TSTP Solution File: GEO565+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO565+1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 23:29:20 EDT 2023
% Result : Theorem 164.50s 21.19s
% Output : Proof 165.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO565+1 : TPTP v8.1.2. Released v7.5.0.
% 0.03/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n003.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 22:54:26 EDT 2023
% 0.12/0.33 % CPUTime :
% 164.50/21.19 Command-line arguments: --flatten
% 164.50/21.19
% 164.50/21.19 % SZS status Theorem
% 164.50/21.19
% 165.13/21.27 % SZS output start Proof
% 165.13/21.27 Take the following subset of the input axioms:
% 165.69/21.28 fof(exemplo6GDDFULL214027, conjecture, ![A, B, C, O, H, OC, OA, OB, MIDPNT1, MIDPNT2, MIDPNT3, MIDPNT4, MIDPNT5, MIDPNT6, MIDPNT7, MIDPNT8, MIDPNT9, MIDPNT01, MIDPNT11, MIDPNT21]: ((perp(A, B, C, H) & (perp(A, C, B, H) & (perp(B, C, A, H) & (midp(MIDPNT1, A, B) & (perp(A, B, MIDPNT1, O) & (midp(MIDPNT2, A, C) & (perp(A, C, MIDPNT2, O) & (midp(MIDPNT3, B, C) & (perp(B, C, MIDPNT3, O) & (midp(MIDPNT4, A, H) & (perp(A, H, MIDPNT4, OC) & (midp(MIDPNT5, A, B) & (perp(A, B, MIDPNT5, OC) & (midp(MIDPNT6, H, B) & (perp(H, B, MIDPNT6, OC) & (midp(MIDPNT7, B, H) & (perp(B, H, MIDPNT7, OA) & (midp(MIDPNT8, B, C) & (perp(B, C, MIDPNT8, OA) & (midp(MIDPNT9, H, C) & (perp(H, C, MIDPNT9, OA) & (midp(MIDPNT01, C, H) & (perp(C, H, MIDPNT01, OB) & (midp(MIDPNT11, C, A) & (perp(C, A, MIDPNT11, OB) & (midp(MIDPNT21, H, A) & perp(H, A, MIDPNT21, OB))))))))))))))))))))))))))) => cong(H, OA, H, OB))).
% 165.69/21.28 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 165.69/21.28 fof(ruleD11, axiom, ![M, B2, A2_2]: (midp(M, B2, A2_2) => midp(M, A2_2, B2))).
% 165.69/21.28 fof(ruleD17, axiom, ![D, E, B2, C2, A2_2]: ((cyclic(A2_2, B2, C2, D) & cyclic(A2_2, B2, C2, E)) => cyclic(B2, C2, D, E))).
% 165.69/21.28 fof(ruleD19, axiom, ![P, Q, U, V, B2, C2, D2, A2_2]: (eqangle(A2_2, B2, C2, D2, P, Q, U, V) => eqangle(C2, D2, A2_2, B2, U, V, P, Q))).
% 165.69/21.28 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 165.69/21.28 fof(ruleD21, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: (eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) => eqangle(A2_2, B2, P2, Q2, C2, D2, U2, V2))).
% 165.69/21.28 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 165.69/21.28 fof(ruleD4, axiom, ![B2, C2, D2, A2_2]: (para(A2_2, B2, C2, D2) => para(A2_2, B2, D2, C2))).
% 165.69/21.28 fof(ruleD40, axiom, ![B2, C2, D2, A2_2, P2, Q2]: (para(A2_2, B2, C2, D2) => eqangle(A2_2, B2, P2, Q2, C2, D2, P2, Q2))).
% 165.69/21.28 fof(ruleD42b, axiom, ![B2, A2_2, P2, Q2]: ((eqangle(P2, A2_2, P2, B2, Q2, A2_2, Q2, B2) & coll(P2, Q2, B2)) => cyclic(A2_2, B2, P2, Q2))).
% 165.69/21.28 fof(ruleD43, axiom, ![R, B2, C2, A2_2, P2, Q2]: ((cyclic(A2_2, B2, C2, P2) & (cyclic(A2_2, B2, C2, Q2) & (cyclic(A2_2, B2, C2, R) & eqangle(C2, A2_2, C2, B2, R, P2, R, Q2)))) => cong(A2_2, B2, P2, Q2))).
% 165.69/21.28 fof(ruleD44, axiom, ![F, B2, C2, E2, A2_2]: ((midp(E2, A2_2, B2) & midp(F, A2_2, C2)) => para(E2, F, B2, C2))).
% 165.69/21.28 fof(ruleD45, axiom, ![B2, C2, E2, F2, A2_2]: ((midp(E2, A2_2, B2) & (para(E2, F2, B2, C2) & coll(F2, A2_2, C2))) => midp(F2, A2_2, C2))).
% 165.69/21.28 fof(ruleD56, axiom, ![B2, A2_2, P2, Q2]: ((cong(A2_2, P2, B2, P2) & cong(A2_2, Q2, B2, Q2)) => perp(A2_2, B2, P2, Q2))).
% 165.69/21.28 fof(ruleD63, axiom, ![B2, C2, D2, A2_2, M2]: ((midp(M2, A2_2, B2) & midp(M2, C2, D2)) => para(A2_2, C2, B2, D2))).
% 165.69/21.28 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 165.69/21.28 fof(ruleD68, axiom, ![B2, C2, A2_2]: (midp(A2_2, B2, C2) => cong(A2_2, B2, A2_2, C2))).
% 165.69/21.28 fof(ruleD73, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: ((eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) & para(P2, Q2, U2, V2)) => para(A2_2, B2, C2, D2))).
