reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th19:
  for A,B being set, R being Subset of [:A,B:] st
  X in dom(.:R) holds (.:R).X = R.:X
proof
  let A,B be set;
  let R be Subset of [:A,B:];
  assume X in dom (.:R);
  then X in bool A by Def1;
  hence thesis by Def1;
end;
