%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : HEN009-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n014.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 10:19:00 AM UTC 2026
% Result : Unsatisfiable 33.20s 4.62s
% Output : Proof 35.59s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : HEN009-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.36 % Computer : n014.cluster.edu
% 0.12/0.36 % Model : x86_64 x86_64
% 0.12/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.36 % Memory : 8046.5625MB
% 0.12/0.36 % OS : Linux 6.8.0-71-generic
% 0.12/0.36 % CPULimit : 300
% 0.12/0.36 % WCLimit : 300
% 0.12/0.36 % DateTime : Sun Sep 27 10:51:01 UTC 2026
% 0.12/0.36 % CPUTime :
% 0.12/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 33.20/4.62 Command-line arguments: --flatten-regeneralise
% 33.20/4.62
% 33.20/4.62 % SZS status Unsatisfiable
% 33.20/4.62
% 34.02/4.78 % SZS output start Proof
% 34.02/4.78 Axiom 1 (identity_is_largest): less_equal(X, identity) = true.
% 34.02/4.78 Axiom 2 (zero_is_smallest): less_equal(zero, X) = true.
% 34.02/4.78 Axiom 3 (identity_divide_a): quotient(identity, a, idQa) = true.
% 34.02/4.78 Axiom 4 (identity_divide_idQ_idQa): quotient(identity, idQ_idQa, idQ__idQ_idQa) = true.
% 34.02/4.78 Axiom 5 (identity_divide_idQa): quotient(identity, idQa, idQ_idQa) = true.
% 34.02/4.78 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 34.02/4.78 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 34.02/4.78 Axiom 8 (closure): quotient(X, Y, divide(X, Y)) = true.
% 34.02/4.78 Axiom 9 (quotient_less_equal): ifeq(less_equal(X, Y), true, quotient(X, Y, zero), true) = true.
% 34.02/4.78 Axiom 10 (divisor_existence): ifeq(quotient(X, Y, Z), true, less_equal(Z, X), true) = true.
% 34.02/4.78 Axiom 11 (less_equal_quotient): ifeq(quotient(X, Y, zero), true, less_equal(X, Y), true) = true.
% 34.02/4.78 Axiom 12 (less_equal_and_equal): ifeq2(less_equal(X, Y), true, ifeq2(less_equal(Y, X), true, Y, X), X) = X.
% 34.02/4.78 Axiom 13 (well_defined): ifeq2(quotient(X, Y, Z), true, ifeq2(quotient(X, Y, W), true, W, Z), Z) = Z.
% 34.02/4.78 Axiom 14 (quotient_property): ifeq(quotient(X, Y, Z), true, ifeq(quotient(W, V, U), true, ifeq(quotient(T, V, Y), true, ifeq(quotient(S, V, X), true, ifeq(quotient(S, T, W), true, less_equal(Z, U), true), true), true), true), true) = true.
% 34.02/4.78
% 34.02/4.78 Lemma 15: divide(idQ__idQ_idQa, idQ__idQ_idQa) = zero.
% 34.02/4.78 Proof:
% 34.02/4.78 divide(idQ__idQ_idQa, idQ__idQ_idQa)
% 34.02/4.78 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.02/4.78 ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.78 ifeq2(ifeq(true, true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 8 (closure) R->L }
% 34.02/4.78 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.78 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(true, true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 4 (identity_divide_idQ_idQa) R->L }
% 34.02/4.78 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.78 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(true, true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 9 (quotient_less_equal) R->L }
% 34.02/4.78 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(ifeq(less_equal(identity, identity), true, quotient(identity, identity, zero), true), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.78 = { by axiom 1 (identity_is_largest) }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(ifeq(true, true, quotient(identity, identity, zero), true), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(true, true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 9 (quotient_less_equal) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(ifeq(less_equal(zero, idQ_idQa), true, quotient(zero, idQ_idQa, zero), true), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 2 (zero_is_smallest) }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(ifeq(true, true, quotient(zero, idQ_idQa, zero), true), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(true, true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 4 (identity_divide_idQ_idQa) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 14 (quotient_property) }
% 34.02/4.79 ifeq2(true, true, ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.79 ifeq2(less_equal(zero, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 2 (zero_is_smallest) }
% 34.02/4.79 ifeq2(true, true, zero, divide(idQ__idQ_idQa, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.79 zero
% 34.02/4.79
% 34.02/4.79 Lemma 16: divide(divide(X, Y), divide(identity, Y)) = zero.