% 165.69/21.28 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 165.69/21.28 fof(ruleD9, axiom, ![B2, C2, D2, E2, F2, A2_2]: ((perp(A2_2, B2, C2, D2) & perp(C2, D2, E2, F2)) => para(A2_2, B2, E2, F2))).
% 165.69/21.28
% 165.69/21.28 Now clausify the problem and encode Horn clauses using encoding 3 of
% 165.69/21.28 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 165.69/21.28 We repeatedly replace C & s=t => u=v by the two clauses:
% 165.69/21.28 fresh(y, y, x1...xn) = u
% 165.69/21.28 C => fresh(s, t, x1...xn) = v
% 165.69/21.28 where fresh is a fresh function symbol and x1..xn are the free
% 165.69/21.28 variables of u and v.
% 165.69/21.28 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 165.69/21.28 input problem has no model of domain size 1).
% 165.69/21.28
% 165.69/21.28 The encoding turns the above axioms into the following unit equations and goals:
% 165.69/21.28
% 165.69/21.28 Axiom 1 (exemplo6GDDFULL214027_15): midp(midpnt1, a, b) = true.
% 165.69/21.28 Axiom 2 (exemplo6GDDFULL214027_19): midp(midpnt11, c, a) = true.
% 165.69/21.28 Axiom 3 (exemplo6GDDFULL214027_1): perp(b, c, midpnt8, oa) = true.
% 165.69/21.28 Axiom 4 (exemplo6GDDFULL214027): perp(b, c, a, h) = true.
% 165.69/21.28 Axiom 5 (ruleD45): fresh181(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 6 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 7 (ruleD11): fresh144(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 8 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 9 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 10 (ruleD45): fresh98(X, X, Y, Z, W) = midp(W, Y, Z).
% 165.69/21.28 Axiom 11 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 12 (ruleD68): fresh63(X, X, Y, Z, W) = true.
% 165.69/21.28 Axiom 13 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 14 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 15 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 165.69/21.28 Axiom 16 (ruleD4): fresh105(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 17 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 165.69/21.28 Axiom 18 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 19 (ruleD44): fresh99(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 20 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 165.69/21.28 Axiom 21 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 22 (ruleD63): fresh69(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 23 (ruleD73): fresh57(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 24 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 25 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 165.69/21.28 Axiom 26 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 165.69/21.28 Axiom 27 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 165.69/21.28 Axiom 28 (ruleD44): fresh100(X, X, Y, Z, W, V, U) = para(V, U, Z, W).
% 165.69/21.28 Axiom 29 (ruleD63): fresh70(X, X, Y, Z, W, V, U) = para(Y, W, Z, V).
% 165.69/21.28 Axiom 30 (ruleD45): fresh180(X, X, Y, Z, W, V, U) = fresh181(coll(U, Y, W), true, Y, W, U).
% 165.69/21.28 Axiom 31 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 165.69/21.28 Axiom 32 (ruleD11): fresh144(midp(X, Y, Z), true, Z, Y, X) = midp(X, Z, Y).
% 165.69/21.28 Axiom 33 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 165.69/21.28 Axiom 34 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 165.69/21.28 Axiom 35 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
% 165.69/21.28 Axiom 36 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 165.69/21.28 Axiom 37 (ruleD50): fresh91(X, X, Y, Z, W, V, U) = eqangle(Y, Z, Y, W, V, Z, V, U).
% 165.69/21.29 Axiom 38 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 165.69/21.29 Axiom 39 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 165.69/21.29 Axiom 40 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 165.69/21.29 Axiom 41 (ruleD45): fresh180(midp(X, Y, Z), true, Y, Z, W, X, V) = fresh98(para(X, V, Z, W), true, Y, W, V).
% 165.69/21.29 Axiom 42 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 165.69/21.29 Axiom 43 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 165.69/21.29 Axiom 44 (ruleD4): fresh105(para(X, Y, Z, W), true, X, Y, Z, W) = para(X, Y, W, Z).
% 165.69/21.29 Axiom 45 (ruleD44): fresh100(midp(X, Y, Z), true, Y, W, Z, V, X) = fresh99(midp(V, Y, W), true, W, Z, V, X).
% 165.69/21.29 Axiom 46 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 165.69/21.29 Axiom 47 (ruleD63): fresh70(midp(X, Y, Z), true, W, V, Y, Z, X) = fresh69(midp(X, W, V), true, W, V, Y, Z).
% 165.69/21.29 Axiom 48 (ruleD73): fresh58(X, X, Y, Z, W, V, U, T, S, X2) = para(Y, Z, W, V).
% 165.69/21.29 Axiom 49 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 165.69/21.29 Axiom 50 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 165.69/21.29 Axiom 51 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 165.69/21.29 Axiom 52 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 165.69/21.29 Axiom 53 (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).
% 165.69/21.29 Axiom 54 (ruleD42b): fresh102(eqangle(X, Y, X, Z, W, Y, W, Z), true, Y, Z, X, W) = fresh101(coll(X, W, Z), true, Y, Z, X, W).
% 165.69/21.29 Axiom 55 (ruleD43): fresh182(eqangle(X, Y, X, Z, W, V, W, U), true, Y, Z, X, V, U, W) = fresh184(cyclic(Y, Z, X, W), true, Y, Z, X, V, U).
% 165.69/21.29 Axiom 56 (ruleD19): fresh134(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).
% 165.69/21.29 Axiom 57 (ruleD21): fresh131(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).
% 165.69/21.29 Axiom 58 (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).
% 165.69/21.29
% 165.69/21.29 Lemma 59: perp(b, c, midpnt8, oa) = midp(midpnt1, a, b).
