%   ORIGINAL: 'h4/thm/transc/DIVISION_SINGLE_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !a b. 'h4/const/real/real_lte' a b ==> 'h4/const/transc/division' ('h4/const/pair/,' a b) (\n. 'h4/const/bool/COND' (n = 'h4/const/num/0') a b)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!a b n. ?v. (v <=> n = 'h4/const/num/0') /\ happ (happ (happ _0 a) b) n = 'h4/const/bool/COND' v a b) ==> (!a b. 'h4/const/real/real_lte' a b ==> 'h4/const/transc/division' ('h4/const/pair/,' a b) (happ (happ _0 a) b))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f72488,V_3f72484]: ![F, G]: (![X]: s(V_3f72484,happ(s(fun(V_3f72488,V_3f72484),F),s(V_3f72488,X))) = s(V_3f72484,happ(s(fun(V_3f72488,V_3f72484),G),s(V_3f72488,X))) => s(fun(V_3f72488,V_3f72484),F) = s(fun(V_3f72488,V_3f72484),G))).
fof('h4/thm/transc/DIVISION_SINGLE_', conjecture, ![V__0]: (![A, B, N]: ?[V]: ((p(s(bool,V)) <=> s('h4/type/num/num',N) = s('h4/type/num/num','h4/const/num/0')) & s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real')),happ(s(fun('h4/type/realax/real',fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real'))),V__0),s('h4/type/realax/real',A))),s('h4/type/realax/real',B))),s('h4/type/num/num',N))) = s('h4/type/realax/real','h4/const/bool/COND'(s(bool,V),s('h4/type/realax/real',A),s('h4/type/realax/real',B)))) => ![A, B]: (p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',A),s('h4/type/realax/real',B)))) => p(s(bool,'h4/const/transc/division'(s('h4/type/pair/prod'('h4/type/realax/real','h4/type/realax/real'),'h4/const/pair/,'(s('h4/type/realax/real',A),s('h4/type/realax/real',B))),s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real')),happ(s(fun('h4/type/realax/real',fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real'))),V__0),s('h4/type/realax/real',A))),s('h4/type/realax/real',B))))))))).
