%   ORIGINAL: 'h4/thm/poly/POLY_ZERO_LEMMA_'
% 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: !h t. 'h4/const/poly/poly' ('h4/const/list/CONS' h t) = 'h4/const/poly/poly' 'h4/const/list/NIL' ==> h = 'h4/const/real/real_of_num' 'h4/const/num/0' /\ 'h4/const/poly/poly' t = 'h4/const/poly/poly' 'h4/const/list/NIL'
%   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: !h t. 'h4/const/poly/poly' ('h4/const/list/CONS' h t) = 'h4/const/poly/poly' 'h4/const/list/NIL' ==> h = 'h4/const/real/real_of_num' 'h4/const/num/0' /\ 'h4/const/poly/poly' t = 'h4/const/poly/poly' 'h4/const/list/NIL'
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_3f47309,V_3f47305]: ![F, G]: (![X]: s(V_3f47305,happ(s(fun(V_3f47309,V_3f47305),F),s(V_3f47309,X))) = s(V_3f47305,happ(s(fun(V_3f47309,V_3f47305),G),s(V_3f47309,X))) => s(fun(V_3f47309,V_3f47305),F) = s(fun(V_3f47309,V_3f47305),G))).
fof('h4/thm/poly/POLY_ZERO_LEMMA_', conjecture, ![H, T]: (s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),'h4/const/list/CONS'(s('h4/type/realax/real',H),s('h4/type/list/list'('h4/type/realax/real'),T))))) = s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),'h4/const/list/NIL'))) => (s('h4/type/realax/real',H) = s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0'))) & s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),T))) = s(fun('h4/type/realax/real','h4/type/realax/real'),'h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),'h4/const/list/NIL')))))).
