.43/92.08	  antidomain(one)
725.43/92.08	= { by axiom 1 (multiplicative_right_identity) R->L }
725.43/92.08	  multiplication(antidomain(one), one)
725.43/92.08	= { by axiom 11 (domain1) R->L }
725.43/92.08	  zero
725.43/92.08	
725.43/92.08	Lemma 27: domain(one) = antidomain(zero).
725.43/92.08	Proof:
725.43/92.08	  domain(one)
725.43/92.08	= { by axiom 7 (domain4) }
725.43/92.08	  antidomain(antidomain(one))
725.43/92.08	= { by lemma 26 }
725.43/92.08	  antidomain(zero)
725.43/92.08	
725.43/92.08	Lemma 28: addition(zero, X) = X.
725.43/92.08	Proof:
725.43/92.08	  addition(zero, X)
725.43/92.08	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.08	  addition(X, zero)
725.43/92.08	= { by axiom 6 (additive_identity) }
725.43/92.08	  X
725.43/92.08	
725.43/92.08	Lemma 29: addition(antidomain(X), domain(X)) = one.
725.43/92.08	Proof:
725.43/92.08	  addition(antidomain(X), domain(X))
725.43/92.08	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.08	  addition(domain(X), antidomain(X))
725.43/92.08	= { by axiom 7 (domain4) }
725.43/92.08	  addition(antidomain(antidomain(X)), antidomain(X))
725.43/92.08	= { by axiom 19 (domain3) }
725.43/92.08	  one
725.43/92.08	
725.43/92.08	Lemma 30: antidomain(zero) = one.
725.43/92.08	Proof:
725.43/92.08	  antidomain(zero)
725.43/92.08	= { by lemma 27 R->L }
725.43/92.08	  domain(one)
725.43/92.08	= { by lemma 28 R->L }
725.43/92.08	  addition(zero, domain(one))
725.43/92.08	= { by lemma 26 R->L }
725.43/92.08	  addition(antidomain(one), domain(one))
725.43/92.08	= { by lemma 29 }
725.43/92.08	  one
725.43/92.08	
725.43/92.08	Lemma 31: coantidomain(one) = zero.
725.43/92.08	Proof:
725.43/92.08	  coantidomain(one)
725.43/92.08	= { by axiom 3 (multiplicative_left_identity) R->L }
725.43/92.08	  multiplication(one, coantidomain(one))
725.43/92.08	= { by axiom 10 (codomain1) R->L }
725.43/92.08	  zero
725.43/92.08	
725.43/92.08	Lemma 32: addition(coantidomain(X), codomain(X)) = one.
725.43/92.08	Proof:
725.43/92.08	  addition(coantidomain(X), codomain(X))
725.43/92.08	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.08	  addition(codomain(X), coantidomain(X))
725.43/92.08	= { by axiom 9 (codomain4) R->L }
725.43/92.08	  addition(coantidomain(coantidomain(X)), coantidomain(X))
725.43/92.08	= { by axiom 20 (codomain3) }
725.43/92.08	  one
725.43/92.08	
725.43/92.08	Lemma 33: coantidomain(zero) = one.
725.43/92.08	Proof:
725.43/92.08	  coantidomain(zero)
725.43/92.08	= { by lemma 31 R->L }
725.43/92.08	  coantidomain(coantidomain(one))
725.43/92.08	= { by axiom 9 (codomain4) }
725.43/92.08	  codomain(one)
725.43/92.08	= { by lemma 28 R->L }
725.43/92.08	  addition(zero, codomain(one))
725.43/92.08	= { by lemma 31 R->L }
725.43/92.08	  addition(coantidomain(one), codomain(one))
725.43/92.08	= { by lemma 32 }
725.43/92.08	  one
725.43/92.08	
725.43/92.08	Lemma 34: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
725.43/92.08	Proof:
725.43/92.08	  c(multiplication(X, domain(Y)))
725.43/92.08	= { by axiom 8 (complement) }
725.43/92.08	  antidomain(domain(multiplication(X, domain(Y))))
725.43/92.08	= { by axiom 14 (forward_diamond) R->L }
725.43/92.08	  antidomain(forward_diamond(X, Y))
725.43/92.08	
725.43/92.08	Lemma 35: antidomain(forward_diamond(X, one)) = c(X).
