%   ORIGINAL: 'h4/thm/realax/TREAL_EQ_EQUIV_'
% 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: !p q. 'h4/const/realax/treal_eq' p q <=> 'h4/const/realax/treal_eq' p = 'h4/const/realax/treal_eq' q
%   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: !p q. happ ('h4/const/realax/treal_eq' p) q <=> 'h4/const/realax/treal_eq' p = 'h4/const/realax/treal_eq' q
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_3f49111,V_3f49107]: ![F, G]: (![X]: s(V_3f49107,happ(s(fun(V_3f49111,V_3f49107),F),s(V_3f49111,X))) = s(V_3f49107,happ(s(fun(V_3f49111,V_3f49107),G),s(V_3f49111,X))) => s(fun(V_3f49111,V_3f49107),F) = s(fun(V_3f49111,V_3f49107),G))).
fof('h4/thm/realax/TREAL_EQ_EQUIV_', conjecture, ![P, Q]: (p(s(bool,happ(s(fun('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),bool),'h4/const/realax/treal_eq'(s('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),P))),s('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),Q)))) <=> s(fun('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),bool),'h4/const/realax/treal_eq'(s('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),P))) = s(fun('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),bool),'h4/const/realax/treal_eq'(s('h4/type/pair/prod'('h4/type/hreal/hreal','h4/type/hreal/hreal'),Q))))).