% 34.02/4.79 Proof:
% 34.02/4.79 divide(divide(X, Y), divide(identity, Y))
% 34.02/4.79 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.02/4.79 ifeq2(less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(true, true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 8 (closure) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(true, true, ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 8 (closure) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(true, true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 8 (closure) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(true, true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 9 (quotient_less_equal) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(ifeq(less_equal(X, identity), true, quotient(X, identity, zero), true), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 1 (identity_is_largest) }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(ifeq(true, true, quotient(X, identity, zero), true), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(quotient(X, identity, zero), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(true, true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(quotient(X, identity, zero), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 9 (quotient_less_equal) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(ifeq(less_equal(zero, Y), true, quotient(zero, Y, zero), true), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(quotient(X, identity, zero), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 2 (zero_is_smallest) }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(ifeq(true, true, quotient(zero, Y, zero), true), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(quotient(X, identity, zero), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(X, Y), divide(identity, Y), divide(divide(X, Y), divide(identity, Y))), true, ifeq(quotient(zero, Y, zero), true, ifeq(quotient(identity, Y, divide(identity, Y)), true, ifeq(quotient(X, Y, divide(X, Y)), true, ifeq(quotient(X, identity, zero), true, less_equal(divide(divide(X, Y), divide(identity, Y)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 14 (quotient_property) }
% 34.02/4.79 ifeq2(true, true, ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y))), divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.79 ifeq2(less_equal(zero, divide(divide(X, Y), divide(identity, Y))), true, zero, divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 2 (zero_is_smallest) }
% 34.02/4.79 ifeq2(true, true, zero, divide(divide(X, Y), divide(identity, Y)))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.79 zero
% 34.02/4.79
% 34.02/4.79 Lemma 17: divide(identity, idQ__idQ_idQa) = divide(idQ_idQa, idQ__idQ_idQa).
% 34.02/4.79 Proof:
% 34.02/4.79 divide(identity, idQ__idQ_idQa)
% 34.02/4.79 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.02/4.79 ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(true, true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 8 (closure) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(true, true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 2 (zero_is_smallest) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(zero, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by lemma 15 R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 6 (ifeq_axiom) R->L }
% 34.02/4.79 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(true, true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.79 = { by axiom 14 (quotient_property) R->L }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 4 (identity_divide_idQ_idQa) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, ifeq(true, true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq(quotient(idQ__idQ_idQa, idQ__idQ_idQa, divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 8 (closure) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq(true, true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 8 (closure) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(true, true, ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, idQ__idQ_idQa, divide(idQ_idQa, idQ__idQ_idQa)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 8 (closure) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(true, true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 8 (closure) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(ifeq(true, true, less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), ifeq2(less_equal(divide(idQ__idQ_idQa, idQ__idQ_idQa), divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa))), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, divide(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa)), divide(idQ__idQ_idQa, idQ__idQ_idQa))), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 12 (less_equal_and_equal) }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), divide(idQ__idQ_idQa, idQ__idQ_idQa)), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by lemma 15 }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa), zero), true, less_equal(divide(identity, idQ__idQ_idQa), divide(idQ_idQa, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 11 (less_equal_quotient) }
% 34.02/4.80 ifeq2(true, true, ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.80 ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.80 ifeq2(ifeq(true, true, less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 8 (closure) R->L }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa), divide(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))), true, less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by lemma 16 }
% 34.02/4.80 ifeq2(ifeq(quotient(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa), zero), true, less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 11 (less_equal_quotient) }
% 34.02/4.80 ifeq2(true, true, divide(idQ_idQa, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))
% 34.02/4.80 = { by axiom 6 (ifeq_axiom) }
% 34.02/4.80 divide(idQ_idQa, idQ__idQ_idQa)
% 34.02/4.80
% 34.02/4.80 Lemma 18: divide(idQa, idQa) = zero.