725.43/92.08	Proof:
725.43/92.08	  antidomain(forward_diamond(X, one))
725.43/92.08	= { by lemma 34 R->L }
725.43/92.08	  c(multiplication(X, domain(one)))
725.43/92.08	= { by lemma 27 }
725.43/92.08	  c(multiplication(X, antidomain(zero)))
725.43/92.08	= { by lemma 30 }
725.43/92.08	  c(multiplication(X, one))
725.43/92.08	= { by axiom 1 (multiplicative_right_identity) }
725.43/92.08	  c(X)
725.43/92.08	
725.43/92.08	Lemma 36: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
725.43/92.08	Proof:
725.43/92.08	  multiplication(antidomain(X), addition(Y, X))
725.43/92.08	= { by axiom 21 (right_distributivity) R->L }
725.43/92.08	  addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
725.43/92.08	= { by axiom 11 (domain1) R->L }
725.43/92.08	  addition(multiplication(antidomain(X), Y), zero)
725.43/92.08	= { by axiom 6 (additive_identity) }
725.43/92.08	  multiplication(antidomain(X), Y)
725.43/92.08	
725.43/92.08	Lemma 37: domain(antidomain(X)) = c(X).
725.43/92.08	Proof:
725.43/92.08	  domain(antidomain(X))
725.43/92.08	= { by axiom 7 (domain4) }
725.43/92.08	  antidomain(antidomain(antidomain(X)))
725.43/92.08	= { by axiom 7 (domain4) R->L }
725.43/92.08	  antidomain(domain(X))
725.43/92.08	= { by axiom 8 (complement) R->L }
725.43/92.08	  c(X)
725.43/92.08	
725.43/92.08	Lemma 38: multiplication(domain(X), X) = X.
725.43/92.08	Proof:
725.43/92.08	  multiplication(domain(X), X)
725.43/92.08	= { by axiom 6 (additive_identity) R->L }
725.43/92.08	  addition(multiplication(domain(X), X), zero)
725.43/92.08	= { by axiom 11 (domain1) }
725.43/92.08	  addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
725.43/92.08	= { by axiom 22 (left_distributivity) }
725.43/92.08	  multiplication(addition(domain(X), antidomain(X)), X)
725.43/92.08	= { by axiom 5 (additive_commutativity) }
725.43/92.08	  multiplication(addition(antidomain(X), domain(X)), X)
725.43/92.08	= { by lemma 29 }
725.43/92.08	  multiplication(one, X)
725.43/92.08	= { by axiom 3 (multiplicative_left_identity) }
725.43/92.08	  X
725.43/92.08	
725.43/92.08	Lemma 39: c(X) = antidomain(X).
725.43/92.08	Proof:
725.43/92.08	  c(X)
725.43/92.08	= { by lemma 35 R->L }
725.43/92.08	  antidomain(forward_diamond(X, one))
725.43/92.08	= { by axiom 1 (multiplicative_right_identity) R->L }
725.43/92.08	  multiplication(antidomain(forward_diamond(X, one)), one)
725.43/92.08	= { by lemma 29 R->L }
725.43/92.08	  multiplication(antidomain(forward_diamond(X, one)), addition(antidomain(multiplication(X, domain(one))), domain(multiplication(X, domain(one)))))
725.43/92.08	= { by axiom 14 (forward_diamond) R->L }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(antidomain(multiplication(X, domain(one))), forward_diamond(X, one)))
725.43/92.09	= { by axiom 5 (additive_commutativity) }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(forward_diamond(X, one), antidomain(multiplication(X, domain(one)))))
725.43/92.09	= { by lemma 27 }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(forward_diamond(X, one), antidomain(multiplication(X, antidomain(zero)))))
725.43/92.09	= { by lemma 30 }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(forward_diamond(X, one), antidomain(multiplication(X, one))))
725.43/92.09	= { by axiom 1 (multiplicative_right_identity) }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(forward_diamond(X, one), antidomain(X)))
725.43/92.09	= { by axiom 5 (additive_commutativity) }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), addition(antidomain(X), forward_diamond(X, one)))
725.43/92.09	= { by lemma 36 }
725.43/92.09	  multiplication(antidomain(forward_diamond(X, one)), antidomain(X))
725.43/92.09	= { by lemma 35 }
725.43/92.09	  multiplication(c(X), antidomain(X))
725.43/92.09	= { by lemma 37 R->L }
725.43/92.09	  multiplication(domain(antidomain(X)), antidomain(X))
725.43/92.09	= { by lemma 38 }
725.43/92.09	  antidomain(X)
725.43/92.09	
725.43/92.09	Lemma 40: antidomain(domain(X)) = antidomain(X).
