reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;
reserve A,B for Ordinal;
reserve n,k for Nat;

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;
