theorem
  (id R)|R is increasing
proof
  now
    let r1,r2;
    assume that
A1: r1 in R /\ dom (id R) and
A2: r2 in R /\ dom (id R) and
A3: r1<r2;
    r1 in R by A1,XBOOLE_0:def 4;
    then
A4: (id R).r1 = r1 by FUNCT_1:18;
    r2 in R by A2,XBOOLE_0:def 4;
    hence (id R).r1 < (id R).r2 by A3,A4,FUNCT_1:18;
  end;
  hence thesis by Th20;
end;
