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 Th15:
  ProperPrefixes <*x*> = { {} }
proof
  thus ProperPrefixes <*x*> c= D
  proof
    let y be object;
    assume y in ProperPrefixes <*x*>;
    then consider p being FinSequence such that
A1: y = p and
A2: p is_a_proper_prefix_of <*x*> by Def2;
A3: len p < len <*x*> by A2,Th5;
 len <*x*> = 1 & 1 = 0+1 by FINSEQ_1:39;
then  p = {} by A3,NAT_1:13;
    hence thesis by A1,TARSKI:def 1;
  end;
  let y be object;
  assume y in D;
then A4: y = {} by TARSKI:def 1;
 {} is_a_prefix_of <*x*>;
then  {} is_a_proper_prefix_of <*x*>;
  hence thesis by A4,Def2;
end;
