0.08/0.14	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.14	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.15/0.35	% Computer   : n031.cluster.edu
0.15/0.35	% Model      : x86_64 x86_64
0.15/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.15/0.35	% Memory     : 8042.1875MB
0.15/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.15/0.35	% CPULimit   : 1200
0.15/0.35	% WCLimit    : 120
0.15/0.35	% DateTime   : Tue Jul 13 16:34:29 EDT 2021
0.15/0.36	% CPUTime    : 
14.74/2.26	% SZS status Theorem
14.74/2.26	
15.36/2.35	% SZS output start Proof
15.36/2.35	Take the following subset of the input axioms:
15.36/2.35	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
15.36/2.35	  fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
15.36/2.35	  fof(kn2, axiom, ![P, Q]: is_a_theorem(implies(and(P, Q), P)) <=> kn2).
15.36/2.36	  fof(kn3, axiom, kn3 <=> ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(not(and(Q, R)), not(and(R, P)))))).
15.36/2.36	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
15.36/2.36	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
15.36/2.36	  fof(op_and, axiom, op_and => ![X, Y]: not(or(not(X), not(Y)))=and(X, Y)).
15.36/2.36	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
15.36/2.36	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
15.36/2.36	  fof(op_implies_or, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y) <= op_implies_or).
15.36/2.36	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
15.36/2.36	  fof(principia_op_and, axiom, op_and).
15.36/2.36	  fof(principia_op_implies_or, axiom, op_implies_or).
15.36/2.36	  fof(principia_r3, conjecture, r3).
15.36/2.36	  fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
15.36/2.36	  fof(rosser_kn1, axiom, kn1).
15.36/2.36	  fof(rosser_kn2, axiom, kn2).
15.36/2.36	  fof(rosser_kn3, axiom, kn3).
15.36/2.36	  fof(rosser_modus_ponens, axiom, modus_ponens).
15.36/2.36	  fof(rosser_op_equiv, axiom, op_equiv).
15.36/2.36	  fof(rosser_op_implies_and, axiom, op_implies_and).
15.36/2.36	  fof(rosser_op_or, axiom, op_or).
15.36/2.36	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
15.36/2.36	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
15.36/2.36	
15.36/2.36	Now clausify the problem and encode Horn clauses using encoding 3 of
15.36/2.36	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
15.36/2.36	We repeatedly replace C & s=t => u=v by the two clauses:
15.36/2.36	  fresh(y, y, x1...xn) = u
15.36/2.36	  C => fresh(s, t, x1...xn) = v
15.36/2.36	where fresh is a fresh function symbol and x1..xn are the free
15.36/2.36	variables of u and v.
15.36/2.36	A predicate p(X) is encoded as p(X)=true (this is sound, because the
15.36/2.36	input problem has no model of domain size 1).
15.36/2.36	
15.36/2.36	The encoding turns the above axioms into the following unit equations and goals:
15.36/2.36	
15.36/2.36	Axiom 1 (rosser_modus_ponens): modus_ponens = true.
15.36/2.36	Axiom 2 (rosser_kn3): kn3 = true.
15.36/2.36	Axiom 3 (rosser_kn1): kn1 = true.
15.36/2.36	Axiom 4 (rosser_kn2): kn2 = true.
15.36/2.36	Axiom 5 (substitution_of_equivalents): substitution_of_equivalents = true.
15.36/2.36	Axiom 6 (rosser_op_equiv): op_equiv = true.
15.36/2.36	Axiom 7 (principia_op_implies_or): op_implies_or = true.
15.36/2.36	Axiom 8 (rosser_op_implies_and): op_implies_and = true.
15.36/2.36	Axiom 9 (rosser_op_or): op_or = true.
15.36/2.36	Axiom 10 (principia_op_and): op_and = true.
15.36/2.36	Axiom 11 (r3): fresh9(X, X) = true.
15.36/2.36	Axiom 12 (modus_ponens_2): fresh60(X, X, Y) = true.
15.36/2.36	Axiom 13 (kn1_1): fresh33(X, X, Y) = true.
15.36/2.36	Axiom 14 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
15.36/2.36	Axiom 15 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(is_a_theorem(Y), true, Z).
15.36/2.36	Axiom 16 (kn2_1): fresh31(X, X, Y, Z) = true.
