reserve A, B, X, Y for set;

theorem
  for R being reflexive non empty RelStr holds id the carrier of R c= (
  the InternalRel of R) /\ the InternalRel of (R~)
proof
  let R be reflexive non empty RelStr;
  let a be object;
  assume
A1: a in id the carrier of R;
  then consider y, z being object such that
A2: a = [y,z] by RELAT_1:def 1;
  reconsider y, z as Element of R by A1,A2,ZFMISC_1:87;
  y <= z by A1,A2,RELAT_1:def 10;
  then
A3: a in the InternalRel of R by A2,ORDERS_2:def 5;
  y = z by A1,A2,RELAT_1:def 10;
  then a in the InternalRel of R~ by A2,A3,RELAT_1:def 7;
  hence thesis by A3,XBOOLE_0:def 4;
end;
