0.11/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.11/0.33	% Computer   : n005.cluster.edu
0.11/0.33	% Model      : x86_64 x86_64
0.11/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.33	% Memory     : 8042.1875MB
0.11/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.33	% CPULimit   : 1200
0.11/0.33	% WCLimit    : 120
0.11/0.33	% DateTime   : Tue Jul 13 14:14:48 EDT 2021
0.11/0.33	% CPUTime    : 
11.61/2.37	% SZS status Theorem
11.61/2.37	
12.55/2.43	% SZS output start Proof
12.55/2.43	Take the following subset of the input axioms:
12.55/2.43	  fof(exemplo6GDDFULL012002, conjecture, ![A, B, C, O, A1, B1, C1]: (perp(O, A1, B1, C1) <= (midp(A1, C, B) & (circle(O, A, B, C) & (midp(C1, B, A) & midp(B1, C, A)))))).
12.55/2.43	  fof(ruleD1, axiom, ![A, B, C]: (coll(A, B, C) => coll(A, C, B))).
12.55/2.43	  fof(ruleD10, axiom, ![A, B, C, D, E, F]: ((perp(C, D, E, F) & para(A, B, C, D)) => perp(A, B, E, F))).
12.55/2.43	  fof(ruleD14, axiom, ![A, B, C, D]: (cyclic(A, B, D, C) <= cyclic(A, B, C, D))).
12.55/2.43	  fof(ruleD15, axiom, ![A, B, C, D]: (cyclic(A, B, C, D) => cyclic(A, C, B, D))).
12.55/2.43	  fof(ruleD16, axiom, ![A, B, C, D]: (cyclic(B, A, C, D) <= cyclic(A, B, C, D))).
12.55/2.43	  fof(ruleD17, axiom, ![A, B, C, D, E]: ((cyclic(A, B, C, D) & cyclic(A, B, C, E)) => cyclic(B, C, D, E))).
12.55/2.43	  fof(ruleD19, axiom, ![A, B, C, P, Q, D, U, V]: (eqangle(C, D, A, B, U, V, P, Q) <= eqangle(A, B, C, D, P, Q, U, V))).
12.55/2.43	  fof(ruleD2, axiom, ![A, B, C]: (coll(B, A, C) <= coll(A, B, C))).
12.55/2.43	  fof(ruleD21, axiom, ![A, B, C, P, Q, D, U, V]: (eqangle(A, B, P, Q, C, D, U, V) <= eqangle(A, B, C, D, P, Q, U, V))).
12.55/2.43	  fof(ruleD23, axiom, ![A, B, C, D]: (cong(A, B, D, C) <= cong(A, B, C, D))).
12.55/2.43	  fof(ruleD24, axiom, ![A, B, C, D]: (cong(A, B, C, D) => cong(C, D, A, B))).
12.55/2.43	  fof(ruleD40, axiom, ![A, B, C, P, Q, D]: (eqangle(A, B, P, Q, C, D, P, Q) <= para(A, B, C, D))).
12.55/2.43	  fof(ruleD42b, axiom, ![A, B, P, Q]: ((eqangle(P, A, P, B, Q, A, Q, B) & coll(P, Q, B)) => cyclic(A, B, P, Q))).
12.55/2.43	  fof(ruleD43, axiom, ![A, B, C, P, Q, R]: (cong(A, B, P, Q) <= (cyclic(A, B, C, P) & (eqangle(C, A, C, B, R, P, R, Q) & (cyclic(A, B, C, R) & cyclic(A, B, C, Q)))))).
12.55/2.43	  fof(ruleD44, axiom, ![A, B, C, E, F]: ((midp(E, A, B) & midp(F, A, C)) => para(E, F, B, C))).
12.55/2.43	  fof(ruleD56, axiom, ![A, B, P, Q]: (perp(A, B, P, Q) <= (cong(A, P, B, P) & cong(A, Q, B, Q)))).
12.55/2.43	  fof(ruleD57, axiom, ![A, B, P, Q]: ((cyclic(A, B, P, Q) & (cong(A, Q, B, Q) & cong(A, P, B, P))) => perp(P, A, A, Q))).
12.55/2.43	  fof(ruleD66, axiom, ![A, B, C]: (coll(A, B, C) <= para(A, B, A, C))).
