 reserve S,T,W,Y for RealNormSpace;
 reserve f,f1,f2 for PartFunc of S,T;
 reserve Z for Subset of S;
 reserve i,n for Nat;

theorem
  for E,F,G be RealNormSpace,
      L be Point of R_NormSpace_of_BoundedLinearOperators([:E,F:],G),
      L11,L12 be Point of R_NormSpace_of_BoundedLinearOperators(E,G),
      L21,L22 be Point of R_NormSpace_of_BoundedLinearOperators(F,G)
  st ( for x be Point of E,
           y be Point of F
       holds L. [x,y] = L11.x + L21.y )
   & ( for x be Point of E,
           y be Point of F
       holds L. [x,y] = L12.x + L22.y )
  holds L11 = L12 & L21 = L22
  proof
    let E,F,G be RealNormSpace,
        L be Point of R_NormSpace_of_BoundedLinearOperators([:E,F:],G),
        L11,L12 be Point of R_NormSpace_of_BoundedLinearOperators(E,G),
        L21,L22 be Point of R_NormSpace_of_BoundedLinearOperators(F,G);
    assume
    A1: for x be Point of E,
             y be Point of F
         holds L. [x,y] = L11.x + L21.y;
    assume
    A2: for x be Point of E,
             y be Point of F
         holds L. [x,y] = L12.x + L22.y;
    reconsider LP = L
      as Lipschitzian LinearOperator of [:E,F:],G by LOPBAN_1:def 9;
    reconsider LP11 = L11, LP12 = L12
      as Lipschitzian LinearOperator of E,G by LOPBAN_1:def 9;
    reconsider LP21 = L21, LP22 = L22
      as Lipschitzian LinearOperator of F,G by LOPBAN_1:def 9;
    A3: for x be Point of E, y be Point of F
        holds LP. [x,y] = LP11.x + LP21.y by A1;
    for x be Point of E, y be Point of F
        holds LP. [x,y] = LP12.x + LP22.y by A2;
    hence L11 = L12 & L21 = L22 by A3,Th0X;
  end;