% 165.69/21.29 Proof:
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by axiom 3 (exemplo6GDDFULL214027_1) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29
% 165.69/21.29 Lemma 60: perp(b, c, a, h) = perp(b, c, midpnt8, oa).
% 165.69/21.29 Proof:
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29 = { by axiom 4 (exemplo6GDDFULL214027) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29
% 165.69/21.29 Lemma 61: fresh144(X, X, Y, Z, W) = perp(b, c, a, h).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh144(X, X, Y, Z, W)
% 165.69/21.29 = { by axiom 7 (ruleD11) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 60 R->L }
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29
% 165.69/21.29 Lemma 62: fresh144(midp(X, Y, Z), perp(b, c, a, h), Z, Y, X) = midp(X, Z, Y).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh144(midp(X, Y, Z), perp(b, c, a, h), Z, Y, X)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh144(midp(X, Y, Z), perp(b, c, midpnt8, oa), Z, Y, X)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh144(midp(X, Y, Z), midp(midpnt1, a, b), Z, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh144(midp(X, Y, Z), true, Z, Y, X)
% 165.69/21.29 = { by axiom 32 (ruleD11) }
% 165.69/21.29 midp(X, Z, Y)
% 165.69/21.29
% 165.69/21.29 Lemma 63: perp(b, c, a, h) = midp(midpnt1, a, b).
% 165.69/21.29 Proof:
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29
% 165.69/21.29 Lemma 64: perp(b, c, a, h) = para(midpnt11, midpnt1, c, b).
% 165.69/21.29 Proof:
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 19 (ruleD44) R->L }
% 165.69/21.29 fresh99(perp(b, c, a, h), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 61 R->L }
% 165.69/21.29 fresh99(fresh144(perp(b, c, a, h), perp(b, c, a, h), a, c, midpnt11), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh99(fresh144(perp(b, c, midpnt8, oa), perp(b, c, a, h), a, c, midpnt11), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh99(fresh144(midp(midpnt1, a, b), perp(b, c, a, h), a, c, midpnt11), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh99(fresh144(true, perp(b, c, a, h), a, c, midpnt11), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 2 (exemplo6GDDFULL214027_19) R->L }
% 165.69/21.29 fresh99(fresh144(midp(midpnt11, c, a), perp(b, c, a, h), a, c, midpnt11), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 62 }
% 165.69/21.29 fresh99(midp(midpnt11, a, c), perp(b, c, a, h), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh99(midp(midpnt11, a, c), perp(b, c, midpnt8, oa), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh99(midp(midpnt11, a, c), midp(midpnt1, a, b), c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh99(midp(midpnt11, a, c), true, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 45 (ruleD44) R->L }
% 165.69/21.29 fresh100(midp(midpnt1, a, b), true, a, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 fresh100(midp(midpnt1, a, b), midp(midpnt1, a, b), a, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.29 fresh100(midp(midpnt1, a, b), perp(b, c, midpnt8, oa), a, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 60 R->L }
% 165.69/21.29 fresh100(midp(midpnt1, a, b), perp(b, c, a, h), a, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by lemma 63 R->L }
% 165.69/21.29 fresh100(perp(b, c, a, h), perp(b, c, a, h), a, c, b, midpnt11, midpnt1)
% 165.69/21.29 = { by axiom 28 (ruleD44) }
% 165.69/21.29 para(midpnt11, midpnt1, c, b)
% 165.69/21.29
% 165.69/21.29 Lemma 65: fresh131(eqangle(X, Y, Z, W, V, U, T, S), perp(b, c, a, h), X, Y, Z, W, V, U, T, S) = eqangle(X, Y, V, U, Z, W, T, S).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh131(eqangle(X, Y, Z, W, V, U, T, S), perp(b, c, a, h), X, Y, Z, W, V, U, T, S)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh131(eqangle(X, Y, Z, W, V, U, T, S), perp(b, c, midpnt8, oa), X, Y, Z, W, V, U, T, S)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh131(eqangle(X, Y, Z, W, V, U, T, S), midp(midpnt1, a, b), X, Y, Z, W, V, U, T, S)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh131(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S)
% 165.69/21.29 = { by axiom 57 (ruleD21) }
% 165.69/21.29 eqangle(X, Y, V, U, Z, W, T, S)
% 165.69/21.29
% 165.69/21.29 Lemma 66: fresh104(para(X, Y, Z, W), perp(b, c, a, h), X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh104(para(X, Y, Z, W), perp(b, c, a, h), X, Y, Z, W, V, U)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh104(para(X, Y, Z, W), perp(b, c, midpnt8, oa), X, Y, Z, W, V, U)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh104(para(X, Y, Z, W), midp(midpnt1, a, b), X, Y, Z, W, V, U)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U)
% 165.69/21.29 = { by axiom 52 (ruleD40) }
% 165.69/21.29 eqangle(X, Y, V, U, Z, W, V, U)
% 165.69/21.29
% 165.69/21.29 Lemma 67: fresh104(X, X, Y, Z, W, V, U, T) = perp(b, c, a, h).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh104(X, X, Y, Z, W, V, U, T)
% 165.69/21.29 = { by axiom 34 (ruleD40) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 60 R->L }
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29
% 165.69/21.29 Lemma 68: fresh131(X, X, Y, Z, W, V, U, T, S, X2) = perp(b, c, a, h).
% 165.69/21.29 Proof:
% 165.69/21.29 fresh131(X, X, Y, Z, W, V, U, T, S, X2)
% 165.69/21.29 = { by axiom 43 (ruleD21) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 60 R->L }
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29
% 165.69/21.29 Lemma 69: perp(b, c, a, h) = coll(X, X, Y).