12.55/2.43	  fof(ruleD68, axiom, ![A, B, C]: (cong(A, B, A, C) <= midp(A, B, C))).
12.55/2.43	  fof(ruleD73, axiom, ![A, B, C, P, Q, D, U, V]: ((para(P, Q, U, V) & eqangle(A, B, C, D, P, Q, U, V)) => para(A, B, C, D))).
12.55/2.43	  fof(ruleD8, axiom, ![A, B, C, D]: (perp(C, D, A, B) <= perp(A, B, C, D))).
12.55/2.43	  fof(ruleD9, axiom, ![A, B, C, D, E, F]: (para(A, B, E, F) <= (perp(C, D, E, F) & perp(A, B, C, D)))).
12.55/2.43	
12.55/2.43	Now clausify the problem and encode Horn clauses using encoding 3 of
12.55/2.43	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
12.55/2.43	We repeatedly replace C & s=t => u=v by the two clauses:
12.55/2.43	  fresh(y, y, x1...xn) = u
12.55/2.43	  C => fresh(s, t, x1...xn) = v
12.55/2.43	where fresh is a fresh function symbol and x1..xn are the free
12.55/2.43	variables of u and v.
12.55/2.43	A predicate p(X) is encoded as p(X)=true (this is sound, because the
12.55/2.43	input problem has no model of domain size 1).
12.55/2.43	
12.55/2.43	The encoding turns the above axioms into the following unit equations and goals:
12.55/2.43	
12.55/2.43	Axiom 1 (exemplo6GDDFULL012002): midp(b1, c, a) = true.
12.55/2.43	Axiom 2 (ruleD57): fresh179(X, X, Y, Z, W) = true.
12.55/2.43	Axiom 3 (ruleD1): fresh147(X, X, Y, Z, W) = true.
12.55/2.43	Axiom 4 (ruleD2): fresh134(X, X, Y, Z, W) = true.
12.55/2.43	Axiom 5 (ruleD66): fresh66(X, X, Y, Z, W) = true.
12.55/2.43	Axiom 6 (ruleD68): fresh63(X, X, Y, Z, W) = true.
12.55/2.43	Axiom 7 (ruleD10): fresh146(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 8 (ruleD14): fresh141(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 9 (ruleD15): fresh140(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 10 (ruleD16): fresh139(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 11 (ruleD17): fresh137(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 12 (ruleD23): fresh129(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 13 (ruleD24): fresh128(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 14 (ruleD42b): fresh103(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
12.55/2.43	Axiom 15 (ruleD42b): fresh102(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 16 (ruleD43): fresh101(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 17 (ruleD44): fresh99(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 18 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
12.55/2.43	Axiom 19 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 20 (ruleD57): fresh78(X, X, Y, Z, W, V) = perp(W, Y, Y, V).
12.55/2.43	Axiom 21 (ruleD73): fresh57(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 22 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 23 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
12.55/2.43	Axiom 24 (ruleD1): fresh147(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
12.55/2.43	Axiom 25 (ruleD17): fresh138(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
12.55/2.43	Axiom 26 (ruleD2): fresh134(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
12.55/2.43	Axiom 27 (ruleD44): fresh100(X, X, Y, Z, W, V, U) = para(V, U, Z, W).
12.55/2.43	Axiom 28 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
12.55/2.43	Axiom 29 (ruleD57): fresh178(X, X, Y, Z, W, V) = fresh179(cong(Y, W, Z, W), true, Y, W, V).
12.55/2.43	Axiom 30 (ruleD43): fresh159(X, X, Y, Z, W, V, U, T) = cong(Y, Z, V, U).
12.55/2.43	Axiom 31 (ruleD10): fresh148(X, X, Y, Z, W, V, U, T) = perp(Y, Z, U, T).
12.55/2.43	Axiom 32 (ruleD40): fresh105(X, X, Y, Z, W, V, U, T) = true.
12.55/2.43	Axiom 33 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
12.55/2.43	Axiom 34 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
12.55/2.43	Axiom 35 (ruleD14): fresh141(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Y, W, Z).
12.55/2.43	Axiom 36 (ruleD15): fresh140(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Z, Y, W).
12.55/2.43	Axiom 37 (ruleD16): fresh139(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(Y, X, Z, W).
12.55/2.43	Axiom 38 (ruleD23): fresh129(cong(X, Y, Z, W), true, X, Y, Z, W) = cong(X, Y, W, Z).
