reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th12:
  for R be RelStr holds the InternalRel of R misses the
  InternalRel of ComplRelStr R
proof
  let R be RelStr;
  assume not thesis;
  then (the InternalRel of R) /\ the InternalRel of ComplRelStr R <> {};
  then consider a being object such that
A1: a in (the InternalRel of R) /\ the InternalRel of ComplRelStr R by
XBOOLE_0:def 1;
  a in the InternalRel of ComplRelStr R by A1,XBOOLE_0:def 4;
  then a in (the InternalRel of R)` \ id (the carrier of R) by NECKLACE:def 8;
  then a in (the InternalRel of R)` by XBOOLE_0:def 5;
  then a in [:the carrier of R,the carrier of R:] \ (the InternalRel of R) by
SUBSET_1:def 4;
  then not a in the InternalRel of R by XBOOLE_0:def 5;
  hence thesis by A1,XBOOLE_0:def 4;
end;
