%   ORIGINAL: 'h4/thm/integral/GAUGE_MIN_FINITE_'
% 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: !s gs n. (!m. 'h4/const/arithmetic/<=' m n ==> 'h4/const/transc/gauge' s (gs m)) ==> (?g. 'h4/const/transc/gauge' s g /\ (!d p. 'h4/const/transc/fine' g ('h4/const/pair/,' d p) ==> (!m. 'h4/const/arithmetic/<=' m n ==> 'h4/const/transc/fine' (gs m) ('h4/const/pair/,' d p))))
%   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: !s gs n. (!m. 'h4/const/arithmetic/<=' m n ==> 'h4/const/transc/gauge' s (happ gs m)) ==> (?g. 'h4/const/transc/gauge' s g /\ (!d p. 'h4/const/transc/fine' g ('h4/const/pair/,' d p) ==> (!m. 'h4/const/arithmetic/<=' m n ==> 'h4/const/transc/fine' (happ gs m) ('h4/const/pair/,' d p))))
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_3f40684,V_3f40680]: ![F, G]: (![X]: s(V_3f40680,happ(s(fun(V_3f40684,V_3f40680),F),s(V_3f40684,X))) = s(V_3f40680,happ(s(fun(V_3f40684,V_3f40680),G),s(V_3f40684,X))) => s(fun(V_3f40684,V_3f40680),F) = s(fun(V_3f40684,V_3f40680),G))).
fof('h4/thm/integral/GAUGE_MIN_FINITE_', conjecture, ![S, Gs, N]: (![M]: (p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',M),s('h4/type/num/num',N)))) => p(s(bool,'h4/const/transc/gauge'(s(fun('h4/type/realax/real',bool),S),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')),Gs),s('h4/type/num/num',M))))))) => ?[G]: (p(s(bool,'h4/const/transc/gauge'(s(fun('h4/type/realax/real',bool),S),s(fun('h4/type/realax/real','h4/type/realax/real'),G)))) & ![D, P]: (p(s(bool,'h4/const/transc/fine'(s(fun('h4/type/realax/real','h4/type/realax/real'),G),s('h4/type/pair/prod'(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),'h4/const/pair/,'(s(fun('h4/type/num/num','h4/type/realax/real'),D),s(fun('h4/type/num/num','h4/type/realax/real'),P)))))) => ![M]: (p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',M),s('h4/type/num/num',N)))) => p(s(bool,'h4/const/transc/fine'(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')),Gs),s('h4/type/num/num',M))),s('h4/type/pair/prod'(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),'h4/const/pair/,'(s(fun('h4/type/num/num','h4/type/realax/real'),D),s(fun('h4/type/num/num','h4/type/realax/real'),P))))))))))).
