theorem
  for S being with_common_domain functional set, f be Function, i be set
     st f in S & i in dom f holds f.i in (product" S).i
 proof let S be with_common_domain functional set, f be Function, i be set;
  assume that
A1: f in S and
A2: i in dom f;
   dom f = dom(product" S) by A1,Th97;
  hence f.i in (product" S).i by A1,A2,Th98;
 end;
