 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,
      L1 be Lipschitzian LinearOperator of E,G,
      L2 be Lipschitzian LinearOperator of F,G
  holds
    ex L be Lipschitzian LinearOperator of [:E,F:],G
    st ( for x be Point of E,
             y be Point of F
         holds L. [x,y] = L1.x + L2.y )
     & ( for x be Point of E holds L1.x = L/. [x,0.F] )
     & ( for y be Point of F holds L2.y = L/. [0.E,y] )
  proof
    let E,F,G be RealNormSpace,
        L1 be Lipschitzian LinearOperator of E,G,
        L2 be Lipschitzian LinearOperator of F,G;
    consider L be LinearOperator of [:E,F:],G such that
    A1: ( for x be Point of E,
              y be Point of F
          holds L. [x,y] = L1.x + L2.y )
      & ( for x be Point of E holds L1.x = L/. [x,0.F] )
      & ( for y be Point of F holds L2.y = L/. [0.E,y] ) by Th1;
    consider K1 being Real such that
    A2: 0 <= K1
      & for x being VECTOR of E
        holds ||. L1.x .|| <= K1 * ||.x.|| by LOPBAN_1:def 8;
    consider K2 being Real such that
    A3: 0 <= K2
      & for y being VECTOR of F
        holds ||. L2.y .|| <= K2 * ||.y.|| by LOPBAN_1:def 8;
    now
      let z be Point of [:E,F:];
      consider x be Point of E,y be Point of F such that
      A5: z = [x,y] by PRVECT_3:18;
      L.z = L1.x + L2.y by A1,A5; then
      A7: ||. L.z .|| <= ||. L1.x .|| + ||. L2.y .|| by NORMSP_1:def 1;
      A8: ||. L1.x .|| <= K1* ||.x.||
        & ||. L2.y .|| <= K2* ||.y.|| by A2,A3;
      A9: K1 * ||.x.|| <= K1 * ||.z.|| by A2,A5,NDIFF_8:21,XREAL_1:64;
      K2 * ||.y.|| <= K2 * ||.z.|| by A3,A5,NDIFF_8:21,XREAL_1:64; then
      ||. L1.x .|| <= K1 * ||.z.||
        & ||. L2.y .|| <= K2 * ||.z.|| by A8,A9,XXREAL_0:2;
      then ||. L1.x .|| + ||. L2.y .||
         <= K1* ||.z.|| + K2* ||.z.|| by XREAL_1:7;
      hence ||. L.z .|| <= (K1 + K2) * ||.z.|| by A7,XXREAL_0:2;
    end; then
    L is Lipschitzian by A2,A3,LOPBAN_1:def 8;
    hence thesis by A1;
  end;
