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 Th3:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
    ex K be Real st 0 <= K
      &
    for z be Point of [:E,F:]
    holds
    ( for x be Point of E
      holds ||. (f * (reproj1 z)).x .|| <= K * ||.z`2.|| * ||.x.|| )
        &
    ( for y be Point of F
      holds ||. (f * (reproj2 z)).y .|| <= K * ||.z`1.|| * ||.y.|| )
  proof
    let f be Lipschitzian BilinearOperator of E,F,G;
    consider K be Real such that
    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:def 3;

    take K;
    thus 0 <= K by A1;
    let z be Point of [:E,F:];
    set L1 = f * (reproj1 z);
    set L2 = f * (reproj2 z);
    set K1 = K * ||.z`2.||;
    set K2 = K * ||.z`1.||;

    thus 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 A1;
      hence thesis;
    end;

    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 A1;
  end;