% 165.69/21.29 Proof:
% 165.69/21.29 perp(b, c, a, h)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 perp(b, c, midpnt8, oa)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 midp(midpnt1, a, b)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 true
% 165.69/21.29 = { by axiom 6 (ruleD1) R->L }
% 165.69/21.29 fresh146(perp(b, c, a, h), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh146(perp(b, c, midpnt8, oa), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh146(midp(midpnt1, a, b), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh146(true, perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 8 (ruleD2) R->L }
% 165.69/21.29 fresh146(fresh133(perp(b, c, a, h), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh146(fresh133(perp(b, c, midpnt8, oa), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh146(fresh133(midp(midpnt1, a, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh146(fresh133(true, perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 11 (ruleD66) R->L }
% 165.69/21.29 fresh146(fresh133(fresh66(perp(b, c, a, h), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh146(fresh133(fresh66(perp(b, c, midpnt8, oa), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh146(fresh133(fresh66(midp(midpnt1, a, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh146(fresh133(fresh66(true, perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 23 (ruleD73) R->L }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh57(perp(b, c, a, h), perp(b, c, a, h), Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 64 }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh57(para(midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 60 }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh57(para(midpnt11, midpnt1, c, b), perp(b, c, midpnt8, oa), Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 59 }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh57(para(midpnt11, midpnt1, c, b), midp(midpnt1, a, b), Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh57(para(midpnt11, midpnt1, c, b), true, Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 58 (ruleD73) R->L }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh58(eqangle(Y, X, Y, X, midpnt11, midpnt1, c, b), true, Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.29 fresh146(fresh133(fresh66(fresh58(eqangle(Y, X, Y, X, midpnt11, midpnt1, c, b), midp(midpnt1, a, b), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.29 = { by lemma 59 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(eqangle(Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, midpnt8, oa), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(eqangle(Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 65 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(eqangle(Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 56 (ruleD19) R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(eqangle(midpnt11, midpnt1, Y, X, c, b, Y, X), true, midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(eqangle(midpnt11, midpnt1, Y, X, c, b, Y, X), midp(midpnt1, a, b), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(eqangle(midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, midpnt8, oa), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(eqangle(midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 66 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(fresh104(para(midpnt11, midpnt1, c, b), perp(b, c, a, h), midpnt11, midpnt1, c, b, Y, X), perp(b, c, a, h), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 64 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(fresh104(perp(b, c, a, h), perp(b, c, a, h), midpnt11, midpnt1, c, b, Y, X), perp(b, c, a, h), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 67 }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(perp(b, c, a, h), perp(b, c, a, h), midpnt11, midpnt1, Y, X, c, b, Y, X), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 42 (ruleD19) }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(true, perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(midp(midpnt1, a, b), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(perp(b, c, midpnt8, oa), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(fresh131(perp(b, c, a, h), perp(b, c, a, h), Y, X, midpnt11, midpnt1, Y, X, c, b), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 68 }
% 165.69/21.30 fresh146(fresh133(fresh66(fresh58(perp(b, c, a, h), perp(b, c, a, h), Y, X, Y, X, midpnt11, midpnt1, c, b), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 48 (ruleD73) }
% 165.69/21.30 fresh146(fresh133(fresh66(para(Y, X, Y, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 fresh146(fresh133(fresh66(para(Y, X, Y, X), perp(b, c, midpnt8, oa), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 fresh146(fresh133(fresh66(para(Y, X, Y, X), midp(midpnt1, a, b), Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 fresh146(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 39 (ruleD66) }
% 165.69/21.30 fresh146(fresh133(coll(Y, X, X), perp(b, c, a, h), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 fresh146(fresh133(coll(Y, X, X), perp(b, c, midpnt8, oa), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 fresh146(fresh133(coll(Y, X, X), midp(midpnt1, a, b), Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 fresh146(fresh133(coll(Y, X, X), true, Y, X, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by axiom 33 (ruleD2) }
% 165.69/21.30 fresh146(coll(X, Y, X), perp(b, c, a, h), X, Y, X)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 fresh146(coll(X, Y, X), perp(b, c, midpnt8, oa), X, Y, X)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 fresh146(coll(X, Y, X), midp(midpnt1, a, b), X, Y, X)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 fresh146(coll(X, Y, X), true, X, Y, X)
% 165.69/21.30 = { by axiom 31 (ruleD1) }
% 165.69/21.30 coll(X, X, Y)
% 165.69/21.30
% 165.69/21.30 Lemma 70: perp(b, c, a, h) = coll(X, Y, Z).
% 165.69/21.30 Proof:
% 165.69/21.30 perp(b, c, a, h)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 perp(b, c, midpnt8, oa)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 midp(midpnt1, a, b)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 true
% 165.69/21.30 = { by axiom 9 (ruleD3) R->L }
% 165.69/21.30 fresh119(perp(b, c, a, h), perp(b, c, a, h), Z, X, Y)
% 165.69/21.30 = { by lemma 69 }
% 165.69/21.30 fresh119(coll(Z, Z, X), perp(b, c, a, h), Z, X, Y)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 fresh119(coll(Z, Z, X), perp(b, c, midpnt8, oa), Z, X, Y)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 fresh119(coll(Z, Z, X), midp(midpnt1, a, b), Z, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 fresh119(coll(Z, Z, X), true, Z, X, Y)
% 165.69/21.30 = { by axiom 38 (ruleD3) R->L }
% 165.69/21.30 fresh120(coll(Z, Z, Y), true, Z, Z, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh120(coll(Z, Z, Y), midp(midpnt1, a, b), Z, Z, X, Y)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh120(coll(Z, Z, Y), perp(b, c, midpnt8, oa), Z, Z, X, Y)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh120(coll(Z, Z, Y), perp(b, c, a, h), Z, Z, X, Y)
% 165.69/21.30 = { by lemma 69 R->L }
% 165.69/21.30 fresh120(perp(b, c, a, h), perp(b, c, a, h), Z, Z, X, Y)
% 165.69/21.30 = { by axiom 15 (ruleD3) }
% 165.69/21.30 coll(X, Y, Z)
% 165.69/21.30
% 165.69/21.30 Lemma 71: fresh136(X, X, Y, Z, W, V) = perp(b, c, a, h).
