 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 Th2:
  for E,F,G be RealNormSpace,
      L be Lipschitzian LinearOperator of [:E,F:],G
  holds
    ex L1 be Lipschitzian LinearOperator of E,G,
       L2 be Lipschitzian LinearOperator of 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,
        L be Lipschitzian LinearOperator of [:E,F:],G;
    consider L1 be LinearOperator of E,G,
             L2 be LinearOperator of 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 Th0;
    consider K being Real such that
    A2: 0 <= K
      & for z being VECTOR of [:E,F:]
        holds ||. L.z .|| <= K * ||.z.|| by LOPBAN_1:def 8;
    now
      let x be Point of E;
      reconsider z=[x,0.F] as Point of [:E,F:];
      A4: L1.x = L/. [x,0.F] by A1
              .= L.z;
      ||.z.|| = ||.x.|| by NDIFF_8:2;
      hence ||. L1.x .|| <= K * ||.x.|| by A2,A4;
    end; then
    A5: L1 is Lipschitzian by A2,LOPBAN_1:def 8;
    now
      let y be Point of F;
      reconsider z = [0.E,y] as Point of [:E,F:];
      A7: L2.y = L /. [0.E,y] by A1
              .= L.z;
      ||.z.|| = ||.y.|| by NDIFF_8:3;
      hence ||. L2.y .|| <= K * ||.y.|| by A2,A7;
    end; then
    L2 is Lipschitzian by A2,LOPBAN_1:def 8;
    hence thesis by A1,A5;
  end;