725.43/92.09	Proof:
725.43/92.09	  antidomain(domain(X))
725.43/92.09	= { by axiom 8 (complement) R->L }
725.43/92.09	  c(X)
725.43/92.09	= { by lemma 39 }
725.43/92.09	  antidomain(X)
725.43/92.09	
725.43/92.09	Lemma 41: domain(domain(X)) = antidomain(c(X)).
725.43/92.09	Proof:
725.43/92.09	  domain(domain(X))
725.43/92.09	= { by axiom 7 (domain4) }
725.43/92.09	  antidomain(antidomain(domain(X)))
725.43/92.09	= { by axiom 8 (complement) R->L }
725.43/92.09	  antidomain(c(X))
725.43/92.09	
725.43/92.09	Lemma 42: addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y))) = coantidomain(multiplication(codomain(X), Y)).
725.43/92.09	Proof:
725.43/92.09	  addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y)))
725.43/92.09	= { by axiom 9 (codomain4) R->L }
725.43/92.09	  addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y)))
725.43/92.09	= { by axiom 25 (codomain2) }
725.43/92.09	  coantidomain(multiplication(coantidomain(coantidomain(X)), Y))
725.43/92.09	= { by axiom 9 (codomain4) }
725.43/92.09	  coantidomain(multiplication(codomain(X), Y))
725.43/92.09	
725.43/92.09	Lemma 43: leq(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y))) = true.
725.43/92.09	Proof:
725.43/92.09	  leq(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y)))
725.43/92.09	= { by axiom 23 (order) R->L }
725.43/92.09	  fresh(coantidomain(multiplication(codomain(X), Y)), addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y))), coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y)))
725.43/92.09	= { by lemma 42 }
725.43/92.09	  fresh(coantidomain(multiplication(codomain(X), Y)), coantidomain(multiplication(codomain(X), Y)), coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y)))
725.43/92.09	= { by axiom 17 (order) }
725.43/92.09	  true
725.43/92.09	
725.43/92.09	Lemma 44: addition(X, addition(X, Y)) = addition(X, Y).
725.43/92.09	Proof:
725.43/92.09	  addition(X, addition(X, Y))
725.43/92.09	= { by axiom 13 (additive_associativity) R->L }
725.43/92.09	  addition(addition(X, X), Y)
725.43/92.09	= { by axiom 4 (additive_idempotence) R->L }
725.43/92.09	  addition(X, Y)
725.43/92.09	
725.43/92.09	Lemma 45: addition(one, coantidomain(X)) = one.
725.43/92.09	Proof:
725.43/92.09	  addition(one, coantidomain(X))
725.43/92.09	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.09	  addition(coantidomain(X), one)
725.43/92.09	= { by lemma 32 R->L }
725.43/92.09	  addition(coantidomain(X), addition(coantidomain(X), codomain(X)))
725.43/92.09	= { by lemma 44 }
725.43/92.09	  addition(coantidomain(X), codomain(X))
725.43/92.09	= { by lemma 32 }
725.43/92.09	  one
725.43/92.09	
725.43/92.09	Lemma 46: addition(domain(X), antidomain(X)) = one.
