reserve F for Field,
  a,b,c,d,e,f,g,h for Element of F;
reserve x,y for Element of [:the carrier of F,the carrier of F,the carrier of
  F:];
reserve F for Field;
reserve PS for non empty ParStr;
reserve x for set,
  a,b,c,d,e,f,g,h,i,j,k,l for Element of [:the carrier of F,
  the carrier of F,the carrier of F:];
reserve a,b,c,d,p,q,r,s for Element of MPS(F);
reserve PS for ParSp,
  a,b,c,d,p,q,r,s for Element of PS;

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;