15.36/2.36	Axiom 17 (modus_ponens_2): fresh28(X, X, Y, Z) = is_a_theorem(Z).
15.36/2.36	Axiom 18 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
15.36/2.36	Axiom 19 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
15.36/2.36	Axiom 20 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
15.36/2.36	Axiom 21 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
15.36/2.36	Axiom 22 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
15.36/2.36	Axiom 23 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
15.36/2.36	Axiom 24 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
15.36/2.36	Axiom 25 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
15.36/2.36	Axiom 26 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
15.36/2.36	Axiom 27 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
15.36/2.36	Axiom 28 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
15.36/2.36	Axiom 29 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
15.36/2.36	Axiom 30 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
15.36/2.36	Axiom 31 (kn3_1): fresh29(X, X, Y, Z, W) = true.
15.36/2.36	Axiom 32 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
15.36/2.36	Axiom 33 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
15.36/2.36	Axiom 34 (modus_ponens_2): fresh59(modus_ponens, true, X, Y) = fresh28(is_a_theorem(implies(X, Y)), true, X, Y).
15.36/2.36	Axiom 35 (modus_tollens_1): fresh25(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
15.36/2.36	Axiom 36 (r3): fresh9(is_a_theorem(implies(or(p8, q6), or(q6, p8))), true) = r3.
15.36/2.36	Axiom 37 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
15.36/2.36	
15.36/2.36	Lemma 38: not(and(X, not(Y))) = implies(X, Y).
15.36/2.36	Proof:
15.36/2.36	  not(and(X, not(Y)))
15.36/2.36	= { by axiom 21 (op_implies_and) R->L }
15.36/2.36	  fresh22(op_implies_and, true, X, Y)
15.36/2.36	= { by axiom 8 (rosser_op_implies_and) }
15.36/2.36	  fresh22(true, true, X, Y)
15.36/2.36	= { by axiom 20 (op_implies_and) }
15.36/2.36	  implies(X, Y)
15.36/2.36	
15.36/2.36	Lemma 39: is_a_theorem(implies(X, and(X, X))) = true.
15.36/2.36	Proof:
15.36/2.36	  is_a_theorem(implies(X, and(X, X)))
15.36/2.36	= { by axiom 26 (kn1_1) R->L }
15.36/2.36	  fresh33(kn1, true, X)
15.36/2.36	= { by axiom 3 (rosser_kn1) }
15.36/2.36	  fresh33(true, true, X)
15.36/2.36	= { by axiom 13 (kn1_1) }
15.36/2.36	  true
15.36/2.36	
15.36/2.36	Lemma 40: fresh28(is_a_theorem(implies(X, Y)), true, X, Y) = fresh60(is_a_theorem(X), true, Y).
15.36/2.36	Proof:
15.36/2.36	  fresh28(is_a_theorem(implies(X, Y)), true, X, Y)
15.36/2.36	= { by axiom 34 (modus_ponens_2) R->L }
15.36/2.36	  fresh59(modus_ponens, true, X, Y)
15.36/2.36	= { by axiom 1 (rosser_modus_ponens) }
15.36/2.36	  fresh59(true, true, X, Y)
15.36/2.36	= { by axiom 15 (modus_ponens_2) }
15.36/2.36	  fresh60(is_a_theorem(X), true, Y)
15.36/2.36	
15.36/2.36	Lemma 41: implies(not(X), Y) = or(X, Y).
15.36/2.36	Proof:
15.36/2.36	  implies(not(X), Y)
15.36/2.36	= { by lemma 38 R->L }
15.36/2.36	  not(and(not(X), not(Y)))
15.36/2.36	= { by axiom 30 (op_or) R->L }
15.36/2.36	  fresh20(op_or, true, X, Y)
15.36/2.36	= { by axiom 9 (rosser_op_or) }
15.36/2.36	  fresh20(true, true, X, Y)
15.36/2.36	= { by axiom 24 (op_or) }
15.36/2.36	  or(X, Y)
15.36/2.36	
15.36/2.36	Lemma 42: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = true.
