%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP776+1 : TPTP v9.3.1. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:11:58 AM UTC 2026
% Result : Theorem 2.07s 0.71s
% Output : Proof 2.07s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRP776+1 : TPTP v9.3.1. Released v4.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.37 % Computer : n017.cluster.edu
% 0.11/0.37 % Model : x86_64 x86_64
% 0.11/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.37 % Memory : 8046.5625MB
% 0.11/0.37 % OS : Linux 6.8.0-71-generic
% 0.11/0.37 % CPULimit : 300
% 0.11/0.37 % WCLimit : 300
% 0.11/0.37 % DateTime : Sun Sep 27 10:39:19 UTC 2026
% 0.11/0.37 % CPUTime :
% 0.11/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 2.07/0.71 Command-line arguments: --no-flatten-goal
% 2.07/0.71
% 2.07/0.71 % SZS status Theorem
% 2.07/0.71
% 2.07/0.75 % SZS output start Proof
% 2.07/0.75 Axiom 1 (sos03): g(eh) = true.
% 2.07/0.75 Axiom 2 (sos11): h(eg) = true.
% 2.07/0.75 Axiom 3 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 2.07/0.75 Axiom 4 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 2.07/0.75 Axiom 5 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 2.07/0.75 Axiom 6 (sos18): f(product(X, Y)) = sum(f(X), f(Y)).
% 2.07/0.75 Axiom 7 (goals_1): ifeq3(f(eh), eg, g(x0), true) = true.
% 2.07/0.75 Axiom 8 (sos02): ifeq(g(X), true, g(inv(X)), true) = true.
% 2.07/0.75 Axiom 9 (sos17): ifeq(g(X), true, h(f(X)), true) = true.
% 2.07/0.75 Axiom 10 (sos06): ifeq2(g(X), true, product(X, eh), X) = X.
% 2.07/0.75 Axiom 11 (sos10): ifeq(h(X), true, h(opp(Y)), true) = true.
% 2.07/0.75 Axiom 12 (sos07): ifeq2(g(X), true, product(X, inv(X)), eh) = eh.
% 2.07/0.75 Axiom 13 (sos14): ifeq2(h(X), true, sum(X, eg), X) = X.
% 2.07/0.75 Axiom 14 (sos13): ifeq2(h(X), true, sum(eg, X), X) = X.
% 2.07/0.75 Axiom 15 (sos15): ifeq2(h(X), true, sum(X, opp(X)), eg) = eg.
% 2.07/0.75 Axiom 16 (sos09): ifeq(h(X), true, ifeq(h(Y), true, h(sum(X, Y)), true), true) = true.
% 2.07/0.75 Axiom 17 (sos12): ifeq2(h(X), true, ifeq2(h(Y), true, ifeq2(h(Z), true, sum(sum(X, Y), Z), sum(X, sum(Y, Z))), sum(X, sum(Y, Z))), sum(X, sum(Y, Z))) = sum(X, sum(Y, Z)).
% 2.07/0.75
% 2.07/0.75 Lemma 18: opp(eg) = eg.
% 2.07/0.75 Proof:
% 2.07/0.75 opp(eg)
% 2.07/0.75 = { by axiom 14 (sos13) R->L }
% 2.07/0.75 ifeq2(h(opp(eg)), true, sum(eg, opp(eg)), opp(eg))
% 2.07/0.75 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.75 ifeq2(h(opp(eg)), true, ifeq2(true, true, sum(eg, opp(eg)), eg), opp(eg))
% 2.07/0.75 = { by axiom 2 (sos11) R->L }
% 2.07/0.75 ifeq2(h(opp(eg)), true, ifeq2(h(eg), true, sum(eg, opp(eg)), eg), opp(eg))
% 2.07/0.75 = { by axiom 15 (sos15) }
% 2.07/0.75 ifeq2(h(opp(eg)), true, eg, opp(eg))
% 2.07/0.75 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.75 ifeq2(ifeq(true, true, h(opp(eg)), true), true, eg, opp(eg))
% 2.07/0.75 = { by axiom 2 (sos11) R->L }
% 2.07/0.75 ifeq2(ifeq(h(eg), true, h(opp(eg)), true), true, eg, opp(eg))
% 2.07/0.75 = { by axiom 11 (sos10) }
% 2.07/0.75 ifeq2(true, true, eg, opp(eg))
% 2.07/0.75 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.75 eg
% 2.07/0.75
% 2.07/0.75 Lemma 19: sum(f(eh), eg) = f(eh).
