reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th72:
  for G1,G2,G3 being Group
  for f1 being Homomorphism of G1,G2
  for f2 being Homomorphism of G2,G3
  for A being Subgroup of G1
  holds the multMagma of f2 .: (f1 .: A) = the multMagma of ((f2 * f1) .: A)
proof
  let G1,G2,G3 be Group;
  let f1 be Homomorphism of G1,G2;
  let f2 be Homomorphism of G2,G3;
  let A be Subgroup of G1;
  for z being Element of G3
  holds z in f2 .: (f1 .: A) iff z in (f2 * f1) .: A
  proof
    let z be Element of G3;
    thus z in f2 .: (f1 .: A) implies z in (f2 * f1) .: A
    proof
      assume z in f2 .: (f1 .: A);
      then z in f2 .: (the carrier of f1 .: A) by GRSOLV_1:8;
      then consider y being object such that
      A2: y in dom f2 and
      A3: y in the carrier of (f1 .: A) and
      A4: z = f2.y by FUNCT_1:def 6;
      y in f1 .: (the carrier of A) by A3,GRSOLV_1:8;
      then consider x being object such that
      A5: x in dom f1 & x in the carrier of A & y = f1.x by FUNCT_1:def 6;
      A6: x in dom(f2 * f1) by A2,A5,FUNCT_1:11;
      then x in the carrier of A & z = (f2 * f1).x by A4,A5,FUNCT_1:12;
      then z in (f2 * f1) .: the carrier of A by A6,FUNCT_1:def 6;
      hence thesis by GRSOLV_1:8;
    end;

    thus z in (f2 * f1) .: A implies z in f2 .: (f1 .: A)
    proof
      assume z in (f2 * f1) .: A;
      then z in (f2 * f1) .: the carrier of A by GRSOLV_1:8;
      then consider x being object such that
      A2: x in dom (f2 * f1) & x in the carrier of A & z = (f2 * f1).x
      by FUNCT_1:def 6;
      A3: x in dom f1 & f1.x in dom f2 by A2,FUNCT_1:11;
      set y = f1.x;
      x in dom f1 & x in the carrier of A & y = f1.x by A2,FUNCT_1:11;
      then A5: y in f1 .: (the carrier of A) by FUNCT_1:def 6;
      z = (f2 * f1).x by A2
       .= f2.(f1.x) by A2,FUNCT_1:12
       .= f2.y;
      then z in f2 .: (f1 .: (the carrier of A)) by A3,A5,FUNCT_1:def 6;
      then z in f2 .: (the carrier of (f1 .: A)) by GRSOLV_1:8;
      hence z in f2 .: (f1 .: A) by GRSOLV_1:8;
    end;
  end;
  hence the multMagma of f2 .: (f1 .: A) = the multMagma of ((f2 * f1) .: A)
  by GROUP_2:60;
end;
