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,y,z*> qua Element of product <*G1,G2,G3*>)" = <*x",y",z" *>
proof
  set G = <*G1,G2,G3*>;
  reconsider lF = <*x,y,z*>, p = <*x",y",z"*> as Element of product Carrier G
  by Def2;
 for i being set st i in {1,2,3} ex H being Group, z being Element of H
  st H = G.i & p.i = z" & z = lF.i
  proof
    let i be set such that
A7: i in {1,2,3};
    per cases by A7,ENUMSET1:def 1;
    suppose
A8:   i = 1;
      reconsider H = G.1 as Group;
      reconsider z = p.1 as Element of H;
      take H, z";
      thus H = G.i by A8;
      thus p.i = z"" by A8;
      thus thesis by A8;
    end;
    suppose
A9:  i = 2;
      reconsider H = G.2 as Group;
      reconsider z = p.2 as Element of H;
      take H, z";
      thus H = G.i by A9;
      thus p.i = z"" by A9;
      thus thesis by A9;
    end;
    suppose
A10:  i = 3;
      reconsider H = G.3 as Group;
      reconsider z = p.3 as Element of H;
      take H, z";
      thus H = G.i by A10;
      thus p.i = z"" by A10;
      thus thesis by A10;
    end;
  end;
  hence thesis by Th7;
end;
