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 Th31:
  not a,b '||' a,c & a,b '||' p,q & a,c '||' p,r & b,c '||' q,r &
  p<>q implies not p,q '||' p,r
proof
  assume that
A1: not a,b '||' a,c and
A2: a,b '||' p,q and
A3: a,c '||' p,r and
A4: b,c '||' q,r and
A5: p<>q;
  now
    assume p=r;
    then
A6: p,q '||' b,c by A4,Th23;
    p,q '||' b,a by A2,Th23;
    then b,a '||' b,c by A5,A6,Def11;
    hence contradiction by A1,Th24;
  end;
  hence thesis by A1,A2,A3,A5,Th30;
end;