% 34.02/4.80 Proof:
% 34.02/4.80 divide(idQa, idQa)
% 34.02/4.80 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.02/4.80 ifeq2(less_equal(divide(idQa, idQa), zero), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.80 ifeq2(ifeq(true, true, less_equal(divide(idQa, idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.02/4.80 = { by axiom 8 (closure) R->L }
% 34.02/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, less_equal(divide(idQa, idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.02/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.02/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(true, true, less_equal(divide(idQa, idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.02/4.80 = { by axiom 3 (identity_divide_a) R->L }
% 34.02/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(identity, a, idQa), true, less_equal(divide(idQa, idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(identity, a, idQa), true, ifeq(true, true, less_equal(divide(idQa, idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(identity, a, idQa), true, ifeq(ifeq(less_equal(identity, identity), true, quotient(identity, identity, zero), true), true, less_equal(divide(idQa, idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 1 (identity_is_largest) }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(identity, a, idQa), true, ifeq(ifeq(true, true, quotient(identity, identity, zero), true), true, less_equal(divide(idQa, idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(true, true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(ifeq(less_equal(zero, a), true, quotient(zero, a, zero), true), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(ifeq(true, true, quotient(zero, a, zero), true), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(zero, a, zero), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(zero, a, zero), true, ifeq(true, true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 3 (identity_divide_a) R->L }
% 34.84/4.80 ifeq2(ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(zero, a, zero), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQa, idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 14 (quotient_property) }
% 34.84/4.80 ifeq2(true, true, ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa)), divide(idQa, idQa))
% 34.84/4.80 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.80 ifeq2(less_equal(zero, divide(idQa, idQa)), true, zero, divide(idQa, idQa))
% 34.84/4.80 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.80 ifeq2(true, true, zero, divide(idQa, idQa))
% 34.84/4.80 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.81 zero
% 34.84/4.81
% 34.84/4.81 Lemma 19: divide(idQ_idQa, idQ_idQa) = zero.
% 34.84/4.81 Proof:
% 34.84/4.81 divide(idQ_idQa, idQ_idQa)
% 34.84/4.81 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.84/4.81 ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), zero), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(true, true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 8 (closure) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(true, true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 5 (identity_divide_idQa) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(true, true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(ifeq(less_equal(identity, identity), true, quotient(identity, identity, zero), true), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 1 (identity_is_largest) }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(ifeq(true, true, quotient(identity, identity, zero), true), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(true, true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(ifeq(less_equal(zero, idQa), true, quotient(zero, idQa, zero), true), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(ifeq(true, true, quotient(zero, idQa, zero), true), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(true, true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 5 (identity_divide_idQa) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, identity, zero), true, less_equal(divide(idQ_idQa, idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 14 (quotient_property) }
% 34.84/4.81 ifeq2(true, true, ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.81 ifeq2(less_equal(zero, divide(idQ_idQa, idQ_idQa)), true, zero, divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.81 ifeq2(true, true, zero, divide(idQ_idQa, idQ_idQa))
% 34.84/4.81 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.81 zero
% 34.84/4.81
% 34.84/4.81 Lemma 20: divide(idQa, idQ_idQa) = idQ__idQ_idQa.
