reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;
reserve s9,w9,v9 for Element of NAT*;

theorem Th16:
  for Z being finite Tree,o being Element of Z st o <> Root Z
  holds card (Z|o) < card Z
proof
  let Z be finite Tree,o be Element of Z such that
A1: o <> Root Z;
  set A = { o^s9 : o^s9 in Z };
A2: Z|o,A are_equipotent by A1,Th15;
  then reconsider A as finite set by CARD_1:38;
  reconsider B = A \/ {Root Z} as finite set;
  now
    let x be object such that
A3: x in B;
    now
      per cases by A3,XBOOLE_0:def 3;
      suppose
        x in { o^s9 : o^s9 in Z };
        then ex v9 st x = o^v9 & o^v9 in Z;
        hence x in Z;
      end;
      suppose
        x in {Root Z};
        hence x in Z;
      end;
    end;
    hence x in Z;
  end;
  then
A4: B c= Z;
  card B = card A + 1 by A1,Th15,CARD_2:41
    .= card (Z|o) + 1 by A2,CARD_1:5;
  then card (Z|o) + 1 <= card Z by A4,NAT_1:43;
  hence thesis by NAT_1:13;
end;
