
theorem Th40:
  for X,Y,Z be RealNormSpace
  for f,g,h be Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  holds
    h = f-g
  iff
    for x be VECTOR of X,y be VECTOR of Y
    holds h.(x,y) = f.(x,y) - g.(x,y)
  proof
    let X, Y, Z be RealNormSpace;
    let f,g,h be Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
    reconsider f9=f,g9=g,h9=h
      as Lipschitzian BilinearOperator of X,Y,Z by Def9;
    hereby
      assume h=f-g; then
      h+g=f-(g-g) by RLVECT_1:29; then
      A1: h+g=f-0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
          by RLVECT_1:15;
      now
        let x be VECTOR of X,y be VECTOR of Y;
        f9.(x,y)=h9.(x,y) + g9.(x,y) by A1,Th35; then
        f9.(x,y)-g9.(x,y)
        = h9.(x,y) + (g9.(x,y)-g9.(x,y)) by RLVECT_1:def 3; then
        f9.(x,y)-g9.(x,y)=h9.(x,y) + 0.Z by RLVECT_1:15;
        hence f9.(x,y)-g9.(x,y)=h9.(x,y);
      end;
      hence for x be VECTOR of X,y be VECTOR of Y
            holds h.(x,y) = f.(x,y) - g.(x,y);
    end;
    assume
    A2: for x be VECTOR of X,y be VECTOR of Y
        holds h.(x,y) = f.(x,y) - g.(x,y);
    now
      let x be VECTOR of X,y be VECTOR of Y;
      h9.(x,y) = f9.(x,y) - g9.(x,y) by A2; then
      h9.(x,y) + g9.(x,y)= f9.(x,y) - (g9.(x,y)- g9.(x,y))
        by RLVECT_1:29; then
      h9.(x,y) + g9.(x,y)= f9.(x,y) - 0.Z by RLVECT_1:15;
      hence h9.(x,y) + g9.(x,y)= f9.(x,y);
    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_BoundedBilinearOperators(X,Y,Z) by RLVECT_1:15;
    hence thesis;
  end;