15.36/2.36	Proof:
15.36/2.36	  is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
15.36/2.36	= { by lemma 41 R->L }
15.36/2.36	  is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
15.36/2.36	= { by axiom 37 (kn3_1) R->L }
15.36/2.36	  fresh29(kn3, true, X, Y, Z)
15.36/2.36	= { by axiom 2 (rosser_kn3) }
15.36/2.36	  fresh29(true, true, X, Y, Z)
15.36/2.36	= { by axiom 31 (kn3_1) }
15.36/2.36	  true
15.36/2.36	
15.36/2.36	Lemma 43: or(not(X), Y) = implies(X, Y).
15.36/2.36	Proof:
15.36/2.36	  or(not(X), Y)
15.36/2.36	= { by axiom 23 (op_implies_or) R->L }
15.36/2.36	  fresh21(op_implies_or, true, X, Y)
15.36/2.36	= { by axiom 7 (principia_op_implies_or) }
15.36/2.36	  fresh21(true, true, X, Y)
15.36/2.36	= { by axiom 22 (op_implies_or) }
15.36/2.36	  implies(X, Y)
15.36/2.36	
15.36/2.36	Lemma 44: not(implies(X, not(Y))) = and(X, Y).
15.36/2.36	Proof:
15.36/2.36	  not(implies(X, not(Y)))
15.36/2.36	= { by lemma 43 R->L }
15.36/2.36	  not(or(not(X), not(Y)))
15.36/2.36	= { by axiom 29 (op_and) R->L }
15.36/2.36	  fresh24(op_and, true, X, Y)
15.36/2.36	= { by axiom 10 (principia_op_and) }
15.36/2.36	  fresh24(true, true, X, Y)
15.36/2.36	= { by axiom 18 (op_and) }
15.36/2.36	  and(X, Y)
15.36/2.36	
15.36/2.36	Lemma 45: fresh58(and_1, true, X, Y) = true.
15.36/2.36	Proof:
15.36/2.36	  fresh58(and_1, true, X, Y)
15.36/2.36	= { by axiom 27 (and_1_1) }
15.36/2.36	  is_a_theorem(implies(and(X, Y), X))
15.36/2.36	= { by axiom 28 (kn2_1) R->L }
15.36/2.36	  fresh31(kn2, true, X, Y)
15.36/2.36	= { by axiom 4 (rosser_kn2) }
15.36/2.36	  fresh31(true, true, X, Y)
15.36/2.36	= { by axiom 16 (kn2_1) }
15.36/2.36	  true
15.36/2.36	
15.36/2.36	Lemma 46: not(or(X, not(Y))) = and(not(X), Y).
15.36/2.36	Proof:
15.36/2.36	  not(or(X, not(Y)))
15.36/2.36	= { by lemma 41 R->L }
15.36/2.36	  not(implies(not(X), not(Y)))
15.36/2.36	= { by lemma 44 }
15.36/2.36	  and(not(X), Y)
15.36/2.36	
15.36/2.36	Lemma 47: and(not(not(X)), Y) = and(X, Y).
15.36/2.36	Proof:
15.36/2.36	  and(not(not(X)), Y)
15.36/2.36	= { by lemma 46 R->L }
15.36/2.36	  not(or(not(X), not(Y)))
15.36/2.36	= { by lemma 43 }
15.36/2.36	  not(implies(X, not(Y)))
15.36/2.36	= { by lemma 44 }
15.36/2.36	  and(X, Y)
15.36/2.36	
15.36/2.36	Lemma 48: fresh60(is_a_theorem(X), true, and(X, X)) = is_a_theorem(and(X, X)).
15.36/2.36	Proof:
15.36/2.36	  fresh60(is_a_theorem(X), true, and(X, X))
15.36/2.36	= { by lemma 40 R->L }
15.36/2.36	  fresh28(is_a_theorem(implies(X, and(X, X))), true, X, and(X, X))
15.36/2.36	= { by lemma 39 }
15.36/2.36	  fresh28(true, true, X, and(X, X))
15.36/2.36	= { by axiom 17 (modus_ponens_2) }
15.36/2.36	  is_a_theorem(and(X, X))
15.36/2.36	
15.36/2.36	Lemma 49: is_a_theorem(implies(X, X)) = true.