725.43/92.09	Proof:
725.43/92.09	  addition(domain(X), antidomain(X))
725.43/92.09	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.09	  addition(antidomain(X), domain(X))
725.43/92.09	= { by lemma 29 }
725.43/92.09	  one
725.43/92.09	
725.43/92.09	Lemma 47: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
725.43/92.09	Proof:
725.43/92.09	  multiplication(addition(X, Y), coantidomain(X))
725.43/92.09	= { by axiom 5 (additive_commutativity) R->L }
725.43/92.09	  multiplication(addition(Y, X), coantidomain(X))
725.43/92.09	= { by axiom 22 (left_distributivity) R->L }
725.43/92.09	  addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
725.43/92.09	= { by axiom 10 (codomain1) R->L }
725.43/92.09	  addition(multiplication(Y, coantidomain(X)), zero)
725.43/92.09	= { by axiom 6 (additive_identity) }
725.43/92.09	  multiplication(Y, coantidomain(X))
725.43/92.09	
725.43/92.09	Lemma 48: coantidomain(antidomain(X)) = domain(X).
725.43/92.09	Proof:
725.43/92.09	  coantidomain(antidomain(X))
725.43/92.09	= { by lemma 40 R->L }
725.43/92.09	  coantidomain(antidomain(domain(X)))
725.43/92.09	= { by axiom 6 (additive_identity) R->L }
725.43/92.09	  addition(coantidomain(antidomain(domain(X))), zero)
725.43/92.09	= { by axiom 10 (codomain1) }
725.43/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X)))))
725.43/92.09	= { by axiom 18 (order_1) R->L }
725.43/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), fresh2(true, true, one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X))))))
726.04/92.09	= { by lemma 43 R->L }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), fresh2(leq(coantidomain(multiplication(antidomain(domain(X)), domain(X))), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X)))), true, one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X))))))
726.04/92.09	= { by axiom 11 (domain1) R->L }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), fresh2(leq(coantidomain(zero), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X)))), true, one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X))))))
726.04/92.09	= { by lemma 33 }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), fresh2(leq(one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X)))), true, one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X))))))
726.04/92.09	= { by axiom 24 (order_1) }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), addition(one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(X))))))
726.04/92.09	= { by lemma 45 }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(X)), one))
726.04/92.09	= { by axiom 12 (multiplicative_associativity) R->L }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(X), one)))
726.04/92.09	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(codomain(antidomain(domain(X))), domain(X)))
726.04/92.09	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.09	  addition(multiplication(codomain(antidomain(domain(X))), domain(X)), coantidomain(antidomain(domain(X))))
726.04/92.09	= { by axiom 1 (multiplicative_right_identity) R->L }
726.04/92.09	  addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), one))
726.04/92.09	= { by lemma 46 R->L }
726.04/92.09	  addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), addition(domain(X), antidomain(X))))
726.04/92.09	= { by lemma 44 R->L }
726.04/92.09	  addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), addition(domain(X), addition(domain(X), antidomain(X)))))
726.04/92.09	= { by lemma 46 }
726.04/92.09	  addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), addition(domain(X), one)))
726.04/92.09	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), addition(domain(X), one)), multiplication(codomain(antidomain(domain(X))), domain(X)))
726.04/92.09	= { by axiom 21 (right_distributivity) R->L }
726.04/92.09	  addition(addition(multiplication(coantidomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), one)), multiplication(codomain(antidomain(domain(X))), domain(X)))
726.04/92.09	= { by axiom 13 (additive_associativity) }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), domain(X)), addition(multiplication(coantidomain(antidomain(domain(X))), one), multiplication(codomain(antidomain(domain(X))), domain(X))))
726.04/92.09	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), domain(X)), addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), one)))
726.04/92.09	= { by axiom 13 (additive_associativity) R->L }
726.04/92.09	  addition(addition(multiplication(coantidomain(antidomain(domain(X))), domain(X)), multiplication(codomain(antidomain(domain(X))), domain(X))), multiplication(coantidomain(antidomain(domain(X))), one))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), one), addition(multiplication(coantidomain(antidomain(domain(X))), domain(X)), multiplication(codomain(antidomain(domain(X))), domain(X))))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), one), addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), domain(X))))