12.55/2.43	Axiom 39 (ruleD24): fresh128(cong(X, Y, Z, W), true, X, Y, Z, W) = cong(Z, W, X, Y).
12.55/2.43	Axiom 40 (ruleD44): fresh100(midp(X, Y, Z), true, Y, W, Z, V, X) = fresh99(midp(V, Y, W), true, W, Z, V, X).
12.55/2.43	Axiom 41 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
12.55/2.43	Axiom 42 (ruleD57): fresh178(cyclic(X, Y, Z, W), true, X, Y, Z, W) = fresh78(cong(X, W, Y, W), true, X, Y, Z, W).
12.55/2.43	Axiom 43 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
12.55/2.43	Axiom 44 (ruleD17): fresh138(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh137(cyclic(X, Y, Z, V), true, Y, Z, V, W).
12.55/2.43	Axiom 45 (ruleD19): fresh135(X, X, Y, Z, W, V, U, T, S, X2) = true.
12.55/2.43	Axiom 46 (ruleD21): fresh132(X, X, Y, Z, W, V, U, T, S, X2) = true.
12.55/2.43	Axiom 47 (ruleD73): fresh58(X, X, Y, Z, W, V, U, T, S, X2) = para(Y, Z, W, V).
12.55/2.43	Axiom 48 (ruleD43): fresh158(X, X, Y, Z, W, V, U, T) = fresh159(cyclic(Y, Z, W, V), true, Y, Z, W, V, U, T).
12.55/2.43	Axiom 49 (ruleD43): fresh157(X, X, Y, Z, W, V, U, T) = fresh158(cyclic(Y, Z, W, U), true, Y, Z, W, V, U, T).
12.55/2.43	Axiom 50 (ruleD10): fresh148(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = fresh146(perp(Z, W, V, U), true, X, Y, V, U).
12.55/2.43	Axiom 51 (ruleD40): fresh105(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
12.55/2.43	Axiom 52 (ruleD9): fresh51(perp(X, Y, Z, W), true, V, U, X, Y, Z, W) = fresh50(perp(V, U, X, Y), true, V, U, Z, W).
12.55/2.43	Axiom 53 (ruleD42b): fresh103(eqangle(X, Y, X, Z, W, Y, W, Z), true, Y, Z, X, W) = fresh102(coll(X, W, Z), true, Y, Z, X, W).
12.55/2.43	Axiom 54 (ruleD43): fresh157(cyclic(X, Y, Z, W), true, X, Y, Z, V, U, W) = fresh101(eqangle(Z, X, Z, Y, W, V, W, U), true, X, Y, V, U).
12.55/2.43	Axiom 55 (ruleD19): fresh135(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(Z, W, X, Y, T, S, V, U).
12.55/2.43	Axiom 56 (ruleD21): fresh132(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(X, Y, V, U, Z, W, T, S).
12.64/2.43	Axiom 57 (ruleD73): fresh58(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = fresh57(para(V, U, T, S), true, X, Y, Z, W).
12.64/2.43	
12.64/2.43	Lemma 58: para(b1, b1, a, a) = true.
12.64/2.43	Proof:
12.64/2.43	  para(b1, b1, a, a)
12.64/2.43	= { by axiom 27 (ruleD44) R->L }
12.64/2.43	  fresh100(true, true, c, a, a, b1, b1)
12.64/2.43	= { by axiom 1 (exemplo6GDDFULL012002) R->L }
12.64/2.43	  fresh100(midp(b1, c, a), true, c, a, a, b1, b1)
12.64/2.43	= { by axiom 40 (ruleD44) }
12.64/2.43	  fresh99(midp(b1, c, a), true, a, a, b1, b1)
12.64/2.43	= { by axiom 1 (exemplo6GDDFULL012002) }
12.64/2.43	  fresh99(true, true, a, a, b1, b1)
12.64/2.43	= { by axiom 17 (ruleD44) }
12.64/2.43	  true
12.64/2.43	
12.64/2.43	Lemma 59: para(X, Y, X, Y) = true.
