reconsider g = f as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
P1: g is continuous by Def2Th;
r (#) g = r (#) f by LM003B;
hence for b₁ being PartFunc of REAL,(REAL n) st b₁ = r (#) f holds
b₁ is continuous by P1, Def2Th; :: thesis: verum