726.04/92.09	= { by axiom 22 (left_distributivity) }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), one), multiplication(addition(codomain(antidomain(domain(X))), coantidomain(antidomain(domain(X)))), domain(X)))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  addition(multiplication(coantidomain(antidomain(domain(X))), one), multiplication(addition(coantidomain(antidomain(domain(X))), codomain(antidomain(domain(X)))), domain(X)))
726.04/92.09	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(addition(coantidomain(antidomain(domain(X))), codomain(antidomain(domain(X)))), domain(X)))
726.04/92.09	= { by lemma 32 }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), multiplication(one, domain(X)))
726.04/92.09	= { by axiom 3 (multiplicative_left_identity) }
726.04/92.09	  addition(coantidomain(antidomain(domain(X))), domain(X))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  addition(domain(X), coantidomain(antidomain(domain(X))))
726.04/92.09	= { by lemma 40 }
726.04/92.09	  addition(domain(X), coantidomain(antidomain(X)))
726.04/92.09	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.09	  addition(coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 24 (order_1) R->L }
726.04/92.09	  fresh2(leq(coantidomain(antidomain(X)), domain(X)), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 3 (multiplicative_left_identity) R->L }
726.04/92.09	  fresh2(leq(multiplication(one, coantidomain(antidomain(X))), domain(X)), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by lemma 29 R->L }
726.04/92.09	  fresh2(leq(multiplication(addition(antidomain(X), domain(X)), coantidomain(antidomain(X))), domain(X)), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by lemma 47 }
726.04/92.09	  fresh2(leq(multiplication(domain(X), coantidomain(antidomain(X))), domain(X)), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 1 (multiplicative_right_identity) R->L }
726.04/92.09	  fresh2(leq(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), one)), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by lemma 32 R->L }
726.04/92.09	  fresh2(leq(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), addition(coantidomain(antidomain(X)), codomain(antidomain(X))))), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 21 (right_distributivity) R->L }
726.04/92.09	  fresh2(leq(multiplication(domain(X), coantidomain(antidomain(X))), addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X))))), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 23 (order) R->L }
726.04/92.09	  fresh2(fresh(addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X)))), addition(multiplication(domain(X), coantidomain(antidomain(X))), addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X))))), multiplication(domain(X), coantidomain(antidomain(X))), addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X))))), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by lemma 44 }
726.04/92.09	  fresh2(fresh(addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X)))), addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X)))), multiplication(domain(X), coantidomain(antidomain(X))), addition(multiplication(domain(X), coantidomain(antidomain(X))), multiplication(domain(X), codomain(antidomain(X))))), true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 17 (order) }
726.04/92.09	  fresh2(true, true, coantidomain(antidomain(X)), domain(X))
726.04/92.09	= { by axiom 18 (order_1) }
726.04/92.09	  domain(X)
726.04/92.09	
726.04/92.09	Lemma 49: forward_diamond(one, X) = antidomain(c(X)).
726.04/92.09	Proof:
726.04/92.09	  forward_diamond(one, X)
726.04/92.09	= { by axiom 14 (forward_diamond) }
726.04/92.09	  domain(multiplication(one, domain(X)))
726.04/92.09	= { by axiom 3 (multiplicative_left_identity) }
726.04/92.09	  domain(domain(X))
726.04/92.09	= { by lemma 41 }
726.04/92.09	  antidomain(c(X))
726.04/92.09	
726.04/92.09	Lemma 50: multiplication(X, codomain(X)) = X.
726.04/92.09	Proof:
726.04/92.09	  multiplication(X, codomain(X))
726.04/92.09	= { by axiom 6 (additive_identity) R->L }
726.04/92.09	  addition(multiplication(X, codomain(X)), zero)
726.04/92.09	= { by axiom 10 (codomain1) }
726.04/92.09	  addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
726.04/92.09	= { by axiom 21 (right_distributivity) }
726.04/92.09	  multiplication(X, addition(codomain(X), coantidomain(X)))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  multiplication(X, addition(coantidomain(X), codomain(X)))
726.04/92.09	= { by lemma 32 }
726.04/92.09	  multiplication(X, one)
726.04/92.09	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.09	  X
726.04/92.09	
726.04/92.09	Lemma 51: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
726.04/92.09	Proof:
726.04/92.09	  multiplication(addition(X, Y), coantidomain(Y))
726.04/92.09	= { by axiom 5 (additive_commutativity) }
726.04/92.09	  multiplication(addition(Y, X), coantidomain(Y))
726.04/92.09	= { by lemma 47 }
726.04/92.09	  multiplication(X, coantidomain(Y))
726.04/92.09	
726.04/92.09	Lemma 52: addition(backward_diamond(X, Y), coantidomain(multiplication(codomain(Y), X))) = one.
