
theorem Th21:
  for X be non empty set, Y be ComplexNormSpace, f,h be Point of
  C_NormSpace_of_BoundedFunctions(X,Y), f9,h9 be bounded Function of X,the
carrier of Y st f9=f & h9=h for c be Complex holds h = c*f iff for x be Element
  of X holds h9.x = c*f9.x
proof
  let X be non empty set;
  let Y be ComplexNormSpace;
  let f,h be Point of C_NormSpace_of_BoundedFunctions(X,Y);
  let f9,h9 be bounded Function of X,the carrier of Y such that
A1: f9=f & h9=h;
  reconsider h1=h as VECTOR of C_VectorSpace_of_BoundedFunctions(X,Y);
  reconsider f1=f as VECTOR of C_VectorSpace_of_BoundedFunctions(X,Y);
  let c be Complex;
A2: now
    assume h1=c*f1;
    hence h=Mult_(ComplexBoundedFunctions(X,Y), ComplexVectSpace(X,Y)) .[c,f1]
    by CLVECT_1:def 1
      .=c*f by CLVECT_1:def 1;
  end;
  now
    assume h=c*f;
    hence h1=Mult_(ComplexBoundedFunctions(X,Y), ComplexVectSpace(X,Y)).[c,f]
    by CLVECT_1:def 1
      .=c*f1 by CLVECT_1:def 1;
  end;
  hence thesis by A1,A2,Th10;
end;
