reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  R.:X = {} iff dom R misses X
proof
  set y = the Element of R.:X;
  thus R.:X = {} implies dom R misses X
  proof
    assume
A1: R.:X = {};
    assume not thesis;
    then consider x be object such that
A2: x in dom R and
A3: x in X by XBOOLE_0:3;
    ex y being object st [x,y] in R by A2,XTUPLE_0:def 12;
    hence contradiction by A1,A2,A3,Th104;
  end;
  assume
A4: dom R /\ X ={};
  assume not thesis;
  then ex x st x in dom R & [x,y] in R & x in X by Th104;
  hence contradiction by A4,XBOOLE_0:def 4;
end;