% 2.07/0.75 Proof:
% 2.07/0.75 sum(f(eh), eg)
% 2.07/0.75 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.75 ifeq2(true, true, sum(f(eh), eg), f(eh))
% 2.07/0.75 = { by axiom 9 (sos17) R->L }
% 2.07/0.75 ifeq2(ifeq(g(eh), true, h(f(eh)), true), true, sum(f(eh), eg), f(eh))
% 2.07/0.75 = { by axiom 1 (sos03) }
% 2.07/0.75 ifeq2(ifeq(true, true, h(f(eh)), true), true, sum(f(eh), eg), f(eh))
% 2.07/0.75 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.75 ifeq2(h(f(eh)), true, sum(f(eh), eg), f(eh))
% 2.07/0.75 = { by axiom 13 (sos14) }
% 2.07/0.76 f(eh)
% 2.07/0.76
% 2.07/0.76 Lemma 20: f(eh) = eg.
% 2.07/0.76 Proof:
% 2.07/0.76 f(eh)
% 2.07/0.76 = { by lemma 19 R->L }
% 2.07/0.76 sum(f(eh), eg)
% 2.07/0.76 = { by axiom 15 (sos15) R->L }
% 2.07/0.76 sum(f(eh), ifeq2(h(f(eh)), true, sum(f(eh), opp(f(eh))), eg))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.76 sum(f(eh), ifeq2(ifeq(true, true, h(f(eh)), true), true, sum(f(eh), opp(f(eh))), eg))
% 2.07/0.76 = { by axiom 1 (sos03) R->L }
% 2.07/0.76 sum(f(eh), ifeq2(ifeq(g(eh), true, h(f(eh)), true), true, sum(f(eh), opp(f(eh))), eg))
% 2.07/0.76 = { by axiom 9 (sos17) }
% 2.07/0.76 sum(f(eh), ifeq2(true, true, sum(f(eh), opp(f(eh))), eg))
% 2.07/0.76 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(f(eh))))
% 2.07/0.76 = { by axiom 10 (sos06) R->L }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(f(ifeq2(g(eh), true, product(eh, eh), eh)))))
% 2.07/0.76 = { by axiom 1 (sos03) }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(f(ifeq2(true, true, product(eh, eh), eh)))))
% 2.07/0.76 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(f(product(eh, eh)))))
% 2.07/0.76 = { by axiom 6 (sos18) }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(sum(f(eh), f(eh)))))
% 2.07/0.76 = { by lemma 19 R->L }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(sum(f(eh), sum(f(eh), eg)))))
% 2.07/0.76 = { by lemma 18 R->L }
% 2.07/0.76 sum(f(eh), sum(f(eh), opp(sum(f(eh), sum(f(eh), opp(eg))))))
% 2.07/0.76 = { by lemma 19 R->L }
% 2.07/0.76 sum(f(eh), sum(sum(f(eh), eg), opp(sum(f(eh), sum(f(eh), opp(eg))))))
% 2.07/0.76 = { by lemma 18 R->L }
% 2.07/0.76 sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))
% 2.07/0.76 = { by axiom 17 (sos12) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(true, true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 16 (sos09) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(h(f(eh)), true, ifeq(h(sum(f(eh), opp(eg))), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(true, true, h(f(eh)), true), true, ifeq(h(sum(f(eh), opp(eg))), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 1 (sos03) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(g(eh), true, h(f(eh)), true), true, ifeq(h(sum(f(eh), opp(eg))), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 9 (sos17) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(true, true, ifeq(h(sum(f(eh), opp(eg))), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(h(sum(f(eh), opp(eg))), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(true, true, h(sum(f(eh), opp(eg))), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 9 (sos17) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(ifeq(g(eh), true, h(f(eh)), true), true, h(sum(f(eh), opp(eg))), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 1 (sos03) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(ifeq(true, true, h(f(eh)), true), true, h(sum(f(eh), opp(eg))), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(h(f(eh)), true, h(sum(f(eh), opp(eg))), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(h(f(eh)), true, ifeq(true, true, h(sum(f(eh), opp(eg))), true), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 