now

let Y be Subset of REAL; :: thesis: ( Y is open implies (f | X) " Y is open )
assume Y is open ; :: thesis: (f | X) " Y is open
then A1: f " Y is open by Th54;
( the carrier of (T | X) = X & (f | X) " Y = X /\ (f " Y) ) by FUNCT_1:70, PRE_TOPC:8;
hence (f | X) " Y is open by A1, TSP_1:def 1; :: thesis: verum

end;

hence for b₁ being RealMap of (T | X) st b₁ = f | X holds
b₁ is continuous by Th54; :: thesis: verum