reconsider g = f as PartFunc of REAL,(REAL-NS n) by REAL_NS1:def 4;
A1: g is continuous by Th23;
r (#) g = r (#) f by Th6;
hence for b₁ being PartFunc of REAL,(REAL n) st b₁ = r (#) f holds
b₁ is continuous by A1, Th23; :: thesis: verum