%   ORIGINAL: 'h4/thm/transc/MCLAURIN_ZERO_'
% 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: !diff n x. x = 'h4/const/real/real_of_num' 'h4/const/num/0' /\ 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (\m. 'h4/const/realax/real_mul' ('h4/const/real/#slash#' (diff m ('h4/const/real/real_of_num' 'h4/const/num/0')) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' m))) ('h4/const/real/pow' x m)) = diff 'h4/const/num/0' ('h4/const/real/real_of_num' 'h4/const/num/0')
%   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. (!diff x m. happ (happ (happ _0 diff) x) m = 'h4/const/realax/real_mul' ('h4/const/real/#slash#' (happ (happ diff m) ('h4/const/real/real_of_num' 'h4/const/num/0')) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' m))) ('h4/const/real/pow' x m)) ==> (!diff n x. x = 'h4/const/real/real_of_num' 'h4/const/num/0' /\ 'h4/const/prim_rec/<' 'h4/const/num/0' n ==> 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (happ (happ _0 diff) x) = happ (happ diff 'h4/const/num/0') ('h4/const/real/real_of_num' 'h4/const/num/0'))
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_3f73064,V_3f73060]: ![F, G]: (![X]: s(V_3f73060,happ(s(fun(V_3f73064,V_3f73060),F),s(V_3f73064,X))) = s(V_3f73060,happ(s(fun(V_3f73064,V_3f73060),G),s(V_3f73064,X))) => s(fun(V_3f73064,V_3f73060),F) = s(fun(V_3f73064,V_3f73060),G))).
fof('h4/thm/transc/MCLAURIN_ZERO_', conjecture, ![V__0]: (![Diff, X, M]: 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(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real'))),V__0),s(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),Diff))),s('h4/type/realax/real',X))),s('h4/type/num/num',M))) = s('h4/type/realax/real','h4/const/realax/real_mul'(s('h4/type/realax/real','h4/const/real/#slash#'(s('h4/type/realax/real',happ(s(fun('h4/type/realax/real','h4/type/realax/real'),happ(s(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),Diff),s('h4/type/num/num',M))),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))))),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/arithmetic/FACT'(s('h4/type/num/num',M))))))),s('h4/type/realax/real','h4/const/real/pow'(s('h4/type/realax/real',X),s('h4/type/num/num',M))))) => ![Diff, N, X]: ((s('h4/type/realax/real',X) = s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))) & p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N))))) => s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N))),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(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),fun('h4/type/realax/real',fun('h4/type/num/num','h4/type/realax/real'))),V__0),s(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),Diff))),s('h4/type/realax/real',X))))) = s('h4/type/realax/real',happ(s(fun('h4/type/realax/real','h4/type/realax/real'),happ(s(fun('h4/type/num/num',fun('h4/type/realax/real','h4/type/realax/real')),Diff),s('h4/type/num/num','h4/const/num/0'))),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0')))))))).
