reserve E, F, G,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 f be Lipschitzian BilinearOperator of E,F,G,
      z be Point of [:E,F:]
  holds
    f * reproj1(z) is Lipschitzian LinearOperator of E,G
  & f * reproj2(z) is Lipschitzian LinearOperator of F,G
  & ex g be Point of (R_NormSpace_of_BoundedBilinearOperators (E,F,G))
    st f = g
     & ( for x be VECTOR of E
         holds ||. (f * reproj1(z)).x .|| <= ||.g.|| * ||.z`2.|| * ||.x.|| )
     & ( for y be VECTOR of F
         holds ||. (f * reproj2(z)).y .|| <= ||.g.|| * ||.z`1.|| * ||.y.|| )
  proof
    let f be Lipschitzian BilinearOperator of E,F,G,
        z be Point of [:E,F:];

    reconsider g = f as Point of R_NormSpace_of_BoundedBilinearOperators(E,F,G)
      by LOPBAN_9:def 4;

    set K = ||.g.||;
    A1: 0 <= K
      & for x be VECTOR of E
        for y be VECTOR of F
        holds ||. f.(x,y) .|| <= K * ||.x.|| * ||.y.|| by LOPBAN_9:16,17;

    reconsider L1 = f * reproj1(z) as LinearOperator of E,G by Th1;
    reconsider L2 = f * reproj2(z) as LinearOperator of F,G by Th1;
    set K1 = K * ||.z`2.||;
    set K2 = K * ||.z`1.||;
    A2: 0 <= ||.z`2.|| by NORMSP_1:4;
    A3: for x be VECTOR of E holds ||. L1.x .|| <= K1 * ||.x.||
    proof
      let x be VECTOR of E;
      f.(x,z`2)
       = f.((reproj1 z).x) by NDIFF_7:def 1
      .= L1.x by FUNCT_2:15;
      then ||.L1.x.|| <= K * ||.x.|| * ||.z`2.|| by LOPBAN_9:16;
      hence thesis;
    end;
    A4: 0 <= ||.z`1.|| by NORMSP_1:4;
    for y be VECTOR of F holds ||. L2.y .|| <= K2 * ||.y.||
    proof
      let y be VECTOR of F;
      f.(z`1,y)
       = f.((reproj2 z).y) by NDIFF_7:def 2
      .= L2.y by FUNCT_2:15;
      hence thesis by LOPBAN_9:16;
    end;
    hence thesis by A1,A2,A3,A4,XREAL_1:127,LOPBAN_1:def 8;
  end;
