reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem
  for T1,T2 being Tree st T1 c= T2 holds ^T1 c= ^T2
proof
  let T1,T2 be Tree such that
A1: T1 c= T2;
  let x be object;
  assume x in ^T1;
  then reconsider p = x as Element of ^T1;
  p = {} or ex q st q in T1 & p = <*0*>^q by Th60;
  hence thesis by A1,Th60;
end;
