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 Th8:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
  ( for z1,z2 be Point of [:E,F:]
    holds partdiff`2(f, z1 + z2) = partdiff`2(f, z1) + partdiff`2(f, z2) )
      &
  ( for z be Point of [:E,F:], a be Real
    holds partdiff`2(f, a * z) = a * partdiff`2(f, z) )
      &
  ( for z1,z2 be Point of [:E,F:]
    holds partdiff`2(f, z1 - z2) = partdiff`2(f, z1) - partdiff`2(f, z2) )
  proof
    let f be Lipschitzian BilinearOperator of E,F,G;
    A1:
    now
      let z1,z2 be Point of [:E,F:];
      for x be VECTOR of F
      holds partdiff`2(f, z1 + z2).x = partdiff`2(f, z1).x + partdiff`2(f,z2).x
      proof
        let x be VECTOR of F;
        A2: partdiff`2(f,z1).x
         = (f * (reproj2 z1)).x by Th4
        .= f.((reproj2 z1).x) by FUNCT_2:15
        .= f.(z1`1,x) by NDIFF_7:def 2;
        A3: (partdiff`2(f,z2)).x
         = (f * (reproj2 z2)).x by Th4
        .= f.((reproj2 z2).x) by FUNCT_2:15
        .= f.(z2`1,x) by NDIFF_7:def 2;
        thus partdiff`2(f,z1 + z2).x
         = (f * (reproj2(z1 + z2))).x by Th4
        .= f.((reproj2(z1 + z2)).x) by FUNCT_2:15
        .= f. [(z1 + z2)`1, x] by NDIFF_7:def 2
        .= f.(z1`1 + z2`1, x) by Th5
        .= partdiff`2(f, z1).x + partdiff`2(f, z2).x by A2,A3,LOPBAN_8:12;
      end;
      hence partdiff`2(f,z1 + z2) = partdiff`2(f, z1) + partdiff`2(f, z2)
        by LOPBAN_1:35;
    end;

    A4:
    now
      let z1 be Point of [:E,F:], a be Real;
      for x be VECTOR of F
      holds partdiff`2(f, a * z1).x = a * partdiff`2(f,z1).x
      proof
        let x be VECTOR of F;
        A5: partdiff`2(f,z1).x
         = (f * (reproj2 z1)).x by Th4
        .= f.((reproj2 z1).x) by FUNCT_2:15
        .= f.(z1`1, x) by NDIFF_7:def 2;
        thus partdiff`2(f,a * z1) . x
         = (f * (reproj2(a * z1))).x by Th4
        .= f.((reproj2(a * z1)).x) by FUNCT_2:15
        .= f.[(a * z1)`1, x] by NDIFF_7:def 2
        .= f.(a * z1`1, x) by Th5
        .= a * partdiff`2(f, z1).x by A5,LOPBAN_8:12;
      end;
      hence partdiff`2(f, a * z1) = a * partdiff`2(f,z1)
        by LOPBAN_1:36;
    end;
    now
      let z1,z2 be Point of [:E,F:];
      thus partdiff`2(f, z1 - z2)
       = partdiff`2(f, z1 + (-1) * z2) by RLVECT_1:16
      .= partdiff`2(f, z1) + partdiff`2(f, (-1) * z2) by A1
      .= partdiff`2(f, z1) + (-1) * partdiff`2(f, z2) by A4
      .= partdiff`2(f, z1) - partdiff`2(f, z2) by RLVECT_1:16;
    end;
    hence thesis by A1,A4;
  end;