15.36/2.36	Proof:
15.36/2.36	  is_a_theorem(implies(X, X))
15.36/2.36	= { by lemma 38 R->L }
15.36/2.36	  is_a_theorem(not(and(X, not(X))))
15.36/2.36	= { by axiom 17 (modus_ponens_2) R->L }
15.36/2.36	  fresh28(true, true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by axiom 12 (modus_ponens_2) R->L }
15.36/2.36	  fresh28(fresh60(true, true, or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by lemma 39 R->L }
15.36/2.36	  fresh28(fresh60(is_a_theorem(implies(not(X), and(not(X), not(X)))), true, or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by lemma 40 R->L }
15.36/2.36	  fresh28(fresh28(is_a_theorem(implies(implies(not(X), and(not(X), not(X))), or(and(and(not(X), not(X)), X), not(and(X, not(X)))))), true, implies(not(X), and(not(X), not(X))), or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by lemma 42 }
15.36/2.36	  fresh28(fresh28(true, true, implies(not(X), and(not(X), not(X))), or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by axiom 17 (modus_ponens_2) }
15.36/2.36	  fresh28(is_a_theorem(or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by lemma 41 R->L }
15.36/2.36	  fresh28(is_a_theorem(implies(not(and(and(not(X), not(X)), X)), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
15.36/2.36	= { by lemma 40 }
15.36/2.36	  fresh60(is_a_theorem(not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
15.36/2.36	= { by lemma 44 R->L }
15.36/2.36	  fresh60(is_a_theorem(not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by axiom 17 (modus_ponens_2) R->L }
15.36/2.37	  fresh60(fresh28(true, true, and(not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X))), not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by lemma 45 R->L }
15.36/2.37	  fresh60(fresh28(fresh58(and_1, true, not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X))), true, and(not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X))), not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by axiom 27 (and_1_1) }
15.36/2.37	  fresh60(fresh28(is_a_theorem(implies(and(not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X))), not(not(implies(and(not(X), not(X)), not(X)))))), true, and(not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X))), not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by lemma 40 }
15.36/2.37	  fresh60(fresh60(is_a_theorem(and(not(not(implies(and(not(X), not(X)), not(X)))), implies(and(not(X), not(X)), not(X)))), true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by lemma 47 }
15.36/2.37	  fresh60(fresh60(is_a_theorem(and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by lemma 48 R->L }
15.36/2.37	  fresh60(fresh60(fresh60(is_a_theorem(implies(and(not(X), not(X)), not(X))), true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by axiom 27 (and_1_1) R->L }
15.36/2.37	  fresh60(fresh60(fresh60(fresh58(and_1, true, not(X), not(X)), true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by lemma 45 }
15.36/2.37	  fresh60(fresh60(fresh60(true, true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by axiom 12 (modus_ponens_2) }
15.36/2.37	  fresh60(fresh60(true, true, not(not(implies(and(not(X), not(X)), not(X))))), true, not(and(X, not(X))))
15.36/2.37	= { by axiom 12 (modus_ponens_2) }
15.36/2.37	  fresh60(true, true, not(and(X, not(X))))
15.36/2.37	= { by axiom 12 (modus_ponens_2) }
15.36/2.37	  true
15.36/2.37	
15.36/2.37	Lemma 50: and(not(and(X, Y)), Z) = and(implies(X, not(Y)), Z).
15.36/2.37	Proof:
15.36/2.37	  and(not(and(X, Y)), Z)
15.36/2.37	= { by lemma 46 R->L }
15.36/2.37	  not(or(and(X, Y), not(Z)))
15.36/2.37	= { by lemma 44 R->L }
15.36/2.37	  not(or(not(implies(X, not(Y))), not(Z)))
15.36/2.37	= { by lemma 43 }
15.36/2.37	  not(implies(implies(X, not(Y)), not(Z)))
15.36/2.37	= { by lemma 44 }
15.36/2.37	  and(implies(X, not(Y)), Z)
15.36/2.37	
15.36/2.37	Lemma 51: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
15.36/2.37	Proof:
15.36/2.37	  and(implies(X, Y), implies(Y, X))
15.36/2.37	= { by axiom 32 (op_equiv) R->L }
15.36/2.37	  fresh23(op_equiv, true, X, Y)
15.36/2.37	= { by axiom 6 (rosser_op_equiv) }
15.36/2.37	  fresh23(true, true, X, Y)
15.36/2.37	= { by axiom 19 (op_equiv) }
15.36/2.37	  equiv(X, Y)
15.36/2.37	
15.36/2.37	Lemma 52: fresh(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
15.36/2.37	Proof:
15.36/2.37	  fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
15.36/2.37	= { by axiom 33 (substitution_of_equivalents_2) R->L }
15.36/2.37	  fresh2(substitution_of_equivalents, true, X, Y)
15.36/2.37	= { by axiom 5 (substitution_of_equivalents) }
15.36/2.37	  fresh2(true, true, X, Y)
15.36/2.37	= { by axiom 25 (substitution_of_equivalents_2) }
15.36/2.37	  X
15.36/2.37	
15.36/2.37	Lemma 53: not(not(X)) = X.
