theorem Th17:
  R is transitive implies R |_2 Y is transitive
proof
  assume
A1: R is transitive;
  now
    let a,b,c;
    assume that
A2: [a,b] in R |_2 Y and
A3: [b,c] in R |_2 Y;
    [a,b] in R & [b,c] in R by A2,A3,XBOOLE_0:def 4;
    then
A4: [a,c] in R by A1,Lm2;
    [b,c] in [:Y,Y:] by A3,XBOOLE_0:def 4;
    then
A5: c in Y by ZFMISC_1:87;
    [a,b] in [:Y,Y:] by A2,XBOOLE_0:def 4;
    then a in Y by ZFMISC_1:87;
    then [a,c] in [:Y,Y:] by A5,ZFMISC_1:87;
    hence [a,c] in R |_2 Y by A4,XBOOLE_0:def 4;
  end;
  hence thesis by Lm2;
end;
