theorem
  not a,b '||' a,c & a,b '||' p,q & a,c '||' p,r & a,c '||' p,s & b,c
  '||' q,r & b,c '||' q,s implies r=s
proof
  assume that
A1: not a,b '||' a,c and
A2: a,b '||' p,q and
A3: a,c '||' p,r and
A4: a,c '||' p,s and
A5: b,c '||' q,r and
A6: b,c '||' q,s;
A7: now
    b<>c by A1,Th18;
    then
A8: q,r '||' q,s by A5,A6,Def11;
    a<>c by A1,Def11;
    then
A9: p,r '||' p,s by A3,A4,Def11;
    assume p<>q;
    then not p,q '||' p,r by A1,A2,A3,A5,Th31;
    hence thesis by A9,A8,Th33;
  end;
  p=q implies p=r & p=s by A1,A3,A4,A5,A6,Th32;
  hence thesis by A7;
end;