726.04/92.09	Proof:
726.04/92.09	  addition(backward_diamond(X, Y), coantidomain(multiplication(codomain(Y), X)))
726.04/92.09	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.09	  addition(coantidomain(multiplication(codomain(Y), X)), backward_diamond(X, Y))
726.04/92.09	= { by axiom 16 (backward_diamond) R->L }
726.04/92.09	  addition(coantidomain(multiplication(codomain(Y), X)), codomain(multiplication(codomain(Y), X)))
726.04/92.09	= { by lemma 32 }
726.04/92.10	  one
726.04/92.10	
726.04/92.10	Goal 1 (goals): addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)) = domain(x1).
726.04/92.10	Proof:
726.04/92.10	  addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1))
726.04/92.10	= { by axiom 5 (additive_commutativity) }
726.04/92.10	  addition(domain(x1), backward_diamond(x0, forward_box(x0, domain(x1))))
726.04/92.10	= { by axiom 1 (multiplicative_right_identity) R->L }
726.04/92.10	  addition(domain(x1), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), one))
726.04/92.10	= { by lemma 32 R->L }
726.04/92.10	  addition(domain(x1), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), addition(coantidomain(domain(x1)), codomain(domain(x1)))))
726.04/92.10	= { by axiom 21 (right_distributivity) R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), coantidomain(domain(x1))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by lemma 47 R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(addition(domain(x1), backward_diamond(x0, forward_box(x0, domain(x1)))), coantidomain(domain(x1))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by axiom 1 (multiplicative_right_identity) R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(coantidomain(domain(x1)), one)), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by axiom 12 (multiplicative_associativity) }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), one), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by lemma 45 R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), addition(one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by axiom 24 (order_1) R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by lemma 33 R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(zero), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by axiom 10 (codomain1) }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1)))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.10	= { by lemma 42 R->L }
726.04/92.10	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), addition(coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 12 (multiplicative_associativity) }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), addition(coantidomain(multiplication(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1)))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 10 (codomain1) R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), addition(coantidomain(zero), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 33 }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), addition(one, coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 45 }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))))), one)), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 12 (multiplicative_associativity) R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1)))))), one))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 12 (multiplicative_associativity) R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), multiplication(coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1))))), one)))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(x1)), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 7 (domain4) }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, antidomain(antidomain(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 39 R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, c(antidomain(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 37 R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(antidomain(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 39 R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, domain(c(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by axiom 7 (domain4) }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, antidomain(antidomain(c(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 39 R->L }
726.04/92.11	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, antidomain(c(c(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.11	= { by lemma 49 R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(forward_box(x0, forward_diamond(one, c(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 15 (forward_box) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(c(forward_diamond(x0, c(forward_diamond(one, c(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 15 (forward_box) R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(c(forward_diamond(x0, forward_box(one, antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 39 }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, forward_box(one, antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 38 R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(domain(forward_box(one, antidomain(x1))), forward_box(one, antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 7 (domain4) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), forward_box(one, antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 36 R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), addition(forward_box(one, antidomain(x1)), antidomain(forward_box(one, antidomain(x1))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 15 (forward_box) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), addition(forward_box(one, antidomain(x1)), antidomain(c(forward_diamond(one, c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 15 (forward_box) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), addition(c(forward_diamond(one, c(antidomain(x1)))), antidomain(c(forward_diamond(one, c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 41 R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), addition(c(forward_diamond(one, c(antidomain(x1)))), domain(domain(forward_diamond(one, c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 8 (complement) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), addition(antidomain(domain(forward_diamond(one, c(antidomain(x1))))), domain(domain(forward_diamond(one, c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 29 }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, multiplication(antidomain(antidomain(forward_box(one, antidomain(x1)))), one))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(forward_box(one, antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 15 (forward_box) }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(c(forward_diamond(one, c(antidomain(x1)))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 49 }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(c(antidomain(c(c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by lemma 37 R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(domain(antidomain(antidomain(c(c(antidomain(x1)))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.12	= { by axiom 7 (domain4) R->L }
