%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Restriction of continuous functions %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
restriction_continuous[T1 : TYPE FROM real ] : THEORY
BEGIN
IMPORTING continuous_functions
f : VAR [real -> real]
restrict_continuous : PROPOSITION
continuous(f) IMPLIES continuous[T1](f)
END restriction_continuous
restriction_continuous2[T1, T2 : TYPE FROM real] : theory
BEGIN
ASSUMING
sub_domain : ASSUMPTION
FORALL (x : T1) : EXISTS (y : T2) : x = y
ENDASSUMING
IMPORTING continuous_functions
f : VAR [T2 -> real]
sub_dom : LEMMA FORALL (u : T1) : T2_pred(u)
restrict2(f) : [T1 -> real] = LAMBDA (u : T1) : f(u)
CONVERSION restrict2
restrict_continuous2 : PROPOSITION
continuous(f) IMPLIES continuous[T1](f)
END restriction_continuous2