11 (sos10) R->L }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(h(f(eh)), true, ifeq(ifeq(h(eg), true, h(opp(eg)), true), true, h(sum(f(eh), opp(eg))), true), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 2 (sos11) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(h(f(eh)), true, ifeq(ifeq(true, true, h(opp(eg)), true), true, h(sum(f(eh), opp(eg))), true), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(ifeq(h(f(eh)), true, ifeq(h(opp(eg)), true, h(sum(f(eh), opp(eg))), true), true), true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 16 (sos09) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(ifeq(true, true, h(sum(f(eh), sum(f(eh), opp(eg)))), true), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, ifeq2(h(sum(f(eh), sum(f(eh), opp(eg)))), true, sum(sum(f(eh), sum(f(eh), opp(eg))), opp(sum(f(eh), sum(f(eh), opp(eg))))), eg), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 15 (sos15) }
% 2.07/0.76 ifeq2(h(f(eh)), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.76 ifeq2(ifeq(true, true, h(f(eh)), true), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 1 (sos03) R->L }
% 2.07/0.76 ifeq2(ifeq(g(eh), true, h(f(eh)), true), true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 9 (sos17) }
% 2.07/0.76 ifeq2(true, true, ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.76 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.76 ifeq2(h(sum(f(eh), opp(eg))), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(ifeq(true, true, h(sum(f(eh), opp(eg))), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 9 (sos17) R->L }
% 2.07/0.77 ifeq2(ifeq(ifeq(g(eh), true, h(f(eh)), true), true, h(sum(f(eh), opp(eg))), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 1 (sos03) }
% 2.07/0.77 ifeq2(ifeq(ifeq(true, true, h(f(eh)), true), true, h(sum(f(eh), opp(eg))), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(ifeq(h(f(eh)), true, h(sum(f(eh), opp(eg))), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(ifeq(h(f(eh)), true, ifeq(true, true, h(sum(f(eh), opp(eg))), true), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 11 (sos10) R->L }
% 2.07/0.77 ifeq2(ifeq(h(f(eh)), true, ifeq(ifeq(h(eg), true, h(opp(eg)), true), true, h(sum(f(eh), opp(eg))), true), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 2 (sos11) }
% 2.07/0.77 ifeq2(ifeq(h(f(eh)), true, ifeq(ifeq(true, true, h(opp(eg)), true), true, h(sum(f(eh), opp(eg))), true), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(ifeq(h(f(eh)), true, ifeq(h(opp(eg)), true, h(sum(f(eh), opp(eg))), true), true), true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 16 (sos09) }
% 2.07/0.77 ifeq2(true, true, ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg))))))), sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(ifeq(true, true, h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 2 (sos11) R->L }
% 2.07/0.77 ifeq2(ifeq(h(eg), true, h(opp(sum(f(eh), sum(f(eh), opp(eg))))), true), true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 11 (sos10) }
% 2.07/0.77 ifeq2(true, true, eg, sum(f(eh), sum(sum(f(eh), opp(eg)), opp(sum(f(eh), sum(f(eh), opp(eg)))))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.77 eg
% 2.07/0.77
% 2.07/0.77 Lemma 21: sum(f(X), eg) = f(product(X, eh)).
% 2.07/0.77 Proof:
% 2.07/0.77 sum(f(X), eg)
% 2.07/0.77 = { by lemma 20 R->L }
% 2.07/0.77 sum(f(X), f(eh))
% 2.07/0.77 = { by axiom 6 (sos18) R->L }
% 2.07/0.77 f(product(X, eh))
% 2.07/0.77
% 2.07/0.77 Lemma 22: sum(opp(X), opp(opp(X))) = eg.
