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 Th24:
  for G,H being strict Group, h being Homomorphism of G,H
  for A being strict Subgroup of G for a,b being Element of G
  for H1 being Subgroup of Image h for a1,b1 being Element of Image h
  st a1 = h.a & b1 = h.b & H1 = h.:A holds a1 * b1 * H1 = h.a * h.b * h.:A
proof
  let G,H be strict Group;
  let h being Homomorphism of G,H;
  let A be strict Subgroup of G;
  let a,b be Element of G;
  let H1 be Subgroup of Image h;
  let a1,b1 be Element of Image h;
  assume that
A1: a1 = h.a and
A2: b1 = h.b and
A3: H1 = h.:A;
A4: a1 * b1 = h.a * h.b by A1,A2,GROUP_2:43;
  set c1 = a1 * b1;
  set c = a * b;
A5: h.c = h.a * h.b by GROUP_6:def 6;
A6: h.:(c * A) = h.c * h.:A by GRSOLV_1:13;
  c1 * H1 = h.c * h.:A
  proof
    now
      let x be object;
      assume x in c1 * H1;
      then consider Z being Element of Image h such that
A7:   x = c1 * Z and
A8:   Z in H1 by GROUP_2:103;
      consider Z1 being Element of A such that
A9:   Z = (h|A).Z1 by A3,A8,GROUP_6:45;
A10:  Z1 in A by STRUCT_0:def 5;
      reconsider Z1 as Element of G by GROUP_2:42;
      Z = h.Z1 by A9,FUNCT_1:49;
      then
A11:  x = h.c * h.Z1 by A4,A5,A7,GROUP_2:43
       .= h.(c * Z1) by GROUP_6:def 6;
      c * Z1 in c * A by A10,GROUP_2:103;
      hence x in h.c * h.:A by A11,A6,FUNCT_2:35;
    end;
    then
A12: c1 * H1 c= h.c * h.:A;
    now
      let x be object;
      assume x in h.c * h.:A;
      then consider y being object such that
A13: y in the carrier of G and
A14: y in c * A and
A15: x = h.y by A6,FUNCT_2:64;
    reconsider y as Element of G by A13;
    consider Z being Element of G such that
A16: y= c * Z and
A17: Z in A by A14,GROUP_2:103;
    Z in the carrier of A by A17,STRUCT_0:def 5;
    then h.Z in h.:(the carrier of A) by FUNCT_2:35;
    then h.Z in the carrier of h.: A by GRSOLV_1:8;
    then
A18: h.Z in H1 by A3,STRUCT_0:def 5;
    then h.Z in Image h by GROUP_2:40;
    then reconsider Z1 = h.Z as Element of Image h by STRUCT_0:def 5;
 x = h.c * h.Z by A15,A16,GROUP_6:def 6;
    then x = c1 * Z1 by A4,A5,GROUP_2:43;
    hence x in c1 * H1 by A18,GROUP_2:103;
    end;
    then h.c * h.:A c= c1 * H1;
    hence thesis by A12,XBOOLE_0:def 10;
  end;
  hence thesis by GROUP_6:def 6;
end;
