reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem
  for R be RelStr st R is asymmetric holds the InternalRel of R misses (
  the InternalRel of R)~
proof
  let R be RelStr;
  assume R is asymmetric;
  then the InternalRel of R is asymmetric;
  then
A1: the InternalRel of R is_asymmetric_in field (the InternalRel of R);
  assume the InternalRel of R meets (the InternalRel of R)~;
  then consider x being object such that
A2: x in the InternalRel of R and
A3: x in (the InternalRel of R)~ by XBOOLE_0:3;
  consider y,z being object such that
A4: x = [y,z] by A3,RELAT_1:def 1;
A5: z in field the InternalRel of R by A2,A4,RELAT_1:15;
  [z,y] in the InternalRel of R & y in field the InternalRel of R by A2,A3,A4,
RELAT_1:15,def 7;
  hence contradiction by A2,A4,A1,A5;
end;
