reserve i, I for set,
  f, g, h for Function,
  s for ManySortedSet of I;
reserve G1, G2, G3 for non empty multMagma,
  x1, x2 for Element of G1,
  y1, y2 for Element of G2,
  z1, z2 for Element of G3;
reserve G1, G2, G3 for Group-like non empty multMagma;
reserve G1, G2, G3 for Group,
  x for Element of G1,
  y for Element of G2,
  z for Element of G3;

theorem
  (<*x*> qua Element of product <*G1*>)" = <*x"*>
proof
  reconsider G = <*G1*> as associative Group-like multMagma-Family of {1};
  reconsider lF = <*x*>, p = <*x"*> as Element of product Carrier G by Def2;
 for i being set st i in {1} ex H being Group, z being Element of H st H
  = G.i & p.i = z" & z = lF.i
  proof
    reconsider H = G.1 as Group;
    reconsider z = p.1 as Element of H;
    let i be set;
    assume
A3: i in {1};
    take H, z";
    thus H = G.i by A3,TARSKI:def 1;
    thus p.i = z"" by A3,TARSKI:def 1;
    i = 1 by A3,TARSKI:def 1;
    hence thesis;
  end;
  hence thesis by Th7;
end;
