%   ORIGINAL: 'h4/thm/transc/DIVISION_THM_'
% 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: !D a b. 'h4/const/transc/division' ('h4/const/pair/,' a b) D <=> D 'h4/const/num/0' = a /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/transc/dsize' D) ==> 'h4/const/realax/real_lt' (D n) (D ('h4/const/num/SUC' n))) /\ (!n. 'h4/const/arithmetic/>=' n ('h4/const/transc/dsize' D) ==> D n = b)
%   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: !D a b. 'h4/const/transc/division' ('h4/const/pair/,' a b) D <=> happ D 'h4/const/num/0' = a /\ (!n. 'h4/const/prim_rec/<' n ('h4/const/transc/dsize' D) ==> 'h4/const/realax/real_lt' (happ D n) (happ D ('h4/const/num/SUC' n))) /\ (!n. 'h4/const/arithmetic/>=' n ('h4/const/transc/dsize' D) ==> happ D n = b)
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_3f72539,V_3f72535]: ![F, G]: (![X]: s(V_3f72535,happ(s(fun(V_3f72539,V_3f72535),F),s(V_3f72539,X))) = s(V_3f72535,happ(s(fun(V_3f72539,V_3f72535),G),s(V_3f72539,X))) => s(fun(V_3f72539,V_3f72535),F) = s(fun(V_3f72539,V_3f72535),G))).
fof('h4/thm/transc/DIVISION_THM_', conjecture, ![D, A, B]: (p(s(bool,'h4/const/transc/division'(s('h4/type/pair/prod'('h4/type/realax/real','h4/type/realax/real'),'h4/const/pair/,'(s('h4/type/realax/real',A),s('h4/type/realax/real',B))),s(fun('h4/type/num/num','h4/type/realax/real'),D)))) <=> (s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num','h4/const/num/0'))) = s('h4/type/realax/real',A) & (![N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/transc/dsize'(s(fun('h4/type/num/num','h4/type/realax/real'),D)))))) => p(s(bool,'h4/const/realax/real_lt'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num',N))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))))))))) & ![N]: (p(s(bool,'h4/const/arithmetic/>='(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/transc/dsize'(s(fun('h4/type/num/num','h4/type/realax/real'),D)))))) => s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num',N))) = s('h4/type/realax/real',B)))))).