% 34.84/4.81 Proof:
% 34.84/4.81 divide(idQa, idQ_idQa)
% 34.84/4.81 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(true, true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, zero), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(true, true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(true, true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 14 (quotient_property) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 4 (identity_divide_idQ_idQa) }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(true, true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(zero, idQ_idQa, zero), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.81 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(ifeq(true, true, quotient(zero, idQ_idQa, zero), true), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.81 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(ifeq(less_equal(zero, idQ_idQa), true, quotient(zero, idQ_idQa, zero), true), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 9 (quotient_less_equal) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(true, true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(idQa, identity, zero), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(ifeq(true, true, quotient(idQa, identity, zero), true), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 1 (identity_is_largest) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(ifeq(less_equal(idQa, identity), true, quotient(idQa, identity, zero), true), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 9 (quotient_less_equal) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(true, true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(true, true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(ifeq(true, true, less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, ifeq2(less_equal(divide(divide(idQa, idQ_idQa), idQ__idQ_idQa), zero), true, ifeq2(less_equal(zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, zero, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), divide(divide(idQa, idQ_idQa), idQ__idQ_idQa))), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.82 ifeq2(ifeq(quotient(divide(idQa, idQ_idQa), idQ__idQ_idQa, divide(divide(idQa, idQ_idQa), idQ__idQ_idQa)), true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) }
% 34.84/4.82 ifeq2(ifeq(true, true, less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(true, true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), zero), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by lemma 19 R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ_idQa, idQ_idQa)), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(true, true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(true, true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(true, true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(true, true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 4 (identity_divide_idQ_idQa) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(true, true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 5 (identity_divide_idQa) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq(quotient(identity, idQ_idQa, idQ__idQ_idQa), true, ifeq(quotient(identity, idQa, idQ_idQa), true, less_equal(divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), true), true), true), true), true), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 14 (quotient_property) }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, ifeq2(true, true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(divide(idQ_idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by lemma 19 }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(less_equal(zero, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), ifeq2(true, true, divide(idQ__idQ_idQa, divide(idQa, idQ_idQa)), divide(idQ_idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.82 ifeq2(ifeq(quotient(idQ__idQ_idQa, divide(idQa, idQ_idQa), divide(idQ__idQ_idQa, divide(idQa, idQ_idQa))), true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 8 (closure) }
% 34.84/4.82 ifeq2(ifeq(true, true, less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.82 ifeq2(less_equal(idQ__idQ_idQa, divide(idQa, idQ_idQa)), true, ifeq2(less_equal(divide(idQa, idQ_idQa), idQ__idQ_idQa), true, divide(idQa, idQ_idQa), idQ__idQ_idQa), idQ__idQ_idQa)
% 34.84/4.82 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.82 idQ__idQ_idQa
% 34.84/4.82
% 34.84/4.82 Lemma 21: divide(divide(divide(X, Y), Z), divide(X, Z)) = zero.
% 34.84/4.82 Proof:
% 34.84/4.82 divide(divide(divide(X, Y), Z), divide(X, Z))
% 34.84/4.82 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.84/4.82 ifeq2(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(true, true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.82 = { by axiom 8 (closure) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(X, Z, divide(X, Z)), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(true, true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.82 = { by axiom 8 (closure) R->L }
% 34.84/4.82 ifeq2(ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.82 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.82 ifeq2(ifeq(true, true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 8 (closure) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(true, true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(ifeq(less_equal(divide(X, Y), X), true, quotient(divide(X, Y), X, zero), true), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(ifeq(ifeq(true, true, less_equal(divide(X, Y), X), true), true, quotient(divide(X, Y), X, zero), true), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 8 (closure) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(ifeq(ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(X, Y), X), true), true, quotient(divide(X, Y), X, zero), true), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 10 (divisor_existence) }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(ifeq(true, true, quotient(divide(X, Y), X, zero), true), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(quotient(divide(X, Y), X, zero), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(true, true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(quotient(divide(X, Y), X, zero), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(ifeq(less_equal(zero, Z), true, quotient(zero, Z, zero), true), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(quotient(divide(X, Y), X, zero), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.83 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(ifeq(true, true, quotient(zero, Z, zero), true), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(quotient(divide(X, Y), X, zero), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.83 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.84 ifeq2(ifeq(quotient(divide(divide(X, Y), Z), divide(X, Z), divide(divide(divide(X, Y), Z), divide(X, Z))), true, ifeq(quotient(zero, Z, zero), true, ifeq(quotient(X, Z, divide(X, Z)), true, ifeq(quotient(divide(X, Y), Z, divide(divide(X, Y), Z)), true, ifeq(quotient(divide(X, Y), X, zero), true, less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.84 = { by axiom 14 (quotient_property) }
% 34.84/4.84 ifeq2(true, true, ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.84 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.84 ifeq2(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.84 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.84 ifeq2(true, true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 34.84/4.84 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.84 zero
% 34.84/4.84
% 34.84/4.84 Lemma 22: divide(idQ_idQa, idQ__idQ_idQa) = divide(a, idQ__idQ_idQa).
