 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 Th0X:
  for E,F,G be RealNormSpace,
      L be LinearOperator of [:E,F:],G,
      L11 be LinearOperator of E,G,
      L12 be LinearOperator of F,G,
      L21 be LinearOperator of E,G,
      L22 be LinearOperator of F,G
  st ( for x be Point of E,
           y be Point of F
       holds L. [x,y] = L11.x + L12.y )
   & ( for x be Point of E,
           y be Point of F
       holds L. [x,y] = L21.x + L22.y )
  holds L11 = L21 & L12 = L22
  proof
    let E,F,G be RealNormSpace,
        L be LinearOperator of [:E,F:],G;
    let L11 be LinearOperator of E,G,
        L12 be LinearOperator of F,G,
        L21 be LinearOperator of E,G,
        L22 be LinearOperator of F,G;
    assume that
    A1: ( for x be Point of E, y be Point of F
           holds L. [x,y] = L11.x + L12.y ) and
    A2: ( for x be Point of E, y be Point of F
           holds L. [x,y] = L21.x + L22.y );
    now
      let x be Point of E;
      A4: L22. (0.F) = 0.G by LOPBAN_7:3;
      L12. (0.F) = 0.G by LOPBAN_7:3;
      hence L11.x = L11.x + L12. (0.F) by RLVECT_1:4
      .= L. [x,0.F] by A1
      .= L21.x + L22. 0.F by A2
      .= L21.x by A4,RLVECT_1:4;
    end;
    hence L11 = L21 by FUNCT_2:def 8;
    now
      let y be Point of F;
      A6: L21.(0.E) = 0.G by LOPBAN_7:3;
      L11.(0.E) = 0.G by LOPBAN_7:3;
      hence L12.y = L11.(0.E) + L12.y by RLVECT_1:4
      .= L. [0.E,y] by A1
      .= L21. 0.E + L22. y by A2
      .= L22.y by A6,RLVECT_1:4;
    end;
    hence L12 = L22 by FUNCT_2:def 8;
  end;