12.64/2.43	Proof:
12.64/2.43	  para(X, Y, X, Y)
12.64/2.44	= { by axiom 47 (ruleD73) R->L }
12.64/2.44	  fresh58(true, true, X, Y, X, Y, b1, b1, a, a)
12.64/2.44	= { by axiom 46 (ruleD21) R->L }
12.64/2.44	  fresh58(fresh132(true, true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.44	= { by axiom 45 (ruleD19) R->L }
12.64/2.44	  fresh58(fresh132(fresh135(true, true, b1, b1, X, Y, a, a, X, Y), true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.44	= { by axiom 32 (ruleD40) R->L }
12.64/2.44	  fresh58(fresh132(fresh135(fresh105(true, true, b1, b1, a, a, X, Y), true, b1, b1, X, Y, a, a, X, Y), true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.44	= { by lemma 58 R->L }
12.64/2.44	  fresh58(fresh132(fresh135(fresh105(para(b1, b1, a, a), true, b1, b1, a, a, X, Y), true, b1, b1, X, Y, a, a, X, Y), true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.44	= { by axiom 51 (ruleD40) }
12.64/2.45	  fresh58(fresh132(fresh135(eqangle(b1, b1, X, Y, a, a, X, Y), true, b1, b1, X, Y, a, a, X, Y), true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.45	= { by axiom 55 (ruleD19) }
12.64/2.45	  fresh58(fresh132(eqangle(X, Y, b1, b1, X, Y, a, a), true, X, Y, b1, b1, X, Y, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.45	= { by axiom 56 (ruleD21) }
12.64/2.45	  fresh58(eqangle(X, Y, X, Y, b1, b1, a, a), true, X, Y, X, Y, b1, b1, a, a)
12.64/2.45	= { by axiom 57 (ruleD73) }
12.64/2.45	  fresh57(para(b1, b1, a, a), true, X, Y, X, Y)
12.64/2.45	= { by lemma 58 }
12.64/2.45	  fresh57(true, true, X, Y, X, Y)
12.64/2.45	= { by axiom 21 (ruleD73) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 60: eqangle(X, Y, Z, W, X, Y, Z, W) = true.
12.64/2.45	Proof:
12.64/2.45	  eqangle(X, Y, Z, W, X, Y, Z, W)
12.64/2.45	= { by axiom 51 (ruleD40) R->L }
12.64/2.45	  fresh105(para(X, Y, X, Y), true, X, Y, X, Y, Z, W)
12.64/2.45	= { by lemma 59 }
12.64/2.45	  fresh105(true, true, X, Y, X, Y, Z, W)
12.64/2.45	= { by axiom 32 (ruleD40) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 61: cyclic(X, Y, X, Z) = true.
12.64/2.45	Proof:
12.64/2.45	  cyclic(X, Y, X, Z)
12.64/2.45	= { by axiom 37 (ruleD16) R->L }
12.64/2.45	  fresh139(cyclic(Y, X, X, Z), true, Y, X, X, Z)
12.64/2.45	= { by axiom 35 (ruleD14) R->L }
12.64/2.45	  fresh139(fresh141(cyclic(Y, X, Z, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 36 (ruleD15) R->L }
12.64/2.45	  fresh139(fresh141(fresh140(cyclic(Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 14 (ruleD42b) R->L }
12.64/2.45	  fresh139(fresh141(fresh140(fresh103(true, true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by lemma 60 R->L }
12.64/2.45	  fresh139(fresh141(fresh140(fresh103(eqangle(X, Y, X, Z, X, Y, X, Z), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 53 (ruleD42b) }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(coll(X, X, Z), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 24 (ruleD1) R->L }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(coll(X, Z, X), true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 26 (ruleD2) R->L }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(fresh134(coll(Z, X, X), true, Z, X, X), true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 33 (ruleD66) R->L }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(fresh134(fresh66(para(Z, X, Z, X), true, Z, X, X), true, Z, X, X), true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by lemma 59 }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(fresh134(fresh66(true, true, Z, X, X), true, Z, X, X), true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 5 (ruleD66) }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(fresh134(true, true, Z, X, X), true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 4 (ruleD2) }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(fresh147(true, true, X, Z, X), true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 3 (ruleD1) }
12.64/2.45	  fresh139(fresh141(fresh140(fresh102(true, true, Y, Z, X, X), true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 15 (ruleD42b) }
12.64/2.45	  fresh139(fresh141(fresh140(true, true, Y, Z, X, X), true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 9 (ruleD15) }
12.64/2.45	  fresh139(fresh141(true, true, Y, X, Z, X), true, Y, X, X, Z)
12.64/2.45	= { by axiom 8 (ruleD14) }
12.64/2.45	  fresh139(true, true, Y, X, X, Z)
12.64/2.45	= { by axiom 10 (ruleD16) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 62: cyclic(X, Y, Z, W) = true.