% 165.69/21.30 Proof:
% 165.69/21.30 fresh136(X, X, Y, Z, W, V)
% 165.69/21.30 = { by axiom 14 (ruleD17) }
% 165.69/21.30 true
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 midp(midpnt1, a, b)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 perp(b, c, midpnt8, oa)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 perp(b, c, a, h)
% 165.69/21.30
% 165.69/21.30 Lemma 72: eqangle(b, a, X, Y, b, a, X, Y) = perp(b, c, a, h).
% 165.69/21.30 Proof:
% 165.69/21.30 eqangle(b, a, X, Y, b, a, X, Y)
% 165.69/21.30 = { by lemma 66 R->L }
% 165.69/21.30 fresh104(para(b, a, b, a), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 44 (ruleD4) R->L }
% 165.69/21.30 fresh104(fresh105(para(b, a, a, b), true, b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh104(fresh105(para(b, a, a, b), midp(midpnt1, a, b), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh104(fresh105(para(b, a, a, b), perp(b, c, midpnt8, oa), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh104(fresh105(para(b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 29 (ruleD63) R->L }
% 165.69/21.30 fresh104(fresh105(fresh70(perp(b, c, a, h), perp(b, c, a, h), b, a, a, b, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 63 }
% 165.69/21.30 fresh104(fresh105(fresh70(midp(midpnt1, a, b), perp(b, c, a, h), b, a, a, b, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 60 }
% 165.69/21.30 fresh104(fresh105(fresh70(midp(midpnt1, a, b), perp(b, c, midpnt8, oa), b, a, a, b, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 59 }
% 165.69/21.30 fresh104(fresh105(fresh70(midp(midpnt1, a, b), midp(midpnt1, a, b), b, a, a, b, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.30 fresh104(fresh105(fresh70(midp(midpnt1, a, b), true, b, a, a, b, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 47 (ruleD63) }
% 165.69/21.30 fresh104(fresh105(fresh69(midp(midpnt1, b, a), true, b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(midp(midpnt1, b, a), midp(midpnt1, a, b), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(midp(midpnt1, b, a), perp(b, c, midpnt8, oa), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(midp(midpnt1, b, a), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 62 R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(fresh144(midp(midpnt1, a, b), perp(b, c, a, h), b, a, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(fresh144(perp(b, c, midpnt8, oa), perp(b, c, a, h), b, a, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh104(fresh105(fresh69(fresh144(perp(b, c, a, h), perp(b, c, a, h), b, a, midpnt1), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 61 }
% 165.69/21.30 fresh104(fresh105(fresh69(perp(b, c, a, h), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 22 (ruleD63) }
% 165.69/21.30 fresh104(fresh105(true, perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.30 fresh104(fresh105(midp(midpnt1, a, b), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 59 R->L }
% 165.69/21.30 fresh104(fresh105(perp(b, c, midpnt8, oa), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by lemma 60 R->L }
% 165.69/21.30 fresh104(fresh105(perp(b, c, a, h), perp(b, c, a, h), b, a, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.30 = { by axiom 16 (ruleD4) }
% 165.69/21.31 fresh104(true, perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh104(midp(midpnt1, a, b), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh104(perp(b, c, midpnt8, oa), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh104(perp(b, c, a, h), perp(b, c, a, h), b, a, b, a, X, Y)
% 165.69/21.31 = { by lemma 67 }
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31
% 165.69/21.31 Lemma 73: perp(b, c, a, h) = cyclic(a, a, b, X).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 perp(b, c, midpnt8, oa)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 midp(midpnt1, a, b)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 true
% 165.69/21.31 = { by axiom 18 (ruleD42b) R->L }
% 165.69/21.31 fresh101(perp(b, c, a, h), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by lemma 70 }
% 165.69/21.31 fresh101(coll(b, X, a), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh101(coll(b, X, a), perp(b, c, midpnt8, oa), a, a, b, X)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh101(coll(b, X, a), midp(midpnt1, a, b), a, a, b, X)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh101(coll(b, X, a), true, a, a, b, X)
% 165.69/21.31 = { by axiom 54 (ruleD42b) R->L }
% 165.69/21.31 fresh102(eqangle(b, a, b, a, X, a, X, a), true, a, a, b, X)
% 165.69/21.31 = { by axiom 37 (ruleD50) R->L }
% 165.69/21.31 fresh102(fresh91(Y, Y, b, a, a, X, a), true, a, a, b, X)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh102(fresh91(Y, Y, b, a, a, X, a), midp(midpnt1, a, b), a, a, b, X)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh102(fresh91(Y, Y, b, a, a, X, a), perp(b, c, midpnt8, oa), a, a, b, X)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh102(fresh91(Y, Y, b, a, a, X, a), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by axiom 37 (ruleD50) }
% 165.69/21.31 fresh102(eqangle(b, a, b, a, X, a, X, a), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by lemma 65 R->L }
% 165.69/21.31 fresh102(fresh131(eqangle(b, a, X, a, b, a, X, a), perp(b, c, a, h), b, a, X, a, b, a, X, a), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by lemma 72 }
% 165.69/21.31 fresh102(fresh131(perp(b, c, a, h), perp(b, c, a, h), b, a, X, a, b, a, X, a), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by lemma 68 }
% 165.69/21.31 fresh102(perp(b, c, a, h), perp(b, c, a, h), a, a, b, X)
% 165.69/21.31 = { by axiom 17 (ruleD42b) }
% 165.69/21.31 cyclic(a, a, b, X)
% 165.69/21.31
% 165.69/21.31 Lemma 74: fresh137(cyclic(X, Y, Z, W), perp(b, c, a, h), X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), perp(b, c, a, h), Y, Z, V, W).
