reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;

theorem Th4:
  ProperPrefixes (v^<*x*>) = ProperPrefixes v \/ {v}
proof
  thus ProperPrefixes (v^<*x*>) c= ProperPrefixes v \/ {v}
  proof
    let y be object;
    assume y in ProperPrefixes (v^<*x*>);
    then consider v1 such that
A1: y = v1 and
A2: v1 is_a_proper_prefix_of v^<*x*> by TREES_1:def 2;
 v1 is_a_prefix_of v & v1 <> v or v1 = v by A2,TREES_1:9;
then
 v1 is_a_proper_prefix_of v or v1 in {v} by TARSKI:def 1;
then  y in ProperPrefixes v or y in {v} by A1,TREES_1:def 2;
    hence thesis by XBOOLE_0:def 3;
  end;
  let y be object;
  assume y in ProperPrefixes v \/ {v};
then A3: y in ProperPrefixes v or y in {v} by XBOOLE_0:def 3;
A4: now
    assume y in ProperPrefixes v;
    then consider v1 such that
A5: y = v1 and
A6: v1 is_a_proper_prefix_of v by TREES_1:def 2;
 v is_a_prefix_of v^<*x*> by TREES_1:1;
then  v1 is_a_proper_prefix_of v^<*x*> by A6,XBOOLE_1:58;
    hence thesis by A5,TREES_1:def 2;
  end;
 v^{} = v by FINSEQ_1:34;
  then
 v is_a_prefix_of v^<*x*> & v <> v^<*x*> by FINSEQ_1:33,TREES_1:1;
then  v is_a_proper_prefix_of v^<*x*>;
then  y in ProperPrefixes v or y = v & v in ProperPrefixes (v^<*x*>)
  by A3,TARSKI:def 1,TREES_1:def 2;
  hence thesis by A4;
end;