% 2.07/0.77 Proof:
% 2.07/0.77 sum(opp(X), opp(opp(X)))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(true, true, sum(opp(X), opp(opp(X))), eg)
% 2.07/0.77 = { by axiom 11 (sos10) R->L }
% 2.07/0.77 ifeq2(ifeq(h(eg), true, h(opp(X)), true), true, sum(opp(X), opp(opp(X))), eg)
% 2.07/0.77 = { by axiom 2 (sos11) }
% 2.07/0.77 ifeq2(ifeq(true, true, h(opp(X)), true), true, sum(opp(X), opp(opp(X))), eg)
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(opp(X)), true, sum(opp(X), opp(opp(X))), eg)
% 2.07/0.77 = { by axiom 15 (sos15) }
% 2.07/0.77 eg
% 2.07/0.77
% 2.07/0.77 Lemma 23: ifeq2(h(X), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)) = sum(X, eg).
% 2.07/0.77 Proof:
% 2.07/0.77 ifeq2(h(X), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg))
% 2.07/0.77 = { by lemma 22 R->L }
% 2.07/0.77 ifeq2(h(X), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(true, true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 11 (sos10) R->L }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(ifeq(h(eg), true, h(opp(opp(Y))), true), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 2 (sos11) }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(ifeq(true, true, h(opp(opp(Y))), true), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(true, true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 11 (sos10) R->L }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(ifeq(h(eg), true, h(opp(Y)), true), true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 2 (sos11) }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(ifeq(true, true, h(opp(Y)), true), true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(h(opp(Y)), true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, eg)), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by lemma 22 R->L }
% 2.07/0.77 ifeq2(h(X), true, ifeq2(h(opp(Y)), true, ifeq2(h(opp(opp(Y))), true, sum(sum(X, opp(Y)), opp(opp(Y))), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y))))), sum(X, sum(opp(Y), opp(opp(Y)))))
% 2.07/0.77 = { by axiom 17 (sos12) }
% 2.07/0.77 sum(X, sum(opp(Y), opp(opp(Y))))
% 2.07/0.77 = { by lemma 22 }
% 2.07/0.77 sum(X, eg)
% 2.07/0.77
% 2.07/0.77 Lemma 24: opp(opp(f(inv(x0)))) = f(inv(x0)).
% 2.07/0.77 Proof:
% 2.07/0.77 opp(opp(f(inv(x0))))
% 2.07/0.77 = { by axiom 14 (sos13) R->L }
% 2.07/0.77 ifeq2(h(opp(opp(f(inv(x0))))), true, sum(eg, opp(opp(f(inv(x0))))), opp(opp(f(inv(x0)))))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(ifeq(true, true, h(opp(opp(f(inv(x0))))), true), true, sum(eg, opp(opp(f(inv(x0))))), opp(opp(f(inv(x0)))))
% 2.07/0.77 = { by axiom 2 (sos11) R->L }
% 2.07/0.77 ifeq2(ifeq(h(eg), true, h(opp(opp(f(inv(x0))))), true), true, sum(eg, opp(opp(f(inv(x0))))), opp(opp(f(inv(x0)))))
% 2.07/0.77 = { by axiom 11 (sos10) }
% 2.07/0.77 ifeq2(true, true, sum(eg, opp(opp(f(inv(x0))))), opp(opp(f(inv(x0)))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.77 sum(eg, opp(opp(f(inv(x0)))))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(true, true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 9 (sos17) R->L }
% 2.07/0.77 ifeq2(ifeq(g(inv(x0)), true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(ifeq(ifeq(true, true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 7 (goals_1) R->L }
% 2.07/0.77 ifeq2(ifeq(ifeq(ifeq3(f(eh), eg, g(x0), true), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by lemma 20 }
% 2.07/0.77 ifeq2(ifeq(ifeq(ifeq3(eg, eg, g(x0), true), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 3 (ifeq_axiom) }
% 2.07/0.77 ifeq2(ifeq(ifeq(g(x0), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 8 (sos02) }