726.04/92.12	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(domain(domain(c(c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 41 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(antidomain(c(c(c(antidomain(x1))))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by axiom 7 (domain4) R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(domain(c(c(c(antidomain(x1)))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(domain(antidomain(c(c(antidomain(x1)))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 37 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(c(c(c(antidomain(x1))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(c(c(antidomain(x1))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(antidomain(c(antidomain(x1))))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by axiom 7 (domain4) R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(domain(c(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(domain(antidomain(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 37 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(c(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(antidomain(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by axiom 7 (domain4) R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, domain(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 39 R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(c(forward_diamond(x0, domain(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 37 R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(antidomain(forward_diamond(x0, domain(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 34 R->L }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(c(multiplication(x0, domain(domain(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.13	= { by lemma 41 }
726.04/92.13	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(c(multiplication(x0, antidomain(c(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 39 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(antidomain(multiplication(x0, antidomain(c(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 39 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(antidomain(multiplication(x0, antidomain(antidomain(antidomain(x1)))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 7 (domain4) R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(domain(antidomain(multiplication(x0, domain(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 37 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(c(multiplication(x0, domain(antidomain(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 34 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(forward_diamond(x0, antidomain(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 14 (forward_diamond) }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(domain(multiplication(x0, domain(antidomain(x1))))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 37 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(domain(multiplication(x0, c(x1)))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 8 (complement) R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(c(multiplication(x0, c(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 39 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, c(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 39 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(x0, coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 3 (multiplicative_left_identity) R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(x0, multiplication(one, coantidomain(domain(x1))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 46 R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(x0, multiplication(addition(domain(x1), antidomain(x1)), coantidomain(domain(x1))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by lemma 47 }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(x0, multiplication(antidomain(x1), coantidomain(domain(x1))))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 12 (multiplicative_associativity) }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(multiplication(x0, antidomain(x1)), coantidomain(domain(x1)))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 12 (multiplicative_associativity) }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(multiplication(antidomain(multiplication(x0, antidomain(x1))), multiplication(x0, antidomain(x1))), coantidomain(domain(x1))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 11 (domain1) R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(multiplication(zero, coantidomain(domain(x1))))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 2 (left_annihilation) R->L }