% 165.69/21.31 Proof:
% 165.69/21.31 fresh137(cyclic(X, Y, Z, W), perp(b, c, a, h), X, Y, Z, V, W)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh137(cyclic(X, Y, Z, W), perp(b, c, midpnt8, oa), X, Y, Z, V, W)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh137(cyclic(X, Y, Z, W), midp(midpnt1, a, b), X, Y, Z, V, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W)
% 165.69/21.31 = { by axiom 51 (ruleD17) }
% 165.69/21.31 fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh136(cyclic(X, Y, Z, V), midp(midpnt1, a, b), Y, Z, V, W)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh136(cyclic(X, Y, Z, V), perp(b, c, midpnt8, oa), Y, Z, V, W)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh136(cyclic(X, Y, Z, V), perp(b, c, a, h), Y, Z, V, W)
% 165.69/21.31
% 165.69/21.31 Lemma 75: perp(b, c, a, h) = cyclic(a, b, X, Y).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 71 R->L }
% 165.69/21.31 fresh136(perp(b, c, a, h), perp(b, c, a, h), a, b, X, Y)
% 165.69/21.31 = { by lemma 73 }
% 165.69/21.31 fresh136(cyclic(a, a, b, X), perp(b, c, a, h), a, b, X, Y)
% 165.69/21.31 = { by lemma 74 R->L }
% 165.69/21.31 fresh137(cyclic(a, a, b, Y), perp(b, c, a, h), a, a, b, X, Y)
% 165.69/21.31 = { by lemma 73 R->L }
% 165.69/21.31 fresh137(perp(b, c, a, h), perp(b, c, a, h), a, a, b, X, Y)
% 165.69/21.31 = { by axiom 27 (ruleD17) }
% 165.69/21.31 cyclic(a, b, X, Y)
% 165.69/21.31
% 165.69/21.31 Lemma 76: perp(b, c, a, h) = cyclic(b, X, Y, Z).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 71 R->L }
% 165.69/21.31 fresh136(perp(b, c, a, h), perp(b, c, a, h), b, X, Y, Z)
% 165.69/21.31 = { by lemma 75 }
% 165.69/21.31 fresh136(cyclic(a, b, X, Y), perp(b, c, a, h), b, X, Y, Z)
% 165.69/21.31 = { by lemma 74 R->L }
% 165.69/21.31 fresh137(cyclic(a, b, X, Z), perp(b, c, a, h), a, b, X, Y, Z)
% 165.69/21.31 = { by lemma 75 R->L }
% 165.69/21.31 fresh137(perp(b, c, a, h), perp(b, c, a, h), a, b, X, Y, Z)
% 165.69/21.31 = { by axiom 27 (ruleD17) }
% 165.69/21.31 cyclic(b, X, Y, Z)
% 165.69/21.31
% 165.69/21.31 Lemma 77: perp(b, c, a, h) = cyclic(X, Y, Z, W).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 71 R->L }
% 165.69/21.31 fresh136(perp(b, c, a, h), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 76 }
% 165.69/21.31 fresh136(cyclic(b, X, Y, Z), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 74 R->L }
% 165.69/21.31 fresh137(cyclic(b, X, Y, W), perp(b, c, a, h), b, X, Y, Z, W)
% 165.69/21.31 = { by lemma 76 R->L }
% 165.69/21.31 fresh137(perp(b, c, a, h), perp(b, c, a, h), b, X, Y, Z, W)
% 165.69/21.31 = { by axiom 27 (ruleD17) }
% 165.69/21.31 cyclic(X, Y, Z, W)
% 165.69/21.31
% 165.69/21.31 Lemma 78: perp(b, c, a, h) = cong(a, X, a, X).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 perp(b, c, midpnt8, oa)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 midp(midpnt1, a, b)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 true
% 165.69/21.31 = { by axiom 13 (ruleD43) R->L }
% 165.69/21.31 fresh185(perp(b, c, a, h), perp(b, c, a, h), a, X, a, X)
% 165.69/21.31 = { by lemma 77 }
% 165.69/21.31 fresh185(cyclic(a, X, b, a), perp(b, c, a, h), a, X, a, X)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh185(cyclic(a, X, b, a), perp(b, c, midpnt8, oa), a, X, a, X)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh185(cyclic(a, X, b, a), midp(midpnt1, a, b), a, X, a, X)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh185(cyclic(a, X, b, a), true, a, X, a, X)
% 165.69/21.31 = { by axiom 40 (ruleD43) R->L }
% 165.69/21.31 fresh184(perp(b, c, a, h), perp(b, c, a, h), a, X, b, a, X)
% 165.69/21.31 = { by lemma 77 }
% 165.69/21.31 fresh184(cyclic(a, X, b, b), perp(b, c, a, h), a, X, b, a, X)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh184(cyclic(a, X, b, b), perp(b, c, midpnt8, oa), a, X, b, a, X)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh184(cyclic(a, X, b, b), midp(midpnt1, a, b), a, X, b, a, X)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh184(cyclic(a, X, b, b), true, a, X, b, a, X)
% 165.69/21.31 = { by axiom 55 (ruleD43) R->L }
% 165.69/21.31 fresh182(eqangle(b, a, b, X, b, a, b, X), true, a, X, b, a, X, b)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh182(eqangle(b, a, b, X, b, a, b, X), midp(midpnt1, a, b), a, X, b, a, X, b)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh182(eqangle(b, a, b, X, b, a, b, X), perp(b, c, midpnt8, oa), a, X, b, a, X, b)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh182(eqangle(b, a, b, X, b, a, b, X), perp(b, c, a, h), a, X, b, a, X, b)
% 165.69/21.31 = { by lemma 72 }
