
theorem Th5:
  for R being RelStr, X being lower Subset of R, Y being Subset of
  R, x being set st Y = X \/ {x} & (the InternalRel of R)-Seg x c= X holds Y is
  lower
proof
  let R be RelStr, X be lower Subset of R, Y be Subset of R, x be set;
  set r = the InternalRel of R;
  assume that
A1: Y = X \/ {x} and
A2: r-Seg x c= X;
  let z, y be object;
  assume that
A3: z in Y and
A4: [y, z] in r;
  per cases by A1,A3,XBOOLE_0:def 3;
  suppose
    z in X;
    then y in X by A4,Def1;
    hence thesis by A1,XBOOLE_0:def 3;
  end;
  suppose
    z in {x} & y = z;
    hence thesis by A3;
  end;
  suppose
A5: z in {x} & y <> z;
    then z = x by TARSKI:def 1;
    then y in r-Seg x by A4,A5,WELLORD1:1;
    hence thesis by A1,A2,XBOOLE_0:def 3;
  end;
end;
