
theorem Th19:
  for V be vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty ModuleStr over F_Complex for F
be RFunctional of V st F is Real_homogeneous for v be Vector of V for r be Real
  holds F.([**0,r**]*v) = r*F.(i_FC*v)
proof
  let V be vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty ModuleStr over F_Complex;
  let F be RFunctional of V;
  assume
A1: F is Real_homogeneous;
  let v be Vector of V;
  let r be Real;
  thus F.([**0,r**]*v) = F.([**r,0**]*i_FC*v)
    .= F.([**r,0**]*(i_FC*v)) by VECTSP_1:def 16
    .= r*F.(i_FC*v) by A1;
end;