726.04/92.14	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(multiplication(x0, coantidomain(domain(x1))), coantidomain(zero)))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.14	= { by axiom 12 (multiplicative_associativity) R->L }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(x0, multiplication(coantidomain(domain(x1)), coantidomain(zero))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by lemma 33 }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(x0, multiplication(coantidomain(domain(x1)), one)))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by axiom 1 (multiplicative_right_identity) }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(x0, coantidomain(domain(x1))))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by lemma 51 }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), multiplication(x0, coantidomain(domain(x1))))), coantidomain(multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by axiom 12 (multiplicative_associativity) }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), x0), coantidomain(domain(x1)))), coantidomain(multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by axiom 16 (backward_diamond) R->L }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(leq(coantidomain(multiplication(multiplication(codomain(forward_box(x0, domain(x1))), x0), coantidomain(domain(x1)))), coantidomain(multiplication(codomain(multiplication(codomain(forward_box(x0, domain(x1))), x0)), coantidomain(domain(x1))))), true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by lemma 43 }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), fresh2(true, true, one, coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1)))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by axiom 18 (order_1) }
726.04/92.15	  addition(domain(x1), addition(multiplication(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))), coantidomain(multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(domain(x1))))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by axiom 10 (codomain1) R->L }
726.04/92.15	  addition(domain(x1), addition(zero, multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1)))))
726.04/92.15	= { by lemma 28 }
726.04/92.15	  addition(domain(x1), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1))))
726.04/92.15	= { by lemma 50 R->L }
726.04/92.15	  addition(multiplication(domain(x1), codomain(domain(x1))), multiplication(backward_diamond(x0, forward_box(x0, domain(x1))), codomain(domain(x1))))
726.04/92.15	= { by axiom 22 (left_distributivity) }
726.04/92.15	  multiplication(addition(domain(x1), backward_diamond(x0, forward_box(x0, domain(x1)))), codomain(domain(x1)))
726.04/92.15	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), codomain(domain(x1)))
726.04/92.15	= { by lemma 48 R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), codomain(coantidomain(antidomain(x1))))
726.04/92.15	= { by axiom 9 (codomain4) R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(coantidomain(coantidomain(antidomain(x1)))))
726.04/92.15	= { by axiom 9 (codomain4) }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(codomain(antidomain(x1))))
726.04/92.15	= { by axiom 3 (multiplicative_left_identity) R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(one, coantidomain(codomain(antidomain(x1)))))
726.04/92.15	= { by lemma 32 R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(addition(coantidomain(antidomain(x1)), codomain(antidomain(x1))), coantidomain(codomain(antidomain(x1)))))
726.04/92.15	= { by lemma 51 }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(coantidomain(antidomain(x1)), coantidomain(codomain(antidomain(x1)))))
726.04/92.15	= { by axiom 9 (codomain4) R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(coantidomain(antidomain(x1)), coantidomain(coantidomain(coantidomain(antidomain(x1))))))
726.04/92.15	= { by axiom 9 (codomain4) }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), multiplication(coantidomain(antidomain(x1)), codomain(coantidomain(antidomain(x1)))))
726.04/92.15	= { by lemma 50 }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), coantidomain(antidomain(x1)))
726.04/92.15	= { by lemma 48 }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), domain(x1))
726.04/92.15	= { by lemma 28 R->L }
726.04/92.15	  addition(zero, multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), domain(x1)))
726.04/92.15	= { by axiom 11 (domain1) }
726.04/92.15	  addition(multiplication(antidomain(domain(x1)), domain(x1)), multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), domain(x1)))
726.04/92.15	= { by axiom 8 (complement) R->L }
726.04/92.15	  addition(multiplication(c(x1), domain(x1)), multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), domain(x1)))
726.04/92.15	= { by axiom 22 (left_distributivity) }
726.04/92.15	  multiplication(addition(c(x1), addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1))), domain(x1))
726.04/92.15	= { by lemma 39 }
726.04/92.15	  multiplication(addition(antidomain(x1), addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1))), domain(x1))
726.04/92.15	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.15	  multiplication(addition(addition(backward_diamond(x0, forward_box(x0, domain(x1))), domain(x1)), antidomain(x1)), domain(x1))
726.04/92.15	= { by axiom 5 (additive_commutativity) }
726.04/92.15	  multiplication(addition(addition(domain(x1), backward_diamond(x0, forward_box(x0, domain(x1)))), antidomain(x1)), domain(x1))
726.04/92.15	= { by axiom 13 (additive_associativity) }
726.04/92.15	  multiplication(addition(domain(x1), addition(backward_diamond(x0, forward_box(x0, domain(x1))), antidomain(x1))), domain(x1))
726.04/92.15	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.15	  multiplication(addition(domain(x1), addition(antidomain(x1), backward_diamond(x0, forward_box(x0, domain(x1))))), domain(x1))
726.04/92.15	= { by axiom 13 (additive_associativity) R->L }
726.04/92.15	  multiplication(addition(addition(domain(x1), antidomain(x1)), backward_diamond(x0, forward_box(x0, domain(x1)))), domain(x1))
726.04/92.15	= { by lemma 46 }
726.04/92.15	  multiplication(addition(one, backward_diamond(x0, forward_box(x0, domain(x1)))), domain(x1))
726.04/92.15	= { by axiom 5 (additive_commutativity) R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), one), domain(x1))
726.04/92.15	= { by lemma 52 R->L }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), addition(backward_diamond(x0, forward_box(x0, domain(x1))), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), x0)))), domain(x1))
726.04/92.15	= { by lemma 44 }
726.04/92.15	  multiplication(addition(backward_diamond(x0, forward_box(x0, domain(x1))), coantidomain(multiplication(codomain(forward_box(x0, domain(x1))), x0))), domain(x1))
726.04/92.15	= { by lemma 52 }
726.04/92.15	  multiplication(one, domain(x1))
726.04/92.15	= { by axiom 3 (multiplicative_left_identity) }
726.04/92.15	  domain(x1)
726.04/92.15	% SZS output end Proof
726.04/92.15	
726.04/92.15	RESULT: Theorem (the conjecture is true).
726.04/92.18	EOF