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 Th24:
for X be RealNormSpace,
    f,g,h be VECTOR of R_VectorSpace_of_BoundedLinearFunctionals X
 holds h = f+g iff for x be VECTOR of X holds h.x = f.x + g.x
proof
  let X be RealNormSpace;
  let f,g,h be VECTOR of R_VectorSpace_of_BoundedLinearFunctionals X;
A1: R_VectorSpace_of_BoundedLinearFunctionals X is Subspace of X*'
      by Th22,RSSPACE:11;
  then reconsider f1=f, h1=h, g1=g as VECTOR of X*' by RLSUB_1:10;
  hereby
    assume
A2: h = f+g;
    let x be Element of X;
    h1=f1+g1 by A1,A2,RLSUB_1:13;
    hence h.x=f.x+g.x by Th20b;
  end;
  assume for x be Element of X holds h.x=f.x+g.x;
  then h1=f1+g1 by Th20b;
  hence thesis by A1,RLSUB_1:13;
end;
