
theorem Th16:
  for X, Y, Z be RealLinearSpace
  for f,g,h be VECTOR of R_VectorSpace_of_BilinearOperators(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 RealLinearSpace;
    let f,g,h be VECTOR of R_VectorSpace_of_BilinearOperators(X,Y,Z);
    reconsider f9=f,g9=g,h9=h as BilinearOperator of X,Y,Z by Def6;
    A1: R_VectorSpace_of_BilinearOperators(X,Y,Z)
        is Subspace of RealVectSpace(the carrier of [:X,Y:],Z)
        by RSSPACE:11; then
    reconsider f1=f as VECTOR
      of RealVectSpace(the carrier of [:X,Y:],Z ) by RLSUB_1:10;
    reconsider h1=h
      as VECTOR of RealVectSpace(the carrier of [:X,Y:],Z) by A1,RLSUB_1:10;
    reconsider g1=g as VECTOR of RealVectSpace(the carrier of [:X,Y:],Z)
      by A1,RLSUB_1:10;
    A2: now
      assume
      A3: h = f+g;
      let x be Element of X,y be Element of Y;
      A4: h1=f1+g1 by A1,A3,RLSUB_1:13;
      [x,y] is Element of [:X,Y:];
      hence h9.(x,y)=f9.(x,y)+g9.(x,y) by A4,LOPBAN_1:1;
    end;
    now
      assume
      A5: for x be Element of X,y be Element of Y
           holds h9.(x,y)=f9.(x,y)+g9.(x,y);
      for z be Element of [:X,Y:] holds h9.z=f9.z+g9.z
      proof
        let z be Element of [:X,Y:];
        consider x be Point of X,y be Point of Y such that
        A6: z=[x,y] by PRVECT_3:9;
        thus h9.z = h9.(x,y) by A6
         .= f9.(x,y)+g9.(x,y) by A5
         .= f9.z+g9.z by A6;
      end; then
      h1=f1+g1 by LOPBAN_1:1;
      hence h =f+g by A1,RLSUB_1:13;
    end;
    hence thesis by A2;
  end;
