
theorem Th9:
  for X be non empty set for Y be RealNormSpace for f,h be VECTOR
  of R_VectorSpace_of_BoundedFunctions(X,Y) for f9,h9 be bounded Function of X,
  the carrier of Y st f9=f & h9=h
for a be Real holds h = a*f iff for x be
  Element of X holds h9.x = a*f9.x
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f,h be VECTOR of R_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 a be Real;
   reconsider a as Real;
A2: R_VectorSpace_of_BoundedFunctions(X,Y) is Subspace of RealVectSpace(X,Y)
  by Th6,RSSPACE:11;
  then reconsider f1=f, h1=h as VECTOR of RealVectSpace(X,Y) by RLSUB_1:10;
A3: now
    assume
A4: h = a*f;
    let x be Element of X;
    h1=a*f1 by A2,A4,RLSUB_1:14;
    hence h9.x=a*f9.x by A1,LOPBAN_1:12;
  end;
  now
    assume for x be Element of X holds h9.x=a*f9.x;
    then h1=a*f1 by A1,LOPBAN_1:12;
    hence h =a*f by A2,RLSUB_1:14;
  end;
  hence thesis by A3;
end;
