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 Th37:
  for f being Function of the carrier of G1,
    the carrier of product <*G1*> st
  for x being Element of G1 holds f.x = <*x*> holds
    f is Homomorphism of G1, product <*G1*>
proof
  let f be Function of the carrier of G1, the carrier of product <*G1*> such
  that
A1: for x being Element of G1 holds f.x = <*x*>;
  now
    let a, b be Element of G1;
    thus f.(a * b) = <*a * b*> by A1
      .= <*a*> * <*b*> by Th28
      .= <*a*> * f.b by A1
      .= f.a * f.b by A1;
  end;
  hence thesis by GROUP_6:def 6;
end;
