reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th40:
  for X, Y be RealNormSpace for f,g,h be Point of
R_NormSpace_of_BoundedLinearOperators(X,Y) holds h = f-g iff for x be VECTOR of
  X holds h.x = f.x - g.x
proof
  let X, Y be RealNormSpace;
  let f,g,h be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  reconsider f9=f,g9=g,h9=h as Lipschitzian LinearOperator of X,Y by Def9;
  hereby
    assume h=f-g;
    then h+g=f-(g-g) by RLVECT_1:29;
    then h+g=f-0.R_NormSpace_of_BoundedLinearOperators(X,Y) by RLVECT_1:15;
    then
A1: h+g=f;
    now
      let x be VECTOR of X;
      f9.x=h9.x + g9.x by A1,Th35;
      then f9.x-g9.x=h9.x + (g9.x-g9.x) by RLVECT_1:def 3;
      then f9.x-g9.x=h9.x + 0.Y by RLVECT_1:15;
      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;
    then h9.x + g9.x= f9.x - (g9.x- g9.x) by RLVECT_1:29;
    then h9.x + g9.x= f9.x - 0.Y by RLVECT_1:15;
    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.R_NormSpace_of_BoundedLinearOperators(X,Y) by RLVECT_1:15;
  hence thesis;
end;