% 2.07/0.77 ifeq2(ifeq(true, true, h(f(inv(x0))), true), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(eg, opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 15 (sos15) R->L }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(h(f(inv(x0))), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(true, true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 8 (sos02) R->L }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(ifeq(g(x0), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 3 (ifeq_axiom) R->L }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(ifeq(ifeq3(eg, eg, g(x0), true), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by lemma 20 R->L }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(ifeq(ifeq3(f(eh), eg, g(x0), true), true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 7 (goals_1) }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(ifeq(true, true, g(inv(x0)), true), true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(ifeq(g(inv(x0)), true, h(f(inv(x0))), true), true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 9 (sos17) }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(ifeq2(true, true, sum(f(inv(x0)), opp(f(inv(x0)))), eg), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.77 ifeq2(h(f(inv(x0))), true, sum(sum(f(inv(x0)), opp(f(inv(x0)))), opp(opp(f(inv(x0))))), sum(f(inv(x0)), eg))
% 2.07/0.77 = { by lemma 23 }
% 2.07/0.77 sum(f(inv(x0)), eg)
% 2.07/0.77 = { by lemma 21 }
% 2.07/0.77 f(product(inv(x0), eh))
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.77 f(ifeq2(true, true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by axiom 8 (sos02) R->L }
% 2.07/0.77 f(ifeq2(ifeq(g(x0), true, g(inv(x0)), true), true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by axiom 3 (ifeq_axiom) R->L }
% 2.07/0.77 f(ifeq2(ifeq(ifeq3(eg, eg, g(x0), true), true, g(inv(x0)), true), true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by lemma 20 R->L }
% 2.07/0.77 f(ifeq2(ifeq(ifeq3(f(eh), eg, g(x0), true), true, g(inv(x0)), true), true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by axiom 7 (goals_1) }
% 2.07/0.77 f(ifeq2(ifeq(true, true, g(inv(x0)), true), true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.77 f(ifeq2(g(inv(x0)), true, product(inv(x0), eh), inv(x0)))
% 2.07/0.77 = { by axiom 10 (sos06) }
% 2.07/0.77 f(inv(x0))
% 2.07/0.77
% 2.07/0.77 Goal 1 (goals): tuple(f(inv(x0)), f(eh)) = tuple(opp(f(x0)), eg).
% 2.07/0.77 Proof:
% 2.07/0.77 tuple(f(inv(x0)), f(eh))
% 2.07/0.77 = { by lemma 20 }
% 2.07/0.77 tuple(f(inv(x0)), eg)
% 2.07/0.77 = { by lemma 24 R->L }
% 2.07/0.77 tuple(opp(opp(f(inv(x0)))), eg)
% 2.07/0.77 = { by axiom 13 (sos14) R->L }
% 2.07/0.77 tuple(opp(ifeq2(h(opp(f(inv(x0)))), true, sum(opp(f(inv(x0))), eg), opp(f(inv(x0))))), eg)
% 2.07/0.77 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.77 tuple(opp(ifeq2(ifeq(true, true, h(opp(f(inv(x0)))), true), true, sum(opp(f(inv(x0))), eg), opp(f(inv(x0))))), eg)
% 2.07/0.77 = { by axiom 2 (sos11) R->L }
% 2.07/0.77 tuple(opp(ifeq2(ifeq(h(eg), true, h(opp(f(inv(x0)))), true), true, sum(opp(f(inv(x0))), eg), opp(f(inv(x0))))), eg)
% 2.07/0.77 = { by axiom 11 (sos10) }
% 2.07/0.77 tuple(opp(ifeq2(true, true, sum(opp(f(inv(x0))), eg), opp(f(inv(x0))))), eg)
% 2.07/0.77 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.77 tuple(opp(sum(opp(f(inv(x0))), eg)), eg)
% 2.07/0.77 = { by lemma 23 R->L }
% 2.07/0.77 tuple(opp(ifeq2(h(opp(f(inv(x0)))), true, sum(sum(opp(f(inv(x0))), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(opp(f(inv(x0))), eg))), eg)
% 2.07/0.77 = { by lemma 22 }
% 2.07/0.77 tuple(opp(ifeq2(h(opp(f(inv(x0)))), true, sum(eg, opp(opp(opp(f(inv(x0)))))), sum(opp(f(inv(x0))), eg))), eg)
