%   ORIGINAL: 'h4/thm/seq/NEST_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: !f g. (!n. 'h4/const/real/real_ge' (f ('h4/const/num/SUC' n)) (f n)) /\ (!n. 'h4/const/real/real_lte' (g ('h4/const/num/SUC' n)) (g n)) /\ (!n. 'h4/const/real/real_lte' (f n) (g n)) ==> (?l m. 'h4/const/real/real_lte' l m /\ ((!n. 'h4/const/real/real_lte' (f n) l) /\ 'h4/const/seq/-->' f l) /\ (!n. 'h4/const/real/real_lte' m (g n)) /\ 'h4/const/seq/-->' g m)
%   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: !f g. (!n. 'h4/const/real/real_ge' (happ f ('h4/const/num/SUC' n)) (happ f n)) /\ (!n. 'h4/const/real/real_lte' (happ g ('h4/const/num/SUC' n)) (happ g n)) /\ (!n. 'h4/const/real/real_lte' (happ f n) (happ g n)) ==> (?l m. 'h4/const/real/real_lte' l m /\ ((!n. 'h4/const/real/real_lte' (happ f n) l) /\ 'h4/const/seq/-->' f l) /\ (!n. 'h4/const/real/real_lte' m (happ g n)) /\ 'h4/const/seq/-->' g m)
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_3f65136,V_3f65132]: ![F, G]: (![X]: s(V_3f65132,happ(s(fun(V_3f65136,V_3f65132),F),s(V_3f65136,X))) = s(V_3f65132,happ(s(fun(V_3f65136,V_3f65132),G),s(V_3f65136,X))) => s(fun(V_3f65136,V_3f65132),F) = s(fun(V_3f65136,V_3f65132),G))).
fof('h4/thm/seq/NEST_LEMMA_', conjecture, ![F, G]: ((![N]: p(s(bool,'h4/const/real/real_ge'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',N)))))) & (![N]: p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num',N)))))) & ![N]: p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',N))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num',N)))))))) => ?[L, M]: (p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',L),s('h4/type/realax/real',M)))) & ((![N]: p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',N))),s('h4/type/realax/real',L)))) & p(s(bool,'h4/const/seq/-->'(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/realax/real',L))))) & (![N]: p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',M),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num',N)))))) & p(s(bool,'h4/const/seq/-->'(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/realax/real',M))))))))).
