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 Th47: :: (14.2)
  for Y being set holds (R~).:Y = (R~).:(Y /\ proj2 R)
proof
  let Y be set;
  thus (R~).:Y c= (R~).:(Y /\ proj2 R)
  proof
    let y be object;
    assume y in (R~).:Y;
    then consider x being object such that
A1: [x,y] in R~ and
A2: x in Y by RELAT_1:def 13;
    [y,x] in R by A1,RELAT_1:def 7;
    then x in proj2 R by XTUPLE_0:def 13;
    then x in Y /\ proj2 R by A2,XBOOLE_0:def 4;
    hence thesis by A1,RELAT_1:def 13;
  end;
  let y be object;
  assume y in (R~).:(Y /\ proj2 R);
  then consider x being object such that
A3: [x,y] in R~ and
A4: x in (Y /\ proj2 R) by RELAT_1:def 13;
  x in Y by A4,XBOOLE_0:def 4;
  hence thesis by A3,RELAT_1:def 13;
end;
