 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 Th31:
  proj (Carrier F, i) is Function of product (Carrier F), the carrier of F.i
proof
  set X = product (Carrier F);
  set Y = the carrier of F.i;
  set f = proj (Carrier F, i);
  A1: dom f = X by CARD_3:def 16;
  for x being object st x in X holds f.x in Y
  proof
    let x be object;
    assume A2: x in X;
    then reconsider y=x as Element of product F by GROUP_7:def 2;
    f.y = y/.i by A1, A2, CARD_3:def 16;
    hence f.x in Y;
  end;
  hence f is Function of X,Y by A1, FUNCT_2:3;
end;
