reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th25:
  for X be RealNormSpace,
      f,h be VECTOR of R_VectorSpace_of_BoundedLinearFunctionals X,
      a be Real
   holds h = a*f iff for x be VECTOR of X holds h.x = a * f.x
proof
  let X be RealNormSpace;
  let f,h be VECTOR of R_VectorSpace_of_BoundedLinearFunctionals X;
  let a be Real;
A1: R_VectorSpace_of_BoundedLinearFunctionals X is Subspace of X*'
      by Th22,RSSPACE:11;
  then reconsider f1=f, h1=h as VECTOR of X*' by RLSUB_1:10;
  hereby
    assume
A2: h = a*f;
    let x be Element of X;
    h1=a*f1 by A1,A2,RLSUB_1:14;
    hence h.x=a*f.x by Th21b;
  end;
  assume for x be Element of X holds h.x=a*f.x;
  then h1=a*f1 by Th21b;
  hence thesis by A1,RLSUB_1:14;
end;
