reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem
  Image 1:(G,H) = (1).H
proof
  set g = 1:(G,H);
A1: the carrier of Image g c= {1_H}
  proof
    let x be object;
    assume x in the carrier of Image g;
    then x in g .: (the carrier of G) by Def10;
    then consider y being object such that
    y in dom g and
A2: y in the carrier of G and
A3: g.y = x by FUNCT_1:def 6;
    reconsider y as Element of G by A2;
    g.y = 1_H;
    hence thesis by A3,TARSKI:def 1;
  end;
  1_H in Image g by GROUP_2:46;
  then 1_H in the carrier of Image g;
  then {1_H} c= the carrier of Image g by ZFMISC_1:31;
  then the carrier of Image g = {1_H} by A1;
  hence thesis by GROUP_2:def 7;
end;
