TSTP Solution File: SWW231+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWW231+1 : TPTP v8.1.2. Released v5.2.0.
% 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 : n002.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 : Fri Sep 1 00:54:28 EDT 2023
% Result : Theorem 146.60s 19.20s
% Output : Proof 146.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.14 % Problem : SWW231+1 : TPTP v8.1.2. Released v5.2.0.
% 0.12/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n002.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Sun Aug 27 18:19:19 EDT 2023
% 0.15/0.36 % CPUTime :
% 146.60/19.20 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 146.60/19.20
% 146.60/19.20 % SZS status Theorem
% 146.60/19.20
% 146.60/19.21 % SZS output start Proof
% 146.60/19.21 Take the following subset of the input axioms:
% 146.60/19.22 fof(arity_Complex__Ocomplex__Rings_Ocomm__ring__1, axiom, class_Rings_Ocomm__ring__1(tc_Complex_Ocomplex)).
% 146.60/19.22 fof(arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs, axiom, class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal)).
% 146.60/19.22 fof(arity_RealDef__Oreal__Rings_Ocomm__ring__1, axiom, class_Rings_Ocomm__ring__1(tc_RealDef_Oreal)).
% 146.60/19.22 fof(arity_RealDef__Oreal__Rings_Ocomm__semiring__1, axiom, class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal)).
% 146.60/19.22 fof(conj_0, conjecture, c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))=c_Complex_Orcis(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), v_a)).
% 146.60/19.22 fof(fact_Im, axiom, ![V_y, V_x]: c_Complex_OIm(c_Complex_Ocomplex_OComplex(V_x, V_y))=V_y).
% 146.60/19.22 fof(fact_Im__rcis, axiom, ![V_a, V_r]: c_Complex_OIm(c_Complex_Orcis(V_r, V_a))=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_r), c_Transcendental_Osin(V_a))).
% 146.60/19.22 fof(fact_Re, axiom, ![V_x2, V_y2]: c_Complex_ORe(c_Complex_Ocomplex_OComplex(V_x2, V_y2))=V_x2).
% 146.60/19.22 fof(fact_Re__rcis, axiom, ![V_r3, V_a2]: c_Complex_ORe(c_Complex_Orcis(V_r3, V_a2))=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_r3), c_Transcendental_Ocos(V_a2))).
% 146.60/19.22 fof(fact_abs__idempotent, axiom, ![T_a, V_a2]: (class_Groups_Oordered__ab__group__add__abs(T_a) => c_Groups_Oabs__class_Oabs(T_a, c_Groups_Oabs__class_Oabs(T_a, V_a2))=c_Groups_Oabs__class_Oabs(T_a, V_a2))).
% 146.60/19.22 fof(fact_cis__def, axiom, ![V_a2]: c_Complex_Ocis(V_a2)=c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(V_a2), c_Transcendental_Osin(V_a2))).
% 146.60/19.22 fof(fact_cis__rcis__eq, axiom, ![V_a2]: c_Complex_Ocis(V_a2)=c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), V_a2)).
% 146.60/19.22 fof(fact_cis__zero, axiom, c_Complex_Ocis(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))=c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)).
% 146.60/19.22 fof(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J, axiom, ![V_x2, T_a2]: (class_Rings_Ocomm__ring__1(T_a2) => c_Groups_Ouminus__class_Ouminus(T_a2, V_x2)=hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), c_Groups_Ouminus__class_Ouminus(T_a2, c_Groups_Oone__class_Oone(T_a2))), V_x2))).
% 146.60/19.22 fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, axiom, ![V_a2, T_a2]: (class_Rings_Ocomm__semiring__1(T_a2) => c_Groups_Oplus__class_Oplus(T_a2, c_Groups_Ozero__class_Ozero(T_a2), V_a2)=V_a2)).
% 146.60/19.22 fof(fact_complex__minus, axiom, ![V_b, V_a2]: c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Ocomplex_OComplex(V_a2, V_b))=c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, V_a2), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, V_b))).