% 165.69/21.31 fresh182(perp(b, c, a, h), perp(b, c, a, h), a, X, b, a, X, b)
% 165.69/21.31 = { by axiom 50 (ruleD43) }
% 165.69/21.31 fresh183(cyclic(a, X, b, X), true, a, X, b, a, X)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh183(cyclic(a, X, b, X), midp(midpnt1, a, b), a, X, b, a, X)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh183(cyclic(a, X, b, X), perp(b, c, midpnt8, oa), a, X, b, a, X)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh183(cyclic(a, X, b, X), perp(b, c, a, h), a, X, b, a, X)
% 165.69/21.31 = { by lemma 77 R->L }
% 165.69/21.31 fresh183(perp(b, c, a, h), perp(b, c, a, h), a, X, b, a, X)
% 165.69/21.31 = { by axiom 26 (ruleD43) }
% 165.69/21.31 cong(a, X, a, X)
% 165.69/21.31
% 165.69/21.31 Lemma 79: perp(a, a, X, Y) = perp(b, c, a, h).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(a, a, X, Y)
% 165.69/21.31 = { by axiom 20 (ruleD56) R->L }
% 165.69/21.31 fresh80(perp(b, c, a, h), perp(b, c, a, h), a, a, X, Y)
% 165.69/21.31 = { by lemma 78 }
% 165.69/21.31 fresh80(cong(a, Y, a, Y), perp(b, c, a, h), a, a, X, Y)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh80(cong(a, Y, a, Y), perp(b, c, midpnt8, oa), a, a, X, Y)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh80(cong(a, Y, a, Y), midp(midpnt1, a, b), a, a, X, Y)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh80(cong(a, Y, a, Y), true, a, a, X, Y)
% 165.69/21.31 = { by axiom 46 (ruleD56) }
% 165.69/21.31 fresh79(cong(a, X, a, X), true, a, a, X, Y)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh79(cong(a, X, a, X), midp(midpnt1, a, b), a, a, X, Y)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh79(cong(a, X, a, X), perp(b, c, midpnt8, oa), a, a, X, Y)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh79(cong(a, X, a, X), perp(b, c, a, h), a, a, X, Y)
% 165.69/21.31 = { by lemma 78 R->L }
% 165.69/21.31 fresh79(perp(b, c, a, h), perp(b, c, a, h), a, a, X, Y)
% 165.69/21.31 = { by axiom 21 (ruleD56) }
% 165.69/21.31 true
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 midp(midpnt1, a, b)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 perp(b, c, midpnt8, oa)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31
% 165.69/21.31 Lemma 80: perp(b, c, a, h) = para(X, Y, Z, W).
% 165.69/21.31 Proof:
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 perp(b, c, midpnt8, oa)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 midp(midpnt1, a, b)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 true
% 165.69/21.31 = { by axiom 25 (ruleD9) R->L }
% 165.69/21.31 fresh50(perp(b, c, a, h), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh50(perp(b, c, midpnt8, oa), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh50(midp(midpnt1, a, b), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh50(true, perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by axiom 24 (ruleD8) R->L }
% 165.69/21.31 fresh50(fresh52(perp(b, c, a, h), perp(b, c, a, h), a, a, X, Y), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 79 R->L }
% 165.69/21.31 fresh50(fresh52(perp(a, a, X, Y), perp(b, c, a, h), a, a, X, Y), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh50(fresh52(perp(a, a, X, Y), perp(b, c, midpnt8, oa), a, a, X, Y), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh50(fresh52(perp(a, a, X, Y), midp(midpnt1, a, b), a, a, X, Y), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh50(fresh52(perp(a, a, X, Y), true, a, a, X, Y), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by axiom 49 (ruleD8) }
% 165.69/21.31 fresh50(perp(X, Y, a, a), perp(b, c, a, h), X, Y, Z, W)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh50(perp(X, Y, a, a), perp(b, c, midpnt8, oa), X, Y, Z, W)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh50(perp(X, Y, a, a), midp(midpnt1, a, b), X, Y, Z, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh50(perp(X, Y, a, a), true, X, Y, Z, W)
% 165.69/21.31 = { by axiom 53 (ruleD9) R->L }
% 165.69/21.31 fresh51(perp(a, a, Z, W), true, X, Y, a, a, Z, W)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh51(perp(a, a, Z, W), midp(midpnt1, a, b), X, Y, a, a, Z, W)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh51(perp(a, a, Z, W), perp(b, c, midpnt8, oa), X, Y, a, a, Z, W)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh51(perp(a, a, Z, W), perp(b, c, a, h), X, Y, a, a, Z, W)
% 165.69/21.31 = { by lemma 79 }
% 165.69/21.31 fresh51(perp(b, c, a, h), perp(b, c, a, h), X, Y, a, a, Z, W)
% 165.69/21.31 = { by axiom 36 (ruleD9) }
% 165.69/21.31 para(X, Y, Z, W)
% 165.69/21.31
% 165.69/21.31 Lemma 81: fresh180(X, X, Y, Z, W, V, U) = perp(b, c, a, h).
