%   ORIGINAL: 'h4/thm/transc/SIN_PAIRED_'
% 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: !x. 'h4/const/seq/sums' (\n. 'h4/const/realax/real_mul' ('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')))) n) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' ('h4/const/arithmetic/+' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))))) ('h4/const/real/pow' x ('h4/const/arithmetic/+' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))))) ('h4/const/transc/sin' x)
%   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. (!x n. happ (happ _0 x) n = 'h4/const/realax/real_mul' ('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')))) n) ('h4/const/real/real_of_num' ('h4/const/arithmetic/FACT' ('h4/const/arithmetic/+' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))))) ('h4/const/real/pow' x ('h4/const/arithmetic/+' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO'))))) ==> (!x. 'h4/const/seq/sums' (happ _0 x) ('h4/const/transc/sin' 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_3f70486,V_3f70482]: ![F, G]: (![X]: s(V_3f70482,happ(s(fun(V_3f70486,V_3f70482),F),s(V_3f70486,X))) = s(V_3f70482,happ(s(fun(V_3f70486,V_3f70482),G),s(V_3f70486,X))) => s(fun(V_3f70486,V_3f70482),F) = s(fun(V_3f70486,V_3f70482),G))).
fof('h4/thm/transc/SIN_PAIRED_', conjecture, ![V__0]: (![X, N]: 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')),V__0),s('h4/type/realax/real',X))),s('h4/type/num/num',N))) = s('h4/type/realax/real','h4/const/realax/real_mul'(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',N))),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','h4/const/arithmetic/+'(s('h4/type/num/num','h4/const/arithmetic/*'(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/num/num',N))),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/realax/real','h4/const/real/pow'(s('h4/type/realax/real',X),s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num','h4/const/arithmetic/*'(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/num/num',N))),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'))))))))))) => ![X]: p(s(bool,'h4/const/seq/sums'(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')),V__0),s('h4/type/realax/real',X))),s('h4/type/realax/real','h4/const/transc/sin'(s('h4/type/realax/real',X)))))))).
