 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem :: TH31
  Image (incl2 (G, A, phi)) /\ Image (incl1 (G, A, phi))
  = (1).(semidirect_product(G,A,phi))
proof
  set ImA = Image (incl2 (G, A, phi)), ImG = Image (incl1 (G, A, phi));
  set S = semidirect_product(G,A,phi);
  for x being object st x in the carrier of (ImA /\ ImG)
  holds x in {1_S}
  proof
    let x be object;
    assume x in the carrier of (ImA /\ ImG);
    then x in ImA /\ ImG;
    then B1: x in ImA & x in ImG by GROUP_2:82;
    then consider g being Element of G such that
    B2: x = (incl1 (G, A, phi)).g by GROUP_6:45;
    consider a being Element of A such that
    B3: x = (incl2 (G, A, phi)).a by B1, GROUP_6:45;
    <*1_G, a*> = x by B3,Def3
              .= <*g,1_A*> by B2,Def2;
    then a = 1_A & 1_G = g by FINSEQ_1:77;
    then x = <*1_G,1_A*> by B2,Def2
          .= 1_S by Th17;
    hence x in {1_S} by TARSKI:def 1;
  end;
  then the carrier of (ImA /\ ImG) c= {1_S};
  then the carrier of (ImA /\ ImG) c= the carrier of (1).S by GROUP_2:def 7;
  then A1: ImA /\ ImG is Subgroup of (1).S by GROUP_2:57;
  (1).S is Subgroup of ImA /\ ImG by GROUP_2:65;
  hence thesis by A1, GROUP_2:55;
end;
