 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
  [i, f.i] in f
proof
  i in I & f is ManySortedSet of I;
  then i in dom f by PARTFUN1:def 2;
  hence [i, f.i] in f by FUNCT_1:def 2;
end;
