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 Th46: :: (14.1)
  for X being set holds R.:X = R.:(X /\ proj1 R)
proof
  let X be set;
  thus R.:X c= R.:(X /\ proj1 R)
  proof
    let y be object;
    assume y in R.:X;
    then consider x being object such that
A1: [x,y] in R and
A2: x in X by RELAT_1:def 13;
    x in proj1 R by A1,XTUPLE_0:def 12;
    then x in X /\ proj1 R by A2,XBOOLE_0:def 4;
    hence thesis by A1,RELAT_1:def 13;
  end;
  let y be object;
  assume y in R.:(X /\ proj1 R);
  then consider x being object such that
A3: [x,y] in R and
A4: x in X /\ proj1 R by RELAT_1:def 13;
  x in X by A4,XBOOLE_0:def 4;
  hence thesis by A3,RELAT_1:def 13;
end;
