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 Th24:
  a,b '||' a,c implies a,c '||' a,b & b,a '||' a,c & a,b '||' c,a
& a,c '||' b,a & b,a '||' c,a & c,a '||' a,b & c,a '||' b,a & b,a '||' b,c & a,
  b '||' b,c & b,a '||' c,b & b,c '||' b,a & a,b '||' c,b & c,b '||' b,a & b,c
'||' a,b & c,b '||' a,b & c,a '||' c,b & a,c '||' c,b & c,a '||' b,c & a,c '||'
  b,c & c,b '||' c,a & b,c '||' c,a & c,b '||' a,c & b,c '||' a,c
proof
  assume
A1: a,b '||' a,c;
  then a,c '||' a,b by Th19;
  then
A2: c,a '||' c,b by Def11;
  b,a '||' b,c by A1,Def11;
  hence thesis by A1,A2,Th23;
end;
