reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem Th26:
  for G,H being strict Group, h being Homomorphism of G,H holds
  for A being strict normal Subgroup of G holds
  h.:A is strict normal Subgroup of Image h
proof
  let G,H be strict Group;
  let h be Homomorphism of G,H;
  let A be strict normal Subgroup of G;
  reconsider C=h.:A as strict Subgroup of Image h by GRSOLV_1:9;
  for b being Element of Image h holds b* C c= C * b
  proof
    let b be Element of Image h;
A1: b in Image h by STRUCT_0:def 5;
    now
      consider b1 being Element of G such that
A2:   b = h.b1 by A1,GROUP_6:45;
      let x be object;
      assume x in b * C;
      then consider g being Element of Image h such that
A3:   x = b * g and
A4:   g in C by GROUP_2:103;
      consider g1 being Element of A such that
A5:   g=(h|A).g1 by A4,GROUP_6:45;
      reconsider g1 as Element of G by GROUP_2:42;
      g=h.g1 by A5,FUNCT_1:49;
      then
A6:   x =h.b1 * h.g1 by A2,A3,GROUP_2:43
        .=h.(b1 *g1) by GROUP_6:def 6;
      g1 in A by STRUCT_0:def 5; then
A7:   b1 * g1 in b1 * A by GROUP_2:103;
A8:   h.:(A * b1) = h.:A * h.b1 by GRSOLV_1:13;
      b1 * A = A * b1 by GROUP_3:117;
      then x in h.:A * h.b1 by A6,A7,A8,FUNCT_2:35;
      hence x in C * b by A2,GROUP_6:2;
    end;
    hence thesis;
  end;
  hence thesis by GROUP_3:118;
end;