15.36/2.37	Proof:
15.36/2.37	  not(not(X))
15.36/2.37	= { by axiom 14 (substitution_of_equivalents_2) R->L }
15.36/2.37	  fresh(true, true, X, not(not(X)))
15.36/2.37	= { by axiom 12 (modus_ponens_2) R->L }
15.36/2.37	  fresh(fresh60(true, true, and(implies(X, X), implies(X, X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 49 R->L }
15.36/2.37	  fresh(fresh60(is_a_theorem(implies(X, X)), true, and(implies(X, X), implies(X, X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 48 }
15.36/2.37	  fresh(is_a_theorem(and(implies(X, X), implies(X, X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 43 R->L }
15.36/2.37	  fresh(is_a_theorem(and(implies(X, X), or(not(X), X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 41 R->L }
15.36/2.37	  fresh(is_a_theorem(and(implies(X, X), implies(not(not(X)), X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 38 R->L }
15.36/2.37	  fresh(is_a_theorem(and(not(and(X, not(X))), implies(not(not(X)), X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 50 }
15.36/2.37	  fresh(is_a_theorem(and(implies(X, not(not(X))), implies(not(not(X)), X))), true, X, not(not(X)))
15.36/2.37	= { by lemma 51 }
15.36/2.37	  fresh(is_a_theorem(equiv(X, not(not(X)))), true, X, not(not(X)))
15.36/2.37	= { by lemma 52 }
15.36/2.37	  X
15.36/2.37	
15.36/2.37	Lemma 54: is_a_theorem(or(X, not(X))) = true.
15.36/2.37	Proof:
15.36/2.37	  is_a_theorem(or(X, not(X)))
15.36/2.37	= { by lemma 41 R->L }
15.36/2.37	  is_a_theorem(implies(not(X), not(X)))
15.36/2.37	= { by lemma 49 }
15.36/2.37	  true
15.36/2.37	
15.36/2.37	Lemma 55: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = fresh25(modus_tollens, true, Y, X).
15.36/2.37	Proof:
15.36/2.37	  is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
15.36/2.37	= { by lemma 41 R->L }
15.36/2.37	  is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
15.36/2.37	= { by axiom 35 (modus_tollens_1) R->L }
15.36/2.37	  fresh25(modus_tollens, true, Y, X)
15.36/2.37	
15.36/2.37	Lemma 56: or(and(not(X), Y), Z) = implies(or(X, not(Y)), Z).
15.36/2.37	Proof:
15.36/2.37	  or(and(not(X), Y), Z)
15.36/2.37	= { by lemma 46 R->L }
15.36/2.37	  or(not(or(X, not(Y))), Z)
15.36/2.37	= { by lemma 43 }
15.36/2.37	  implies(or(X, not(Y)), Z)
15.36/2.37	
15.36/2.37	Lemma 57: fresh25(modus_tollens, true, X, Y) = true.
