
theorem Th10:
  for X be non empty set for Y be ComplexNormSpace, f,h be VECTOR
  of C_VectorSpace_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 VECTOR of C_VectorSpace_of_BoundedFunctions(X,Y);
  let f9,h9 be bounded Function of X,the carrier of Y such that
A1: f9=f & h9=h;
  let c be Complex;
A2: C_VectorSpace_of_BoundedFunctions(X,Y) is Subspace of ComplexVectSpace(
  X,Y) by Th7,CSSPACE:11;
  then reconsider f1=f, h1=h as VECTOR of ComplexVectSpace(X,Y) by CLVECT_1:29;
A3: now
    assume
A4: h = c*f;
    let x be Element of X;
    h1=c*f1 by A2,A4,CLVECT_1:33;
    hence h9.x=c*f9.x by A1,CLOPBAN1:12;
  end;
  now
    assume for x be Element of X holds h9.x=c*f9.x;
    then h1=c*f1 by A1,CLOPBAN1:12;
    hence h =c*f by A2,CLVECT_1:33;
  end;
  hence thesis by A3;
end;
