now :: thesis: for x0 being Real st x0 in dom (f | X) holds
f | X is_continuous_in x0

let x0 be Real; :: thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 )
assume A1: x0 in dom (f | X) ; :: thesis: f | X is_continuous_in x0
then x0 in dom f by RELAT_1:57;
then A2: f is_continuous_in x0 by Def2;
x0 in X by A1;
hence f | X is_continuous_in x0 by A2, Th6; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL, the carrier of S st b₁ = f | X holds
b₁ is continuous ; :: thesis: verum