15.36/2.37	Proof:
15.36/2.37	  fresh25(modus_tollens, true, X, Y)
15.36/2.37	= { by lemma 55 R->L }
15.36/2.37	  is_a_theorem(implies(or(Y, not(X)), implies(X, Y)))
15.36/2.37	= { by lemma 56 R->L }
15.36/2.37	  is_a_theorem(or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by axiom 17 (modus_ponens_2) R->L }
15.36/2.37	  fresh28(true, true, or(Y, not(Y)), or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by lemma 42 R->L }
15.36/2.37	  fresh28(is_a_theorem(implies(implies(not(Y), not(Y)), or(and(not(Y), X), not(and(X, not(Y)))))), true, or(Y, not(Y)), or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by lemma 38 }
15.36/2.37	  fresh28(is_a_theorem(implies(implies(not(Y), not(Y)), or(and(not(Y), X), implies(X, Y)))), true, or(Y, not(Y)), or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by lemma 41 }
15.36/2.37	  fresh28(is_a_theorem(implies(or(Y, not(Y)), or(and(not(Y), X), implies(X, Y)))), true, or(Y, not(Y)), or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by lemma 40 }
15.36/2.37	  fresh60(is_a_theorem(or(Y, not(Y))), true, or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by lemma 54 }
15.36/2.37	  fresh60(true, true, or(and(not(Y), X), implies(X, Y)))
15.36/2.37	= { by axiom 12 (modus_ponens_2) }
15.36/2.37	  true
15.36/2.37	
15.36/2.37	Lemma 58: implies(and(X, not(Y)), Z) = or(implies(X, Y), Z).
15.36/2.37	Proof:
15.36/2.37	  implies(and(X, not(Y)), Z)
15.36/2.37	= { by lemma 43 R->L }
15.36/2.37	  or(not(and(X, not(Y))), Z)
15.36/2.37	= { by lemma 38 }
15.36/2.37	  or(implies(X, Y), Z)
15.36/2.37	
15.36/2.37	Lemma 59: fresh60(is_a_theorem(implies(X, not(Y))), true, implies(Y, not(X))) = is_a_theorem(implies(Y, not(X))).
15.36/2.37	Proof:
15.36/2.37	  fresh60(is_a_theorem(implies(X, not(Y))), true, implies(Y, not(X)))
15.36/2.37	= { by lemma 40 R->L }
15.36/2.37	  fresh28(is_a_theorem(implies(implies(X, not(Y)), implies(Y, not(X)))), true, implies(X, not(Y)), implies(Y, not(X)))
15.36/2.37	= { by lemma 43 R->L }
15.36/2.37	  fresh28(is_a_theorem(implies(or(not(X), not(Y)), implies(Y, not(X)))), true, implies(X, not(Y)), implies(Y, not(X)))
15.36/2.37	= { by lemma 55 }
15.36/2.37	  fresh28(fresh25(modus_tollens, true, Y, not(X)), true, implies(X, not(Y)), implies(Y, not(X)))
15.36/2.37	= { by lemma 57 }
15.36/2.37	  fresh28(true, true, implies(X, not(Y)), implies(Y, not(X)))
15.36/2.37	= { by axiom 17 (modus_ponens_2) }
15.36/2.38	  is_a_theorem(implies(Y, not(X)))
15.36/2.38	
15.36/2.38	Goal 1 (principia_r3): r3 = true.
15.36/2.38	Proof:
15.36/2.38	  r3
15.36/2.38	= { by axiom 36 (r3) R->L }
15.36/2.38	  fresh9(is_a_theorem(implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by lemma 53 R->L }
15.36/2.38	  fresh9(is_a_theorem(implies(or(p8, q6), not(not(or(q6, p8))))), true)
15.36/2.38	= { by lemma 59 R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(implies(not(or(q6, p8)), not(or(p8, q6)))), true, implies(or(p8, q6), not(not(or(q6, p8))))), true)
15.36/2.38	= { by lemma 41 }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), not(or(p8, q6)))), true, implies(or(p8, q6), not(not(or(q6, p8))))), true)
15.36/2.38	= { by lemma 53 }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), not(or(p8, q6)))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by lemma 41 R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), not(implies(not(p8), q6)))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by axiom 14 (substitution_of_equivalents_2) R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(true, true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by axiom 12 (modus_ponens_2) R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh60(true, true, equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by lemma 54 R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh60(is_a_theorem(or(implies(not(p8), q6), not(implies(not(p8), q6)))), true, equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.38	= { by lemma 40 R->L }
15.36/2.38	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 43 R->L }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(or(not(not(p8)), q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 51 R->L }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(implies(and(not(p8), not(q6)), not(or(not(not(p8)), q6))), implies(not(or(not(not(p8)), q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 41 }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(implies(and(not(p8), not(q6)), not(or(not(not(p8)), q6))), or(or(not(not(p8)), q6), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 41 R->L }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(implies(and(not(p8), not(q6)), not(or(not(not(p8)), q6))), or(implies(not(not(not(p8))), q6), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 58 R->L }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(implies(and(not(p8), not(q6)), not(or(not(not(p8)), q6))), implies(and(not(not(not(p8))), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 50 R->L }
