%   ORIGINAL: 'h4/thm/transc/COS_FDIFF_'
% 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: 'h4/const/powser/diffs' (\n. 'h4/const/bool/COND' ('h4/const/arithmetic/EVEN' n) ('h4/const/real/#slash#' ('h4/const/real/pow' ('h4/const/realax/real_neg' ('h4/const/real/real_of_num' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) ('h4/const/arithmetic/DIV' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')))) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' n))) ('h4/const/real/real_of_num' 'h4/const/num/0')) = (\n. 'h4/const/realax/real_neg' ((\n0. 'h4/const/bool/COND' ('h4/const/arithmetic/EVEN' n0) ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/real/#slash#' ('h4/const/real/pow' ('h4/const/realax/real_neg' ('h4/const/real/real_of_num' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) ('h4/const/arithmetic/DIV' ('h4/const/arithmetic/-' n0 ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')))) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' n0)))) n))
%   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. (!n. happ _0 n = 'h4/const/bool/COND' ('h4/const/arithmetic/EVEN' n) ('h4/const/real/#slash#' ('h4/const/real/pow' ('h4/const/realax/real_neg' ('h4/const/real/real_of_num' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) ('h4/const/arithmetic/DIV' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')))) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' n))) ('h4/const/real/real_of_num' 'h4/const/num/0')) ==> (!x. 'h4/const/powser/diffs' _0 x = 'h4/const/realax/real_neg' ('h4/const/bool/COND' ('h4/const/arithmetic/EVEN' x) ('h4/const/real/real_of_num' 'h4/const/num/0') ('h4/const/real/#slash#' ('h4/const/real/pow' ('h4/const/realax/real_neg' ('h4/const/real/real_of_num' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) ('h4/const/arithmetic/DIV' ('h4/const/arithmetic/-' x ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')))) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' x)))))
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_3f68186,V_3f68182]: ![F, G]: (![X]: s(V_3f68182,happ(s(fun(V_3f68186,V_3f68182),F),s(V_3f68186,X))) = s(V_3f68182,happ(s(fun(V_3f68186,V_3f68182),G),s(V_3f68186,X))) => s(fun(V_3f68186,V_3f68182),F) = s(fun(V_3f68186,V_3f68182),G))).
fof('h4/thm/transc/COS_FDIFF_', conjecture, ![V__0]: (![N]: s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),V__0),s('h4/type/num/num',N))) = s('h4/type/realax/real','h4/const/bool/COND'(s(bool,'h4/const/arithmetic/EVEN'(s('h4/type/num/num',N))),s('h4/type/realax/real','h4/const/real/#slash#'(s('h4/type/realax/real','h4/const/real/pow'(s('h4/type/realax/real','h4/const/realax/real_neg'(s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),s('h4/type/num/num','h4/const/arithmetic/DIV'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),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',N))))))),s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))))) => ![X]: s('h4/type/realax/real','h4/const/powser/diffs'(s(fun('h4/type/num/num','h4/type/realax/real'),V__0),s('h4/type/num/num',X))) = s('h4/type/realax/real','h4/const/realax/real_neg'(s('h4/type/realax/real','h4/const/bool/COND'(s(bool,'h4/const/arithmetic/EVEN'(s('h4/type/num/num',X))),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/#slash#'(s('h4/type/realax/real','h4/const/real/pow'(s('h4/type/realax/real','h4/const/realax/real_neg'(s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),s('h4/type/num/num','h4/const/arithmetic/DIV'(s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num',X),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),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',X))))))))))))).
