 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem
  proj (F, i) = proj (Carrier F, i)
proof
  set X = product (Carrier F);
  set Y = the carrier of F.i;
  product (Carrier F) = the carrier of (product F) by GROUP_7:def 2;
  then A1: proj (Carrier F, i) is Function of X,Y
           & proj (F, i) is Function of X,Y by Th31;
  for x being Element of X
  holds (proj (F, i)) . x = (proj (Carrier F, i)) . x
  proof
    let x be Element of X;
    x is Element of product F by GROUP_7:def 2;
    hence thesis by Th32;
  end;
  hence thesis by A1, FUNCT_2:63;
end;
