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 x being object, X being non empty functional set
  st for f being Function st f in X holds x in dom f
  holds x in DOM X
 proof let x be object, X be non empty functional set such that
A1: for f being Function st f in X holds x in dom f;
  set A = the set of all  dom f where f is Element of X;
   consider Y being object such that
A2:  Y in X by XBOOLE_0:def 1;
    reconsider Y as Function by A2;
A3:  dom Y in A by A2;
   for Y holds Y in A implies x in Y
    proof let Y;
     assume Y in A;
      then ex f being Element of X st Y = dom f;
     hence x in Y by A1;
   end;
  hence x in DOM X by A3,SETFAM_1:def 1;
 end;