12.64/2.45	Proof:
12.64/2.45	  cyclic(X, Y, Z, W)
12.64/2.45	= { by axiom 25 (ruleD17) R->L }
12.64/2.45	  fresh138(true, true, Y, X, Y, Z, W)
12.64/2.45	= { by lemma 61 R->L }
12.64/2.45	  fresh138(cyclic(Y, X, Y, W), true, Y, X, Y, Z, W)
12.64/2.45	= { by axiom 44 (ruleD17) }
12.64/2.45	  fresh137(cyclic(Y, X, Y, Z), true, X, Y, Z, W)
12.64/2.45	= { by lemma 61 }
12.64/2.45	  fresh137(true, true, X, Y, Z, W)
12.64/2.45	= { by axiom 11 (ruleD17) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 63: cong(X, Y, X, Y) = true.
12.64/2.45	Proof:
12.64/2.45	  cong(X, Y, X, Y)
12.64/2.45	= { by axiom 30 (ruleD43) R->L }
12.64/2.45	  fresh159(true, true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by lemma 62 R->L }
12.64/2.45	  fresh159(cyclic(X, Y, Z, X), true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by axiom 48 (ruleD43) R->L }
12.64/2.45	  fresh158(true, true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by lemma 62 R->L }
12.64/2.45	  fresh158(cyclic(X, Y, Z, Y), true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by axiom 49 (ruleD43) R->L }
12.64/2.45	  fresh157(true, true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by lemma 62 R->L }
12.64/2.45	  fresh157(cyclic(X, Y, Z, Z), true, X, Y, Z, X, Y, Z)
12.64/2.45	= { by axiom 54 (ruleD43) }
12.64/2.45	  fresh101(eqangle(Z, X, Z, Y, Z, X, Z, Y), true, X, Y, X, Y)
12.64/2.45	= { by lemma 60 }
12.64/2.45	  fresh101(true, true, X, Y, X, Y)
12.64/2.45	= { by axiom 16 (ruleD43) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 64: perp(X, X, Y, Z) = true.
12.64/2.45	Proof:
12.64/2.45	  perp(X, X, Y, Z)
12.64/2.45	= { by axiom 18 (ruleD56) R->L }
12.64/2.45	  fresh80(true, true, X, X, Y, Z)
12.64/2.45	= { by lemma 63 R->L }
12.64/2.45	  fresh80(cong(X, Z, X, Z), true, X, X, Y, Z)
12.64/2.45	= { by axiom 41 (ruleD56) }
12.64/2.45	  fresh79(cong(X, Y, X, Y), true, X, X, Y, Z)
12.64/2.45	= { by lemma 63 }
12.64/2.45	  fresh79(true, true, X, X, Y, Z)
12.64/2.45	= { by axiom 19 (ruleD56) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 65: cong(a, b1, c, b1) = true.
12.64/2.45	Proof:
12.64/2.45	  cong(a, b1, c, b1)
12.64/2.45	= { by axiom 38 (ruleD23) R->L }
12.64/2.45	  fresh129(cong(a, b1, b1, c), true, a, b1, b1, c)
12.64/2.45	= { by axiom 39 (ruleD24) R->L }
12.64/2.45	  fresh129(fresh128(cong(b1, c, a, b1), true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 38 (ruleD23) R->L }
12.64/2.45	  fresh129(fresh128(fresh129(cong(b1, c, b1, a), true, b1, c, b1, a), true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 28 (ruleD68) R->L }
12.64/2.45	  fresh129(fresh128(fresh129(fresh63(midp(b1, c, a), true, b1, c, a), true, b1, c, b1, a), true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 1 (exemplo6GDDFULL012002) }
12.64/2.45	  fresh129(fresh128(fresh129(fresh63(true, true, b1, c, a), true, b1, c, b1, a), true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 6 (ruleD68) }
12.64/2.45	  fresh129(fresh128(fresh129(true, true, b1, c, b1, a), true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 12 (ruleD23) }
12.64/2.45	  fresh129(fresh128(true, true, b1, c, a, b1), true, a, b1, b1, c)
12.64/2.45	= { by axiom 13 (ruleD24) }
12.64/2.45	  fresh129(true, true, a, b1, b1, c)
12.64/2.45	= { by axiom 12 (ruleD23) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Lemma 66: para(X, Y, Z, W) = true.
