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
  for H being Group st G,H are_isomorphic & G is commutative
  holds H is commutative
proof
  let H be Group;
  assume that
A1: G,H are_isomorphic and
A2: G is commutative;
  consider h being Homomorphism of G,H such that
A3: h is bijective by A1;
  now
    let c,d be Element of H;
    consider a such that
A4: h.a = c by A3,Th58;
    consider b such that
A5: h.b = d by A3,Th58;
    thus c * d = h.(a * b) by A4,A5,Def6
      .= h.(b * a) by A2
      .= d * c by A4,A5,Def6;
  end;
  hence thesis;
end;
