
theorem Th24:
  for X, Y, Z be RealNormSpace
  for f,g,h be VECTOR of R_VectorSpace_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 VECTOR of R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z);
    A1: R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z)
      is Subspace of R_VectorSpace_of_BilinearOperators(X,Y,Z)
        by RSSPACE:11; then
    reconsider f1=f as
      VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z) by RLSUB_1:10;
    reconsider h1=h as VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z)
      by A1,RLSUB_1:10;
    reconsider g1=g as VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z)
      by A1,RLSUB_1:10;
    hereby
      assume
      A2: h = f+g;
      let x be Element of X,y be Element of Y;
      h1 = f1+g1 by A1,A2,RLSUB_1:13;
      hence h.(x,y) = f.(x,y) + g.(x,y) by Th16;
    end;
    assume for x be Element of X,y be Element of Y
    holds h.(x,y) = f.(x,y)+g.(x,y);
    then h1=f1+g1 by Th16;
    hence thesis by A1,RLSUB_1:13;
  end;
