0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.34	% Computer   : n020.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 180
0.14/0.34	% DateTime   : Thu Aug 29 12:56:55 EDT 2019
0.14/0.34	% CPUTime    : 
7.35/7.51	% SZS status Unsatisfiable
7.35/7.51	
7.35/7.51	% SZS output start Proof
7.35/7.51	Take the following subset of the input axioms:
7.50/7.67	  fof(condensed_detachment, axiom, ![A, B]: ifeq(is_a_theorem(equivalent(A, B)), true, ifeq(is_a_theorem(A), true, is_a_theorem(B), true), true)=true).
7.50/7.67	  fof(ifeq_axiom, axiom, ![A, B, C]: B=ifeq(A, A, B, C)).
7.50/7.67	  fof(prove_reflexivity, negated_conjecture, true!=is_a_theorem(equivalent(a, a))).
7.50/7.67	  fof(xcb, axiom, ![A, B, C]: true=is_a_theorem(equivalent(A, equivalent(equivalent(equivalent(A, B), equivalent(C, B)), C)))).
7.50/7.67	
7.50/7.67	Now clausify the problem and encode Horn clauses using encoding 3 of
7.50/7.67	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
7.50/7.67	We repeatedly replace C & s=t => u=v by the two clauses:
7.50/7.67	  fresh(y, y, x1...xn) = u
7.50/7.67	  C => fresh(s, t, x1...xn) = v
7.50/7.67	where fresh is a fresh function symbol and x1..xn are the free
7.50/7.67	variables of u and v.
7.50/7.67	A predicate p(X) is encoded as p(X)=true (this is sound, because the
7.50/7.67	input problem has no model of domain size 1).
7.50/7.67	
7.50/7.67	The encoding turns the above axioms into the following unit equations and goals:
7.50/7.67	
7.50/7.67	Axiom 1 (xcb): true = is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z))).
7.50/7.67	Axiom 2 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
7.50/7.68	Axiom 3 (condensed_detachment): ifeq(is_a_theorem(equivalent(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true) = true.
7.50/7.68	
7.50/7.68	Lemma 4: ifeq(is_a_theorem(X), true, is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), true) = true.
7.50/7.68	Proof:
7.50/7.68	  ifeq(is_a_theorem(X), true, is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), true)
7.50/7.68	= { by axiom 2 (ifeq_axiom) }
7.50/7.68	  ifeq(true, true, ifeq(is_a_theorem(X), true, is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), true), true)
7.50/7.68	= { by axiom 1 (xcb) }
7.50/7.68	  ifeq(is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z))), true, ifeq(is_a_theorem(X), true, is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), true), true)
7.50/7.68	= { by axiom 3 (condensed_detachment) }
7.50/7.68	  true
7.50/7.68	
7.50/7.68	Lemma 5: is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), equivalent(V, W)), V)) = true.
7.50/7.68	Proof:
7.50/7.68	  is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), equivalent(V, W)), V))
7.50/7.68	= { by axiom 2 (ifeq_axiom) }
7.50/7.68	  ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), equivalent(V, W)), V)), true)
7.50/7.68	= { by axiom 1 (xcb) }
7.50/7.68	  ifeq(is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), equivalent(V, W)), V)), true)
7.50/7.68	= { by lemma 4 }
7.50/7.68	  true
7.50/7.68	
7.50/7.68	Lemma 6: ifeq(is_a_theorem(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W)), true, is_a_theorem(W), true) = true.
7.50/7.68	Proof:
7.50/7.68	  ifeq(is_a_theorem(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W)), true, is_a_theorem(W), true)
7.50/7.68	= { by axiom 2 (ifeq_axiom) }
7.50/7.68	  ifeq(is_a_theorem(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W)), true, ifeq(true, true, is_a_theorem(W), true), true)
7.50/7.68	= { by axiom 1 (xcb) }
7.50/7.68	  ifeq(is_a_theorem(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W)), true, ifeq(is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z))), true, is_a_theorem(W), true), true)
7.50/7.68	= { by axiom 3 (condensed_detachment) }
7.50/7.68	  true
7.50/7.68	
7.50/7.68	Lemma 7: is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V))) = true.
7.50/7.68	Proof:
7.50/7.68	  is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V)))
7.50/7.68	= { by axiom 2 (ifeq_axiom) }
7.50/7.68	  ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V))), true)
7.50/7.68	= { by lemma 5 }
7.50/7.68	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), equivalent(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V)), W)), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V)))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(Z, Y)), Z)), W), V), equivalent(W, V))), true)
7.50/7.68	= { by lemma 6 }
7.50/7.69	  true
7.50/7.69	
7.50/7.69	Lemma 8: is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)) = true.
