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 Th97:
  for S being with_common_domain functional set, f be Function
     st f in S  holds dom f = dom product" S
  proof let S be with_common_domain functional set, f be Function such that
A1: f in S;
   thus dom f = DOM S by A1,Lm2
     .= dom product" S by Def12;
  end;