% 34.84/4.84 Proof:
% 34.84/4.84 divide(idQ_idQa, idQ__idQ_idQa)
% 34.84/4.84 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.84 ifeq2(true, true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.84 ifeq2(ifeq(quotient(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa), zero), true, less_equal(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by lemma 21 R->L }
% 34.84/4.84 ifeq2(ifeq(quotient(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa), divide(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa))), true, less_equal(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 8 (closure) }
% 34.84/4.84 ifeq2(ifeq(true, true, less_equal(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.84 ifeq2(less_equal(divide(divide(a, idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.84 ifeq2(less_equal(divide(ifeq2(true, true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.84 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, zero), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.84 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(true, true, zero, divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.84 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.84 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(true, true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 14 (quotient_property) R->L }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 5 (identity_divide_idQa) }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(true, true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(zero, idQa, zero), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(ifeq(true, true, quotient(zero, idQa, zero), true), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(ifeq(less_equal(zero, idQa), true, quotient(zero, idQa, zero), true), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 9 (quotient_less_equal) }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(true, true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(a, identity, zero), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.85 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(ifeq(true, true, quotient(a, identity, zero), true), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.85 = { by axiom 1 (identity_is_largest) R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(ifeq(less_equal(a, identity), true, quotient(a, identity, zero), true), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 9 (quotient_less_equal) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(true, true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 8 (closure) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(true, true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 8 (closure) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(ifeq(true, true, less_equal(divide(divide(a, idQa), idQ_idQa), zero), true), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, ifeq2(less_equal(divide(divide(a, idQa), idQ_idQa), zero), true, ifeq2(less_equal(zero, divide(divide(a, idQa), idQ_idQa)), true, zero, divide(divide(a, idQa), idQ_idQa)), divide(divide(a, idQa), idQ_idQa))), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(divide(a, idQa), idQ_idQa, divide(divide(a, idQa), idQ_idQa)), true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 8 (closure) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(true, true, less_equal(divide(a, idQa), idQ_idQa), true), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(true, true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), zero), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by lemma 18 R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQa, idQa)), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 12 (less_equal_and_equal) R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.86 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(true, true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.86 = { by axiom 8 (closure) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(true, true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 8 (closure) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(true, true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 8 (closure) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(true, true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 5 (identity_divide_idQa) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.87 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(true, true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.87 = { by axiom 3 (identity_divide_a) R->L }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq(quotient(idQa, idQa, divide(idQa, idQa)), true, ifeq(quotient(a, idQa, divide(a, idQa)), true, ifeq(quotient(identity, idQa, idQ_idQa), true, ifeq(quotient(identity, a, idQa), true, less_equal(divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), true), true), true), true), true), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 14 (quotient_property) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, ifeq2(true, true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(divide(idQa, idQa), divide(idQ_idQa, divide(a, idQa))), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by lemma 18 }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(less_equal(zero, divide(idQ_idQa, divide(a, idQa))), true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 2 (zero_is_smallest) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), ifeq2(true, true, divide(idQ_idQa, divide(a, idQa)), divide(idQa, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(quotient(idQ_idQa, divide(a, idQa), divide(idQ_idQa, divide(a, idQa))), true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 8 (closure) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(ifeq(true, true, less_equal(idQ_idQa, divide(a, idQa)), true), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.88 ifeq2(less_equal(divide(ifeq2(less_equal(idQ_idQa, divide(a, idQa)), true, ifeq2(less_equal(divide(a, idQa), idQ_idQa), true, divide(a, idQa), idQ_idQa), idQ_idQa), idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.88 ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by lemma 17 R->L }
% 34.84/4.88 ifeq2(less_equal(divide(idQ_idQa, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by lemma 17 R->L }
% 34.84/4.88 ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.88 ifeq2(true, true, ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.88 ifeq2(ifeq(quotient(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa), zero), true, less_equal(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by lemma 16 R->L }
% 34.84/4.88 ifeq2(ifeq(quotient(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa), divide(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa))), true, less_equal(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 8 (closure) }
% 34.84/4.88 ifeq2(ifeq(true, true, less_equal(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true), true, ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.88 ifeq2(less_equal(divide(a, idQ__idQ_idQa), divide(identity, idQ__idQ_idQa)), true, ifeq2(less_equal(divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), true, divide(identity, idQ__idQ_idQa), divide(a, idQ__idQ_idQa)), divide(a, idQ__idQ_idQa))
% 34.84/4.88 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.88 divide(a, idQ__idQ_idQa)
% 34.84/4.88
% 34.84/4.88 Lemma 23: divide(divide(X, Y), X) = zero.
