 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem Th50:
  Image (proj (F, i)) = the multMagma of F.i
proof
  for g being object st g in the carrier of F.i
  holds g in the carrier of Image (proj (F, i))
  proof
    let g be object;
    assume g in the carrier of F.i;
    then reconsider x=g as Element of F.i;
    ex h being Element of product F st x = (proj (F, i)).h
    proof
      dom (1_(product F)) = I by GROUP_19:3;
      then B1: ((1_(product F)) +* (i,x)).i = x by FUNCT_7:31;
      (1_(product F)) +* (i,x) in ProjSet (F, i) by GROUP_12:def 1;
      then reconsider h = ((1_(product F)) +* (i,x)) as Element of product F;
      take h;
      (proj (F, i)).((1_(product F)) +* (i,x))
       = (proj (F, i)).h
      .= x by B1, Def13;
      hence thesis;
    end;
    then x in Image (proj (F, i)) by GROUP_6:45;
    hence g in the carrier of Image (proj (F, i));
  end;
  hence thesis by GROUP_2:61, TARSKI:def 3;
end;