% 146.60/19.22 fof(fact_complex__of__real__def, axiom, ![V_r3]: c_RealVector_Oof__real(tc_Complex_Ocomplex, V_r3)=c_Complex_Ocomplex_OComplex(V_r3, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))).
% 146.60/19.22 fof(fact_complex__surj, axiom, ![V_z]: c_Complex_Ocomplex_OComplex(c_Complex_ORe(V_z), c_Complex_OIm(V_z))=V_z).
% 146.60/19.22 fof(fact_cos__periodic__pi, axiom, ![V_x2]: c_Transcendental_Ocos(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, V_x2, c_Transcendental_Opi))=c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Ocos(V_x2))).
% 146.60/19.22 fof(fact_cos__zero, axiom, c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))=c_Groups_Oone__class_Oone(tc_RealDef_Oreal)).
% 146.60/19.22 fof(fact_rcis__def, axiom, ![V_r3, V_a2]: c_Complex_Orcis(V_r3, V_a2)=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_RealVector_Oof__real(tc_Complex_Ocomplex, V_r3)), c_Complex_Ocis(V_a2))).
% 146.60/19.22 fof(fact_rcis__mult, axiom, ![V_r2, V_r1, V_a2, V_b2]: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Orcis(V_r1, V_a2)), c_Complex_Orcis(V_r2, V_b2))=c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_r1), V_r2), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, V_a2, V_b2))).
% 146.60/19.22 fof(fact_rcis__zero__arg, axiom, ![V_r3]: c_Complex_Orcis(V_r3, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))=c_RealVector_Oof__real(tc_Complex_Ocomplex, V_r3)).
% 146.60/19.22 fof(fact_real__mult__1, axiom, ![V_z2]: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), V_z2)=V_z2).
% 146.60/19.22 fof(fact_real__mult__commute, axiom, ![V_w, V_z2]: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_z2), V_w)=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_w), V_z2)).
% 146.60/19.22 fof(fact_real__sqrt__abs2, axiom, ![V_x2]: c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_x2), V_x2))=c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, V_x2)).
% 146.60/19.22 fof(fact_real__sqrt__mult, axiom, ![V_x2, V_y2]: c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_x2), V_y2))=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(V_x2)), c_NthRoot_Osqrt(V_y2))).
% 146.60/19.22 fof(fact_sin__periodic__pi, axiom, ![V_x2]: c_Transcendental_Osin(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, V_x2, c_Transcendental_Opi))=c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Osin(V_x2))).
% 146.60/19.22 fof(fact_sin__pi, axiom, c_Transcendental_Osin(c_Transcendental_Opi)=c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)).
% 146.60/19.22 fof(fact_sin__zero, axiom, c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))=c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)).
% 146.60/19.22
% 146.60/19.22 Now clausify the problem and encode Horn clauses using encoding 3 of
% 146.60/19.22 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 146.60/19.22 We repeatedly replace C & s=t => u=v by the two clauses:
% 146.60/19.22 fresh(y, y, x1...xn) = u
% 146.60/19.22 C => fresh(s, t, x1...xn) = v
% 146.60/19.22 where fresh is a fresh function symbol and x1..xn are the free
% 146.60/19.22 variables of u and v.
% 146.60/19.22 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 146.60/19.22 input problem has no model of domain size 1).
% 146.60/19.22
% 146.60/19.22 The encoding turns the above axioms into the following unit equations and goals:
% 146.60/19.22
% 146.60/19.22 Axiom 1 (fact_sin__pi): c_Transcendental_Osin(c_Transcendental_Opi) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal).
% 146.60/19.22 Axiom 2 (arity_RealDef__Oreal__Rings_Ocomm__semiring__1): class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal) = true2.
% 146.60/19.22 Axiom 3 (arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs): class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal) = true2.
% 146.60/19.22 Axiom 4 (arity_RealDef__Oreal__Rings_Ocomm__ring__1): class_Rings_Ocomm__ring__1(tc_RealDef_Oreal) = true2.
