reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  A misses X implies P|A = {}
proof
  assume A misses X;
  then A misses dom P by XBOOLE_1:63;
  hence thesis by RELAT_1:152;
end;
