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:];
reserve R for Relation of A,B;
reserve S for Relation of B,C;

theorem :: (15.2)
  Y c= proj2 R iff Y c= ((R~)*R).:Y
proof
  thus Y c= proj2 R implies Y c= ((R~)*R).:Y
  proof
    assume
A1: Y c= proj2 R;
    let y be object;
    assume
A2: y in Y;
    then consider x being object such that
A3: [x,y] in R by A1,XTUPLE_0:def 13;
A4: [y,x] in R~ by A3,RELAT_1:def 7;
A5: y in Im(R,x) by A3,Th9;
    R.:{x} c= ((R~)*R).:Y
    proof
      let a be object;
      assume a in R.:{x};
      then ex b being object st ( [b,a] in R)&( b in {x}) by RELAT_1:def 13;
      then [x,a] in R by TARSKI:def 1;
      then [y,a] in ((R~)*R) by A4,RELAT_1:def 8;
      hence thesis by A2,RELAT_1:def 13;
    end;
    hence thesis by A5;
  end;
  assume
A6: Y c= ((R~)*R).:Y;
  let x be object;
  assume x in Y;
  then
A7: x in ((R~)*R).:Y by A6;
A8: ((R~)*R).:Y = R.:((R~).:Y) by RELAT_1:126;
  R.:((R~).:Y) c= R.:A by RELAT_1:123;
  then ((R~)*R).:Y c= proj2 R by A8,Th50;
  hence thesis by A7;
end;