12.64/2.45	Proof:
12.64/2.45	  para(X, Y, Z, W)
12.64/2.45	= { by axiom 34 (ruleD9) R->L }
12.64/2.45	  fresh51(true, true, X, Y, V, V, Z, W)
12.64/2.45	= { by lemma 64 R->L }
12.64/2.45	  fresh51(perp(V, V, Z, W), true, X, Y, V, V, Z, W)
12.64/2.45	= { by axiom 52 (ruleD9) }
12.64/2.45	  fresh50(perp(X, Y, V, V), true, X, Y, Z, W)
12.64/2.45	= { by axiom 43 (ruleD8) R->L }
12.64/2.45	  fresh50(fresh52(perp(V, V, X, Y), true, V, V, X, Y), true, X, Y, Z, W)
12.64/2.45	= { by lemma 64 }
12.64/2.45	  fresh50(fresh52(true, true, V, V, X, Y), true, X, Y, Z, W)
12.64/2.45	= { by axiom 22 (ruleD8) }
12.64/2.45	  fresh50(true, true, X, Y, Z, W)
12.64/2.45	= { by axiom 23 (ruleD9) }
12.64/2.45	  true
12.64/2.45	
12.64/2.45	Goal 1 (exemplo6GDDFULL012002_4): perp(o, a1, b1, c1) = true.
12.64/2.45	Proof:
12.64/2.45	  perp(o, a1, b1, c1)
12.64/2.45	= { by axiom 31 (ruleD10) R->L }
12.64/2.45	  fresh148(true, true, o, a1, a, b1, b1, c1)
12.64/2.45	= { by lemma 66 R->L }
12.64/2.45	  fresh148(para(o, a1, a, b1), true, o, a1, a, b1, b1, c1)
12.64/2.45	= { by axiom 50 (ruleD10) }
12.64/2.45	  fresh146(perp(a, b1, b1, c1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 43 (ruleD8) R->L }
12.64/2.45	  fresh146(fresh52(perp(b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 31 (ruleD10) R->L }
12.64/2.45	  fresh146(fresh52(fresh148(true, true, b1, c1, b1, a, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by lemma 66 R->L }
12.64/2.45	  fresh146(fresh52(fresh148(para(b1, c1, b1, a), true, b1, c1, b1, a, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 50 (ruleD10) }
12.64/2.45	  fresh146(fresh52(fresh146(perp(b1, a, a, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 20 (ruleD57) R->L }
12.64/2.45	  fresh146(fresh52(fresh146(fresh78(true, true, a, c, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by lemma 65 R->L }
12.64/2.45	  fresh146(fresh52(fresh146(fresh78(cong(a, b1, c, b1), true, a, c, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 42 (ruleD57) R->L }
12.64/2.45	  fresh146(fresh52(fresh146(fresh178(cyclic(a, c, b1, b1), true, a, c, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by lemma 62 }
12.64/2.45	  fresh146(fresh52(fresh146(fresh178(true, true, a, c, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 29 (ruleD57) }
12.64/2.45	  fresh146(fresh52(fresh146(fresh179(cong(a, b1, c, b1), true, a, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by lemma 65 }
12.64/2.45	  fresh146(fresh52(fresh146(fresh179(true, true, a, b1, b1), true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 2 (ruleD57) }
12.64/2.45	  fresh146(fresh52(fresh146(true, true, b1, c1, a, b1), true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 7 (ruleD10) }
12.64/2.45	  fresh146(fresh52(true, true, b1, c1, a, b1), true, o, a1, b1, c1)
12.64/2.45	= { by axiom 22 (ruleD8) }
12.64/2.45	  fresh146(true, true, o, a1, b1, c1)
12.64/2.45	= { by axiom 7 (ruleD10) }
12.64/2.45	  true
12.64/2.45	% SZS output end Proof
12.64/2.45	
12.64/2.45	RESULT: Theorem (the conjecture is true).
12.64/2.46	EOF