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 Th13:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
    ( for z1,z2 be Point of [:E,F:]
      holds diff(f, z1 + z2) = diff(f,z1) + diff(f,z2) )
  & ( for z be Point of [:E,F:], a be Real
      holds diff(f, a * z) = a * diff(f,z) )
  & ( for z1,z2 be Point of [:E,F:]
      holds diff(f, z1 - z2) = diff(f,z1) - diff(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 [:E,F:]
      holds diff(f, z1 + z2).x = diff(f,z1).x + diff(f,z2).x
      proof
        let x be VECTOR of [:E,F:];
        consider dx be Point of E, dy be Point of F such that
        A2: x = [dx,dy] by PRVECT_3:18;
        A3: diff(f, z1 + z2).x
         = diff(f, z1 + z2).(dx,dy) by A2
        .= partdiff`1(f, z1 + z2).dx + partdiff`2(f, z1 + z2).dy by Th12
        .= (partdiff`1(f, z1) + partdiff`1(f, z2)).dx
          + partdiff`2(f, z1 + z2).dy by Th7
        .= (partdiff`1(f, z1) + partdiff`1(f, z2)).dx
          + (partdiff`2(f, z1) + partdiff`2(f, z2)).dy by Th8
        .= partdiff`1(f, z1).dx + partdiff`1(f, z2).dx
          + (partdiff`2(f, z1) + partdiff`2(f, z2)).dy by LOPBAN_1:35
        .= partdiff`1(f, z1).dx + partdiff`1(f, z2).dx
          + (partdiff`2(f, z1).dy + partdiff`2(f, z2).dy) by LOPBAN_1:35
        .= partdiff`1(f,z1).dx + partdiff`1(f,z2).dx
          + partdiff`2(f,z1).dy + partdiff`2(f,z2).dy by RLVECT_1:def 3
        .= partdiff`1(f,z1).dx + partdiff`2(f,z1).dy
          + partdiff`1(f,z2).dx + partdiff`2(f,z2).dy by RLVECT_1:def 3
        .= partdiff`1(f,z1).dx + partdiff`2(f,z1).dy
          + (partdiff`1(f,z2).dx + partdiff`2(f,z2).dy) by RLVECT_1:def 3;

        A4: partdiff`1(f,z1).dx + partdiff`2(f,z1).dy
         = diff(f,z1).(dx,dy) by Th12
        .= diff(f,z1).x by A2;

        partdiff`1(f,z2).dx + partdiff`2(f,z2).dy
         = diff(f,z2).(dx,dy) by Th12
        .= diff(f,z2).x by A2;
        hence thesis by A3,A4;
      end;
      hence diff(f, z1 + z2) = diff(f, z1) + diff (f, z2) by LOPBAN_1:35;
    end;

    A5:
    now
      let z1 be Point of [:E,F:], a be Real;

      for x be VECTOR of [:E,F:]
      holds diff(f, a * z1).x = a * diff(f, z1).x
      proof
        let x be VECTOR of [:E,F:];
        consider dx be Point of E, dy be Point of F such that
        A6: x = [dx,dy] by PRVECT_3:18;
        A7: partdiff`1(f,z1).dx + partdiff`2(f,z1).dy
         = diff(f,z1).(dx,dy) by Th12
        .= diff(f,z1).x by A6;

        thus diff(f, a * z1).x
         = diff(f, a * z1).(dx,dy) by A6
        .= partdiff`1(f, a * z1).dx + partdiff`2(f, a * z1).dy by Th12
        .= (a * partdiff`1(f, z1)).dx + partdiff`2(f, a * z1).dy by Th7
        .= (a * partdiff`1(f, z1)).dx + (a * partdiff`2(f, z1)).dy by Th8
        .= a * partdiff`1(f,z1).dx + (a * partdiff`2(f,z1)).dy by LOPBAN_1:36
        .= a * partdiff`1(f,z1).dx + a * partdiff`2(f,z1).dy by LOPBAN_1:36
        .= a * (diff(f, z1).x) by A7,RLVECT_1:def 5;
      end;
      hence diff(f, a * z1) = a * diff(f, z1) by LOPBAN_1:36;
    end;
    now
      let z1,z2 be Point of [:E,F:];
      thus diff(f, z1 - z2)
       = diff(f, z1 + (-1)*z2) by RLVECT_1:16
      .= diff(f, z1) + diff(f, (-1) * z2) by A1
      .= diff(f, z1) + (-1) * diff(f, z2) by A5
      .= diff(f, z1) - diff(f, z2) by RLVECT_1:16;
    end;
    hence thesis by A1,A5;
  end;