% 2.07/0.78 = { by axiom 4 (ifeq_axiom) R->L }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(true, true, h(opp(f(inv(x0)))), true), true, sum(eg, opp(opp(opp(f(inv(x0)))))), sum(opp(f(inv(x0))), eg))), eg)
% 2.07/0.78 = { by axiom 2 (sos11) R->L }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(h(eg), true, h(opp(f(inv(x0)))), true), true, sum(eg, opp(opp(opp(f(inv(x0)))))), sum(opp(f(inv(x0))), eg))), eg)
% 2.07/0.78 = { by axiom 11 (sos10) }
% 2.07/0.78 tuple(opp(ifeq2(true, true, sum(eg, opp(opp(opp(f(inv(x0)))))), sum(opp(f(inv(x0))), eg))), eg)
% 2.07/0.78 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.78 tuple(opp(sum(eg, opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by lemma 20 R->L }
% 2.07/0.78 tuple(opp(sum(f(eh), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 12 (sos07) R->L }
% 2.07/0.78 tuple(opp(sum(f(ifeq2(g(x0), true, product(x0, inv(x0)), eh)), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 3 (ifeq_axiom) R->L }
% 2.07/0.78 tuple(opp(sum(f(ifeq2(ifeq3(eg, eg, g(x0), true), true, product(x0, inv(x0)), eh)), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by lemma 20 R->L }
% 2.07/0.78 tuple(opp(sum(f(ifeq2(ifeq3(f(eh), eg, g(x0), true), true, product(x0, inv(x0)), eh)), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 7 (goals_1) }
% 2.07/0.78 tuple(opp(sum(f(ifeq2(true, true, product(x0, inv(x0)), eh)), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 5 (ifeq_axiom) }
% 2.07/0.78 tuple(opp(sum(f(product(x0, inv(x0))), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 6 (sos18) }
% 2.07/0.78 tuple(opp(sum(sum(f(x0), f(inv(x0))), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by lemma 24 R->L }
% 2.07/0.78 tuple(opp(sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0))))))), eg)
% 2.07/0.78 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.78 tuple(opp(ifeq2(true, true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by axiom 9 (sos17) R->L }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(g(x0), true, h(f(x0)), true), true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by axiom 3 (ifeq_axiom) R->L }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(ifeq3(eg, eg, g(x0), true), true, h(f(x0)), true), true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by lemma 20 R->L }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(ifeq3(f(eh), eg, g(x0), true), true, h(f(x0)), true), true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by axiom 7 (goals_1) }
% 2.07/0.78 tuple(opp(ifeq2(ifeq(true, true, h(f(x0)), true), true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by axiom 4 (ifeq_axiom) }
% 2.07/0.78 tuple(opp(ifeq2(h(f(x0)), true, sum(sum(f(x0), opp(opp(f(inv(x0))))), opp(opp(opp(f(inv(x0)))))), sum(f(x0), eg))), eg)
% 2.07/0.78 = { by lemma 23 }
% 2.07/0.78 tuple(opp(sum(f(x0), eg)), eg)
% 2.07/0.78 = { by lemma 21 }
% 2.07/0.78 tuple(opp(f(product(x0, eh))), eg)
% 2.07/0.78 = { by axiom 5 (ifeq_axiom) R->L }
% 2.07/0.78 tuple(opp(f(ifeq2(true, true, product(x0, eh), x0))), eg)
% 2.07/0.78 = { by axiom 7 (goals_1) R->L }
% 2.07/0.78 tuple(opp(f(ifeq2(ifeq3(f(eh), eg, g(x0), true), true, product(x0, eh), x0))), eg)
% 2.07/0.78 = { by lemma 20 }
% 2.07/0.78 tuple(opp(f(ifeq2(ifeq3(eg, eg, g(x0), true), true, product(x0, eh), x0))), eg)
% 2.07/0.78 = { by axiom 3 (ifeq_axiom) }
% 2.07/0.78 tuple(opp(f(ifeq2(g(x0), true, product(x0, eh), x0))), eg)
% 2.07/0.78 = { by axiom 10 (sos06) }
% 2.07/0.78 tuple(opp(f(x0)), eg)
% 2.07/0.78 % SZS output end Proof
% 2.07/0.78
% 2.07/0.78 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------