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 Th4:
  for f be Lipschitzian BilinearOperator of E,F,G,
      z be Point of [:E,F:]
  holds
      f is_partial_differentiable_in`1 z
    & partdiff`1(f,z) = f * (reproj1 z)
    & f is_partial_differentiable_in`2 z
    & partdiff`2(f,z) = f * (reproj2 z)
  proof
    let f be Lipschitzian BilinearOperator of E,F,G,
        z be Point of [:E,F:];
    reconsider L1 = f * (reproj1 z) as Lipschitzian LinearOperator of E,G
      by Th2;
    reconsider L2 = f * (reproj2 z) as Lipschitzian LinearOperator of F,G
      by Th2;
    A1: L1 is_differentiable_in z `1
      & diff(L1, z `1) = L1 by NDIFF_7:26;

    hence f is_partial_differentiable_in`1 z by NDIFF_7:def 4;
    thus partdiff`1(f, z) = f * (reproj1 z) by A1,NDIFF_7:def 6;

    A2: L2 is_differentiable_in z `2
      & diff(L2, z `2) = L2 by NDIFF_7:26;
    hence f is_partial_differentiable_in`2 z by NDIFF_7:def 5;
    thus partdiff`2(f, z) = f * (reproj2 z) by A2,NDIFF_7:def 7;
  end;
