theorem Th23:
  a,b '||' c,d implies b,a '||' c,d & a,b '||' d,c & b,a '||' d,c
  & c,d '||' a,b & d,c '||' a,b & c,d '||' b,a & d,c '||' b,a
proof
  assume a,b '||' c,d;
  then c,d '||' a,b by Th19;
  then
A1: d,c '||' a,b by Th21;
  then
A2: d,c '||' b,a by Th22;
  then c,d '||' b,a by Th21;
  hence thesis by A1,A2,Th19,Th21;
end;
