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 Th40:
  for X be RealNormSpace, f,g,h be Point of DualSp 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 Point of DualSp X;
  reconsider f9=f,g9=g,h9=h as Lipschitzian linear-Functional of X by Def9;
  hereby
    assume h=f-g; then
    h+g=f-(g-g) by RLVECT_1:29;
    then
A11: h+g=f-0.DualSp X by RLVECT_1:15;
    now
      let x be VECTOR of X;
      f9.x=h9.x + g9.x by A11,Th35;
      hence f9.x-g9.x=h9.x;
    end;
    hence for x be VECTOR of X holds h.x = f.x - g.x;
  end;
  assume
A2: for x be VECTOR of X holds h.x = f.x - g.x;
  now
    let x be VECTOR of X;
    h9.x = f9.x - g9.x by A2;
    hence h9.x + g9.x= f9.x;
  end;
  then
  f=h+g by Th35; then
  f-g=h+(g-g) by RLVECT_1:def 3;
  then f-g=h+0.(DualSp X) by RLVECT_1:15;
  hence thesis;
end;
