reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;

theorem Th17:
  for p,q,r being FinSequence st
  q in ProperPrefixes p & r in ProperPrefixes p holds q,r are_c=-comparable
proof
  let p,q,r be FinSequence;
  assume q in ProperPrefixes p;
then
A1: ex q1 being FinSequence st q = q1 & q1 is_a_proper_prefix_of p by Def2;
  assume r in ProperPrefixes p;
then
A2: ex r1 being FinSequence st r = r1 & r1 is_a_proper_prefix_of p by Def2;
 q is_a_prefix_of p by A1;
  then consider n such that
A3: q = p|Seg n;
 r is_a_prefix_of p by A2;
  then consider k such that
A4: r = p|Seg k;
A5: n <= k implies thesis
  proof
    assume n <= k;
then  Seg n c= Seg k by FINSEQ_1:5;
then  q = r|Seg n by A3,A4,FUNCT_1:51;
then  q is_a_prefix_of r;
    hence thesis;
  end;
 k <= n implies thesis
  proof
    assume k <= n;
then  Seg k c= Seg n by FINSEQ_1:5;
then  r = q|Seg k by A3,A4,FUNCT_1:51;
then  r is_a_prefix_of q;
    hence thesis;
  end;
  hence thesis by A5;
end;