% 146.60/19.22 Axiom 5 (arity_Complex__Ocomplex__Rings_Ocomm__ring__1): class_Rings_Ocomm__ring__1(tc_Complex_Ocomplex) = true2.
% 146.60/19.22 Axiom 6 (fact_complex__of__real__def): c_RealVector_Oof__real(tc_Complex_Ocomplex, X) = c_Complex_Ocomplex_OComplex(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)).
% 146.60/19.22 Axiom 7 (fact_cos__zero): c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_RealDef_Oreal).
% 146.60/19.22 Axiom 8 (fact_Im): c_Complex_OIm(c_Complex_Ocomplex_OComplex(X, Y)) = Y.
% 146.60/19.22 Axiom 9 (fact_Re): c_Complex_ORe(c_Complex_Ocomplex_OComplex(X, Y)) = X.
% 146.60/19.22 Axiom 10 (fact_sin__zero): c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal).
% 146.60/19.22 Axiom 11 (fact_rcis__zero__arg): c_Complex_Orcis(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_RealVector_Oof__real(tc_Complex_Ocomplex, X).
% 146.60/19.22 Axiom 12 (fact_cis__rcis__eq): c_Complex_Ocis(X) = c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), X).
% 146.60/19.22 Axiom 13 (fact_cis__zero): c_Complex_Ocis(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Groups_Oone__class_Oone(tc_Complex_Ocomplex).
% 146.60/19.22 Axiom 14 (fact_real__mult__commute): hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), Y) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), Y), X).
% 146.60/19.22 Axiom 15 (fact_complex__minus): c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Ocomplex_OComplex(X, Y)) = c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, X), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, Y)).
% 146.60/19.22 Axiom 16 (fact_cis__def): c_Complex_Ocis(X) = c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(X), c_Transcendental_Osin(X)).
% 146.60/19.22 Axiom 17 (fact_complex__surj): c_Complex_Ocomplex_OComplex(c_Complex_ORe(X), c_Complex_OIm(X)) = X.
% 146.60/19.22 Axiom 18 (fact_cos__periodic__pi): c_Transcendental_Ocos(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, X, c_Transcendental_Opi)) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Ocos(X)).
% 146.60/19.22 Axiom 19 (fact_sin__periodic__pi): c_Transcendental_Osin(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, X, c_Transcendental_Opi)) = c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Osin(X)).
% 146.60/19.22 Axiom 20 (fact_abs__idempotent): fresh1007(X, X, Y, Z) = c_Groups_Oabs__class_Oabs(Z, Y).
% 146.60/19.22 Axiom 21 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J): fresh889(X, X, Y, Z) = c_Groups_Ouminus__class_Ouminus(Z, Y).
% 146.60/19.22 Axiom 22 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J): fresh131(X, X, Y, Z) = Y.
% 146.60/19.22 Axiom 23 (fact_Re__rcis): c_Complex_ORe(c_Complex_Orcis(X, Y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), c_Transcendental_Ocos(Y)).
% 146.60/19.22 Axiom 24 (fact_Im__rcis): c_Complex_OIm(c_Complex_Orcis(X, Y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), c_Transcendental_Osin(Y)).
% 146.60/19.22 Axiom 25 (fact_real__mult__1): hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), X) = X.
% 146.60/19.22 Axiom 26 (fact_real__sqrt__abs2): c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), X)) = c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, X).
% 146.60/19.22 Axiom 27 (fact_abs__idempotent): fresh1007(class_Groups_Oordered__ab__group__add__abs(X), true2, Y, X) = c_Groups_Oabs__class_Oabs(X, c_Groups_Oabs__class_Oabs(X, Y)).
% 146.60/19.22 Axiom 28 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J): fresh131(class_Rings_Ocomm__semiring__1(X), true2, Y, X) = c_Groups_Oplus__class_Oplus(X, c_Groups_Ozero__class_Ozero(X), Y).
% 146.60/19.22 Axiom 29 (fact_real__sqrt__mult): c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), Y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(X)), c_NthRoot_Osqrt(Y)).