15.36/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(not(and(and(not(p8), not(q6)), or(not(not(p8)), q6))), implies(and(not(not(not(p8))), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
15.36/2.39	= { by lemma 41 R->L }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(not(and(and(not(p8), not(q6)), implies(not(not(not(p8))), q6))), implies(and(not(not(not(p8))), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 38 R->L }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(not(and(and(not(p8), not(q6)), not(and(not(not(not(p8))), not(q6))))), implies(and(not(not(not(p8))), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 38 }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), and(implies(and(not(p8), not(q6)), and(not(not(not(p8))), not(q6))), implies(and(not(not(not(p8))), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 51 }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), and(not(not(not(p8))), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 47 }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 56 R->L }
16.03/2.39	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(or(and(not(implies(not(p8), q6)), implies(not(p8), q6)), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.39	= { by lemma 38 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(or(and(not(implies(not(p8), q6)), not(and(not(p8), not(q6)))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 41 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(not(and(not(implies(not(p8), q6)), not(and(not(p8), not(q6))))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 38 }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(implies(not(implies(not(p8), q6)), and(not(p8), not(q6))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 41 }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(or(implies(not(p8), q6), and(not(p8), not(q6))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 58 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(implies(and(not(p8), not(q6)), and(not(p8), not(q6))), equiv(and(not(p8), not(q6)), and(not(p8), not(q6))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 51 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(is_a_theorem(implies(implies(and(not(p8), not(q6)), and(not(p8), not(q6))), and(implies(and(not(p8), not(q6)), and(not(p8), not(q6))), implies(and(not(p8), not(q6)), and(not(p8), not(q6)))))), true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 39 }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(fresh28(true, true, or(implies(not(p8), q6), not(implies(not(p8), q6))), equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by axiom 17 (modus_ponens_2) }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), fresh(is_a_theorem(equiv(and(not(p8), not(q6)), not(implies(not(p8), q6)))), true, and(not(p8), not(q6)), not(implies(not(p8), q6))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 52 }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(or(q6, p8), and(not(p8), not(q6)))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 41 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(implies(not(q6), p8), and(not(p8), not(q6)))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 46 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(or(implies(not(q6), p8), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 41 R->L }
16.03/2.40	  fresh9(fresh60(is_a_theorem(implies(not(implies(not(q6), p8)), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 59 R->L }
16.03/2.40	  fresh9(fresh60(fresh60(is_a_theorem(implies(or(p8, not(not(q6))), not(not(implies(not(q6), p8))))), true, implies(not(implies(not(q6), p8)), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 41 }
16.03/2.40	  fresh9(fresh60(fresh60(is_a_theorem(implies(or(p8, not(not(q6))), not(not(implies(not(q6), p8))))), true, or(implies(not(q6), p8), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 53 }
16.03/2.40	  fresh9(fresh60(fresh60(is_a_theorem(implies(or(p8, not(not(q6))), implies(not(q6), p8))), true, or(implies(not(q6), p8), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 55 }
16.03/2.40	  fresh9(fresh60(fresh60(fresh25(modus_tollens, true, not(q6), p8), true, or(implies(not(q6), p8), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by lemma 57 }
16.03/2.40	  fresh9(fresh60(fresh60(true, true, or(implies(not(q6), p8), not(or(p8, not(not(q6)))))), true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by axiom 12 (modus_ponens_2) }
16.03/2.40	  fresh9(fresh60(true, true, implies(or(p8, q6), or(q6, p8))), true)
16.03/2.40	= { by axiom 12 (modus_ponens_2) }
16.03/2.40	  fresh9(true, true)
16.03/2.40	= { by axiom 11 (r3) }
16.03/2.40	  true
16.03/2.40	% SZS output end Proof
16.03/2.40	
16.03/2.40	RESULT: Theorem (the conjecture is true).
16.03/2.41	EOF