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
  h1 * 1:(G,H) = 1:(G,I) & 1:(H,I) * h = 1:(G,I)
proof
  thus h1 * 1:(G,H) = 1:(G,I)
  proof
    let a be Element of G;
    thus (h1 * 1:(G,H)).a = h1.(1:(G,H).a) by FUNCT_2:15
    .= 1_I by Th31
    .= 1:(G,I).a;
  end;
  let a;
  thus (1:(H,I) * h).a = 1:(H,I).(h.a) by FUNCT_2:15
  .= 1:(G,I).a;
end;