% 146.60/19.22 Axiom 30 (fact_rcis__def): c_Complex_Orcis(X, Y) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_RealVector_Oof__real(tc_Complex_Ocomplex, X)), c_Complex_Ocis(Y)).
% 146.60/19.22 Axiom 31 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J): fresh889(class_Rings_Ocomm__ring__1(X), true2, Y, X) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(X), c_Groups_Ouminus__class_Ouminus(X, c_Groups_Oone__class_Oone(X))), Y).
% 146.60/19.22 Axiom 32 (fact_rcis__mult): hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Orcis(X, Y)), c_Complex_Orcis(Z, W)) = c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), X), Z), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, Y, W)).
% 146.60/19.22
% 146.60/19.22 Lemma 33: c_Complex_Orcis(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)) = c_Complex_Ocomplex_OComplex(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)).
% 146.60/19.22 Proof:
% 146.60/19.22 c_Complex_Orcis(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
% 146.60/19.22 = { by axiom 11 (fact_rcis__zero__arg) }
% 146.60/19.22 c_RealVector_Oof__real(tc_Complex_Ocomplex, X)
% 146.60/19.22 = { by axiom 6 (fact_complex__of__real__def) }
% 146.60/19.22 c_Complex_Ocomplex_OComplex(X, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))
% 146.60/19.22
% 146.60/19.22 Lemma 34: c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(X), c_Transcendental_Osin(X)) = c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), X).
% 146.60/19.22 Proof:
% 146.60/19.22 c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(X), c_Transcendental_Osin(X))
% 146.60/19.22 = { by axiom 16 (fact_cis__def) R->L }
% 146.60/19.22 c_Complex_Ocis(X)
% 146.60/19.22 = { by axiom 12 (fact_cis__rcis__eq) }
% 146.60/19.22 c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), X)
% 146.60/19.22
% 146.60/19.22 Lemma 35: c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), X) = X.
% 146.60/19.22 Proof:
% 146.60/19.22 c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), X)
% 146.60/19.22 = { by axiom 28 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) R->L }
% 146.60/19.22 fresh131(class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal), true2, X, tc_RealDef_Oreal)
% 146.60/19.22 = { by axiom 2 (arity_RealDef__Oreal__Rings_Ocomm__semiring__1) }
% 146.60/19.22 fresh131(true2, true2, X, tc_RealDef_Oreal)
% 146.60/19.22 = { by axiom 22 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) }
% 146.60/19.22 X
% 146.60/19.22
% 146.60/19.22 Lemma 36: c_Complex_Orcis(X, c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, Y)) = c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(X, Y)).
% 146.60/19.22 Proof:
% 146.60/19.22 c_Complex_Orcis(X, c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, Y))
% 146.60/19.22 = { by axiom 25 (fact_real__mult__1) R->L }
% 146.60/19.22 c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), X), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, Y))
% 146.60/19.22 = { by axiom 32 (fact_rcis__mult) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), c_Transcendental_Opi)), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by lemma 35 R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), c_Transcendental_Opi))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by lemma 34 R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), c_Transcendental_Opi)), c_Transcendental_Osin(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), c_Transcendental_Opi)))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 19 (fact_sin__periodic__pi) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), c_Transcendental_Opi)), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 18 (fact_cos__periodic__pi) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 15 (fact_complex__minus) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Ocomplex_OComplex(c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by lemma 34 }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 12 (fact_cis__rcis__eq) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Ocis(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 13 (fact_cis__zero) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Groups_Oone__class_Oone(tc_Complex_Ocomplex))), c_Complex_Orcis(X, Y))
% 146.60/19.22 = { by axiom 31 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) R->L }
% 146.60/19.22 fresh889(class_Rings_Ocomm__ring__1(tc_Complex_Ocomplex), true2, c_Complex_Orcis(X, Y), tc_Complex_Ocomplex)
% 146.60/19.22 = { by axiom 5 (arity_Complex__Ocomplex__Rings_Ocomm__ring__1) }
% 146.60/19.22 fresh889(true2, true2, c_Complex_Orcis(X, Y), tc_Complex_Ocomplex)
% 146.60/19.22 = { by axiom 21 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(X, Y))
% 146.60/19.22
% 146.60/19.22 Goal 1 (conj_0): c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a)) = c_Complex_Orcis(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), v_a).