% 34.84/4.88 Proof:
% 34.84/4.88 divide(divide(X, Y), X)
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.88 ifeq2(true, true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 9 (quotient_less_equal) R->L }
% 34.84/4.88 ifeq2(ifeq(less_equal(divide(X, Y), X), true, quotient(divide(X, Y), X, zero), true), true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 7 (ifeq_axiom) R->L }
% 34.84/4.88 ifeq2(ifeq(ifeq(true, true, less_equal(divide(X, Y), X), true), true, quotient(divide(X, Y), X, zero), true), true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 8 (closure) R->L }
% 34.84/4.88 ifeq2(ifeq(ifeq(quotient(X, Y, divide(X, Y)), true, less_equal(divide(X, Y), X), true), true, quotient(divide(X, Y), X, zero), true), true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 10 (divisor_existence) }
% 34.84/4.88 ifeq2(ifeq(true, true, quotient(divide(X, Y), X, zero), true), true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.88 ifeq2(quotient(divide(X, Y), X, zero), true, divide(divide(X, Y), X), zero)
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.88 ifeq2(quotient(divide(X, Y), X, zero), true, ifeq2(true, true, divide(divide(X, Y), X), zero), zero)
% 34.84/4.88 = { by axiom 8 (closure) R->L }
% 34.84/4.88 ifeq2(quotient(divide(X, Y), X, zero), true, ifeq2(quotient(divide(X, Y), X, divide(divide(X, Y), X)), true, divide(divide(X, Y), X), zero), zero)
% 34.84/4.88 = { by axiom 13 (well_defined) }
% 34.84/4.88 zero
% 34.84/4.88
% 34.84/4.88 Lemma 24: divide(idQ__idQ_idQa, a) = idQa.
% 34.84/4.88 Proof:
% 34.84/4.88 divide(idQ__idQ_idQa, a)
% 34.84/4.88 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.88 ifeq2(true, true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, zero), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(true, true, zero, divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(true, true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(ifeq(quotient(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa), zero), true, less_equal(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa)), true), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by lemma 21 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(ifeq(quotient(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa), divide(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa))), true, less_equal(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa)), true), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 8 (closure) }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(ifeq(true, true, less_equal(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa)), true), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(less_equal(divide(divide(idQ__idQ_idQa, a), idQa), divide(idQ__idQ_idQa, idQa)), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by lemma 20 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(less_equal(divide(divide(idQ__idQ_idQa, a), idQa), divide(divide(idQa, idQ_idQa), idQa)), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by lemma 23 }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, ifeq2(less_equal(divide(divide(idQ__idQ_idQa, a), idQa), zero), true, ifeq2(less_equal(zero, divide(divide(idQ__idQ_idQa, a), idQa)), true, zero, divide(divide(idQ__idQ_idQa, a), idQa)), divide(divide(idQ__idQ_idQa, a), idQa))), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 12 (less_equal_and_equal) }
% 34.84/4.89 ifeq2(ifeq(quotient(divide(idQ__idQ_idQa, a), idQa, divide(divide(idQ__idQ_idQa, a), idQa)), true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 8 (closure) }
% 34.84/4.89 ifeq2(ifeq(true, true, less_equal(divide(idQ__idQ_idQa, a), idQa), true), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 7 (ifeq_axiom) }
% 34.84/4.89 ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa)
% 34.84/4.89 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.89 ifeq2(true, true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by axiom 11 (less_equal_quotient) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), zero), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by lemma 23 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(divide(a, idQ__idQ_idQa), a)), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by lemma 22 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(divide(idQ_idQa, idQ__idQ_idQa), a)), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(true, true, divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by axiom 2 (zero_is_smallest) R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(zero, divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by lemma 23 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(divide(divide(a, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by lemma 22 R->L }
% 34.84/4.89 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.89 = { by lemma 17 R->L }
% 34.84/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(divide(divide(idQ_idQa, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.90 = { by lemma 17 R->L }
% 34.84/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.90 = { by axiom 6 (ifeq_axiom) R->L }
