
theorem Th8:
  for p, q being FinSequence holds not p is_a_prefix_of q iff len
  maxPrefix(p,q) < len p
proof
  let p, q be FinSequence;
A1: maxPrefix(p,q) c= p by Def1;
  hereby
    assume
A2: not p c= q;
A3: now
      assume len maxPrefix(p,q) = len p;
      then
A4:   dom maxPrefix(p,q) = dom p by FINSEQ_3:29;
      maxPrefix(p,q) c= p by Def1;
      then maxPrefix(p,q) = p by A4,GRFUNC_1:3;
      hence contradiction by A2,Def1;
    end;
    maxPrefix(p,q) c= p by Def1;
    then len maxPrefix(p,q) <= len p by FINSEQ_1:63;
    hence len maxPrefix(p,q) < len p by A3,XXREAL_0:1;
  end;
  assume that
A5: len maxPrefix(p,q) < len p and
A6: p c= q;
  p c= maxPrefix(p,q) by A6,Def1;
  hence contradiction by A5,A1,XBOOLE_0:def 10;
end;