% 146.60/19.22 Proof:
% 146.60/19.22 c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by lemma 36 }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), c_NthRoot_Osqrt(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), v_a))
% 146.60/19.22 = { by axiom 29 (fact_real__sqrt__mult) R->L }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(c_NthRoot_Osqrt(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r))), v_a))
% 146.60/19.22 = { by axiom 26 (fact_real__sqrt__abs2) }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), v_a))
% 146.60/19.22 = { by axiom 27 (fact_abs__idempotent) R->L }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(fresh1007(class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal), true2, v_r, tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 3 (arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(fresh1007(true2, true2, v_r, tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 20 (fact_abs__idempotent) }
% 146.60/19.22 c_Groups_Ouminus__class_Ouminus(tc_Complex_Ocomplex, c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), v_a))
% 146.60/19.22 = { by lemma 36 R->L }
% 146.60/19.22 c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by axiom 9 (fact_Re) R->L }
% 146.60/19.22 c_Complex_Orcis(c_Complex_ORe(c_Complex_Ocomplex_OComplex(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by lemma 33 R->L }
% 146.60/19.22 c_Complex_Orcis(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by axiom 23 (fact_Re__rcis) }
% 146.60/19.22 c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by axiom 7 (fact_cos__zero) }
% 146.60/19.22 c_Complex_Orcis(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Transcendental_Opi, v_a))
% 146.60/19.22 = { by axiom 32 (fact_rcis__mult) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 17 (fact_complex__surj) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), c_Complex_OIm(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 24 (fact_Im__rcis) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Transcendental_Osin(c_Transcendental_Opi)))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 1 (fact_sin__pi) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 10 (fact_sin__zero) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Transcendental_Osin(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 24 (fact_Im__rcis) R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), c_Complex_OIm(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by lemma 33 }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), c_Complex_OIm(c_Complex_Ocomplex_OComplex(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 8 (fact_Im) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Complex_ORe(c_Complex_Orcis(c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Transcendental_Opi)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 23 (fact_Re__rcis) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Transcendental_Ocos(c_Transcendental_Opi)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by lemma 35 R->L }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Transcendental_Ocos(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), c_Transcendental_Opi))), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 18 (fact_cos__periodic__pi) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Transcendental_Ocos(c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.22 = { by axiom 7 (fact_cos__zero) }
% 146.60/19.22 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.23 = { by axiom 14 (fact_real__mult__commute) R->L }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.23 = { by axiom 31 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) R->L }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(fresh889(class_Rings_Ocomm__ring__1(tc_RealDef_Oreal), true2, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), tc_RealDef_Oreal), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.23 = { by axiom 4 (arity_RealDef__Oreal__Rings_Ocomm__ring__1) }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(fresh889(true2, true2, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), tc_RealDef_Oreal), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.23 = { by axiom 21 (fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Orcis(c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_a))
% 146.60/19.23 = { by axiom 12 (fact_cis__rcis__eq) R->L }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Complex_Ocomplex_OComplex(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))), c_Complex_Ocis(v_a))
% 146.60/19.23 = { by axiom 6 (fact_complex__of__real__def) R->L }
% 146.60/19.23 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_RealVector_Oof__real(tc_Complex_Ocomplex, c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)))), c_Complex_Ocis(v_a))
% 146.60/19.23 = { by axiom 30 (fact_rcis__def) R->L }
% 146.60/19.23 c_Complex_Orcis(c_Groups_Ouminus__class_Ouminus(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), v_a)
% 146.60/19.23 % SZS output end Proof
% 146.60/19.23
% 146.60/19.23 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------