reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem :: (17)
  X <> {} implies R.:^X c= R.:X
proof
  assume
A1: X <> {};
  let y be object;
  assume
A2: y in R.:^X;
  consider x being object such that
A3: x in X by A1,XBOOLE_0:def 1;
  y in Im(R,x) by A2,A3,Th24;
  then [x,y] in R by Th9;
  hence thesis by A3,RELAT_1:def 13;
end;