7.50/7.69	Proof:
7.50/7.69	  is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(true, true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by lemma 6 }
7.50/7.69	  ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by lemma 7 }
7.50/7.69	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)))), true, is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by lemma 6 }
7.50/7.69	  ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true)
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by lemma 6 }
7.50/7.69	  ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by lemma 5 }
7.50/7.69	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by lemma 4 }
7.50/7.69	  ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by axiom 2 (ifeq_axiom) }
7.50/7.69	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X), equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?)), equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)), ?), equivalent(?, ?)), ?))), equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W)))), X)), true, is_a_theorem(equivalent(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Y, Z), equivalent(W, Z)), W))), X)), true), true)
7.50/7.69	= { by axiom 3 (condensed_detachment) }
7.50/7.70	  true
7.50/7.70	
7.50/7.70	Lemma 9: is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)) = true.
7.50/7.70	Proof:
7.50/7.70	  is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W))
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 3 (condensed_detachment) }
7.50/7.70	  ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)), equivalent(equivalent(X, Y), equivalent(X, Y)))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by lemma 5 }
7.50/7.70	  ifeq(ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by lemma 8 }
7.50/7.70	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)), equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)))), true, is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)), equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)))), true, ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true), true), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by lemma 7 }
7.50/7.70	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)), equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)), equivalent(X, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X)))), true, is_a_theorem(equivalent(equivalent(equivalent(?, equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), ?)), X), equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), X))), true), true), true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 3 (condensed_detachment) }
7.50/7.70	  ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true)
7.50/7.70	= { by lemma 4 }
7.50/7.70	  true
7.50/7.70	
7.50/7.70	Lemma 10: is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W))) = true.
7.50/7.70	Proof:
7.50/7.70	  is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W)))
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W))), true)
7.50/7.70	= { by lemma 9 }
7.50/7.70	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W)), Z)), equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W)))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), W), equivalent(Z, W))), true)
7.50/7.70	= { by lemma 6 }
7.50/7.70	  true
7.50/7.70	
7.50/7.70	Lemma 11: ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z))), true, is_a_theorem(W), true) = true.
7.50/7.70	Proof:
7.50/7.70	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z))), true, is_a_theorem(W), true)
7.50/7.70	= { by axiom 2 (ifeq_axiom) }
7.50/7.70	  ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z))), true, is_a_theorem(W), true), true)
7.50/7.70	= { by lemma 9 }
7.50/7.70	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z)), W)), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), Z), equivalent(W, Z))), true, is_a_theorem(W), true), true)
7.50/7.70	= { by axiom 3 (condensed_detachment) }
7.81/7.98	  true
7.81/7.98	
7.81/7.98	Goal 1 (prove_reflexivity): true = is_a_theorem(equivalent(a, a)).
7.81/7.98	Proof:
7.81/7.98	  true
7.81/7.98	= { by axiom 3 (condensed_detachment) }
7.81/7.98	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(a, a))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true, is_a_theorem(equivalent(a, a)), true), true)
7.81/7.98	= { by lemma 10 }
7.81/7.98	  ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true, is_a_theorem(equivalent(a, a)), true), true)
7.81/7.98	= { by axiom 2 (ifeq_axiom) }
7.81/7.98	  ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.98	= { by axiom 2 (ifeq_axiom) }
7.81/7.98	  ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.98	= { by axiom 3 (condensed_detachment) }
7.81/7.98	  ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.98	= { by axiom 2 (ifeq_axiom) }
7.81/7.98	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.98	= { by lemma 4 }
7.81/7.99	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.99	= { by axiom 2 (ifeq_axiom) }
7.81/7.99	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/7.99	= { by axiom 3 (condensed_detachment) }
7.81/8.00	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/8.00	= { by lemma 8 }
7.81/8.01	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/8.01	= { by axiom 2 (ifeq_axiom) }
7.81/8.01	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/8.01	= { by axiom 2 (ifeq_axiom) }
7.81/8.02	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.81/8.02	= { by lemma 6 }
7.81/8.03	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.03	= { by lemma 10 }
7.89/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.04	= { by axiom 2 (ifeq_axiom) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.05	= { by lemma 11 }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.05	= { by axiom 2 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.06	= { by lemma 11 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.06	= { by axiom 2 (ifeq_axiom) }
7.89/8.07	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)))), true), true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by lemma 6 }
7.89/8.07	  ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by axiom 2 (ifeq_axiom) }
7.89/8.07	  ifeq(ifeq(ifeq(true, true, ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by lemma 9 }
7.89/8.07	  ifeq(ifeq(ifeq(true, true, ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a)))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by lemma 4 }
7.89/8.07	  ifeq(ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by axiom 2 (ifeq_axiom) }
7.89/8.07	  ifeq(ifeq(ifeq(true, true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by axiom 2 (ifeq_axiom) }
7.89/8.07	  ifeq(ifeq(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a))), equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a))), true, is_a_theorem(equivalent(equivalent(equivalent(equivalent(?, ?), equivalent(?, ?)), a), a)), true), true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by lemma 6 }
7.89/8.07	  ifeq(true, true, is_a_theorem(equivalent(a, a)), true)
7.89/8.07	= { by axiom 2 (ifeq_axiom) }
7.89/8.07	  is_a_theorem(equivalent(a, a))
7.89/8.07	% SZS output end Proof
7.89/8.07	
7.89/8.07	RESULT: Unsatisfiable (the axioms are contradictory).
7.93/8.07	EOF