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;
