
theorem Th22:
for f be PartFunc of REAL,REAL, x0, r be Real st
 f is_left_differentiable_in x0 holds
  r(#)f is_left_differentiable_in x0 & Ldiff(r(#)f,x0) = r*Ldiff(f,x0)
proof
    let f be PartFunc of REAL,REAL, x0, r be Real;
    assume A1: f is_left_differentiable_in x0; then
    r = 0 implies
     r(#)f is_left_differentiable_in x0 &
      Ldiff(r(#)f,x0) = r*Ldiff(f,x0) by Lm17;
    hence thesis by A1,Lm19;
end;