% 165.69/21.31 Proof:
% 165.69/21.31 fresh180(X, X, Y, Z, W, V, U)
% 165.69/21.31 = { by axiom 30 (ruleD45) }
% 165.69/21.31 fresh181(coll(U, Y, W), true, Y, W, U)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh181(coll(U, Y, W), midp(midpnt1, a, b), Y, W, U)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh181(coll(U, Y, W), perp(b, c, midpnt8, oa), Y, W, U)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 fresh181(coll(U, Y, W), perp(b, c, a, h), Y, W, U)
% 165.69/21.31 = { by lemma 70 R->L }
% 165.69/21.31 fresh181(perp(b, c, a, h), perp(b, c, a, h), Y, W, U)
% 165.69/21.31 = { by axiom 5 (ruleD45) }
% 165.69/21.31 true
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 midp(midpnt1, a, b)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 perp(b, c, midpnt8, oa)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.31 perp(b, c, a, h)
% 165.69/21.31
% 165.69/21.31 Lemma 82: fresh180(midp(X, Y, Z), perp(b, c, a, h), Y, Z, W, X, V) = fresh98(para(X, V, Z, W), perp(b, c, a, h), Y, W, V).
% 165.69/21.31 Proof:
% 165.69/21.31 fresh180(midp(X, Y, Z), perp(b, c, a, h), Y, Z, W, X, V)
% 165.69/21.31 = { by lemma 60 }
% 165.69/21.31 fresh180(midp(X, Y, Z), perp(b, c, midpnt8, oa), Y, Z, W, X, V)
% 165.69/21.31 = { by lemma 59 }
% 165.69/21.31 fresh180(midp(X, Y, Z), midp(midpnt1, a, b), Y, Z, W, X, V)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) }
% 165.69/21.31 fresh180(midp(X, Y, Z), true, Y, Z, W, X, V)
% 165.69/21.31 = { by axiom 41 (ruleD45) }
% 165.69/21.31 fresh98(para(X, V, Z, W), true, Y, W, V)
% 165.69/21.31 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.31 fresh98(para(X, V, Z, W), midp(midpnt1, a, b), Y, W, V)
% 165.69/21.31 = { by lemma 59 R->L }
% 165.69/21.31 fresh98(para(X, V, Z, W), perp(b, c, midpnt8, oa), Y, W, V)
% 165.69/21.31 = { by lemma 60 R->L }
% 165.69/21.32 fresh98(para(X, V, Z, W), perp(b, c, a, h), Y, W, V)
% 165.69/21.32
% 165.69/21.32 Goal 1 (exemplo6GDDFULL214027_27): cong(h, oa, h, ob) = true.
% 165.69/21.32 Proof:
% 165.69/21.32 cong(h, oa, h, ob)
% 165.69/21.32 = { by axiom 35 (ruleD68) R->L }
% 165.69/21.32 fresh63(midp(h, oa, ob), true, h, oa, ob)
% 165.69/21.32 = { by axiom 1 (exemplo6GDDFULL214027_15) R->L }
% 165.69/21.32 fresh63(midp(h, oa, ob), midp(midpnt1, a, b), h, oa, ob)
% 165.69/21.32 = { by lemma 59 R->L }
% 165.69/21.32 fresh63(midp(h, oa, ob), perp(b, c, midpnt8, oa), h, oa, ob)
% 165.69/21.32 = { by lemma 60 R->L }
% 165.69/21.32 fresh63(midp(h, oa, ob), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by axiom 10 (ruleD45) R->L }
% 165.69/21.32 fresh63(fresh98(perp(b, c, a, h), perp(b, c, a, h), oa, ob, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 80 }
% 165.69/21.32 fresh63(fresh98(para(X, h, a, ob), perp(b, c, a, h), oa, ob, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 82 R->L }
% 165.69/21.32 fresh63(fresh180(midp(X, oa, a), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 62 R->L }
% 165.69/21.32 fresh63(fresh180(fresh144(midp(X, a, oa), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by axiom 10 (ruleD45) R->L }
% 165.69/21.32 fresh63(fresh180(fresh144(fresh98(perp(b, c, a, h), perp(b, c, a, h), a, oa, X), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 80 }
% 165.69/21.32 fresh63(fresh180(fresh144(fresh98(para(midpnt1, X, b, oa), perp(b, c, a, h), a, oa, X), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 82 R->L }
% 165.69/21.32 fresh63(fresh180(fresh144(fresh180(midp(midpnt1, a, b), perp(b, c, a, h), a, b, oa, midpnt1, X), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 63 R->L }
% 165.69/21.32 fresh63(fresh180(fresh144(fresh180(perp(b, c, a, h), perp(b, c, a, h), a, b, oa, midpnt1, X), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 81 }
% 165.69/21.32 fresh63(fresh180(fresh144(perp(b, c, a, h), perp(b, c, a, h), oa, a, X), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 61 }
% 165.69/21.32 fresh63(fresh180(perp(b, c, a, h), perp(b, c, a, h), oa, a, ob, X, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by lemma 81 }
% 165.69/21.32 fresh63(perp(b, c, a, h), perp(b, c, a, h), h, oa, ob)
% 165.69/21.32 = { by axiom 12 (ruleD68) }
% 165.69/21.32 true
% 165.69/21.32 % SZS output end Proof
% 165.69/21.32
% 165.69/21.32 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------