% 34.84/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(true, true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.90 = { by axiom 14 (quotient_property) R->L }
% 34.84/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(idQa, divide(idQ__idQ_idQa, a))), true, ifeq(quotient(divide(identity, idQ__idQ_idQa), a, divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, a, idQa), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.90 = { by axiom 3 (identity_divide_a) }
% 34.84/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(idQa, divide(idQ__idQ_idQa, a))), true, ifeq(quotient(divide(identity, idQ__idQ_idQa), a, divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(true, true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 34.84/4.90 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(idQa, divide(idQ__idQ_idQa, a))), true, ifeq(quotient(divide(identity, idQ__idQ_idQa), a, divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.90 = { by axiom 8 (closure) }
% 35.59/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(true, true, ifeq(quotient(divide(identity, idQ__idQ_idQa), a, divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.90 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(divide(identity, idQ__idQ_idQa), a, divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.90 = { by axiom 8 (closure) }
% 35.59/4.90 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(true, true, ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.90 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 8 (closure) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(true, true, ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(quotient(identity, idQ__idQ_idQa, divide(identity, idQ__idQ_idQa)), true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 8 (closure) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(ifeq(true, true, less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), ifeq2(less_equal(divide(idQa, divide(idQ__idQ_idQa, a)), divide(divide(identity, idQ__idQ_idQa), a)), true, ifeq2(less_equal(divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), true, divide(divide(identity, idQ__idQ_idQa), a), divide(idQa, divide(idQ__idQ_idQa, a))), divide(idQa, divide(idQ__idQ_idQa, a)))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 12 (less_equal_and_equal) }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, divide(idQ__idQ_idQa, a), divide(idQa, divide(idQ__idQ_idQa, a))), true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 8 (closure) }
% 35.59/4.91 ifeq2(ifeq(true, true, less_equal(idQa, divide(idQ__idQ_idQa, a)), true), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(less_equal(idQa, divide(idQ__idQ_idQa, a)), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQa), true, divide(idQ__idQ_idQa, a), idQa), idQa)
% 35.59/4.91 = { by axiom 12 (less_equal_and_equal) }
% 35.59/4.91 idQa
% 35.59/4.91
% 35.59/4.91 Goal 1 (prove_one_inversion_equals_three): idQa = idQ__idQ_idQa.
% 35.59/4.91 Proof:
% 35.59/4.91 idQa
% 35.59/4.91 = { by lemma 24 R->L }
% 35.59/4.91 divide(idQ__idQ_idQa, a)
% 35.59/4.91 = { by axiom 6 (ifeq_axiom) R->L }
% 35.59/4.91 ifeq2(true, true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 10 (divisor_existence) R->L }
% 35.59/4.91 ifeq2(ifeq(quotient(idQa, idQ_idQa, divide(idQa, idQ_idQa)), true, less_equal(divide(idQa, idQ_idQa), idQa), true), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 8 (closure) }
% 35.59/4.91 ifeq2(ifeq(true, true, less_equal(divide(idQa, idQ_idQa), idQa), true), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(less_equal(divide(idQa, idQ_idQa), idQa), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by lemma 20 }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, idQa), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by lemma 24 R->L }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, divide(idQ__idQ_idQa, a)), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 6 (ifeq_axiom) R->L }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, divide(idQ__idQ_idQa, a)), true, ifeq2(true, true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 10 (divisor_existence) R->L }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, divide(idQ__idQ_idQa, a)), true, ifeq2(ifeq(quotient(idQ__idQ_idQa, a, divide(idQ__idQ_idQa, a)), true, less_equal(divide(idQ__idQ_idQa, a), idQ__idQ_idQa), true), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 8 (closure) }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, divide(idQ__idQ_idQa, a)), true, ifeq2(ifeq(true, true, less_equal(divide(idQ__idQ_idQa, a), idQ__idQ_idQa), true), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 7 (ifeq_axiom) }
% 35.59/4.91 ifeq2(less_equal(idQ__idQ_idQa, divide(idQ__idQ_idQa, a)), true, ifeq2(less_equal(divide(idQ__idQ_idQa, a), idQ__idQ_idQa), true, divide(idQ__idQ_idQa, a), idQ__idQ_idQa), idQ__idQ_idQa)
% 35.59/4.91 = { by axiom 12 (less_equal_and_equal) }
% 35.59/4.91 idQ__idQ_idQa
% 35.59/4.91 % SZS output end Proof
% 35.59/4.91
% 35.59/4.91 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------