reserve i,j,x,y for object,
  f,g for Function;
reserve T,T1 for finite Tree,
  t,p for Element of T,
  t1 for Element of T1;

theorem Th23:
  for D being finite set, k being Nat st card D = k+1
  ex x being Element of D,C being Subset of D st D = C \/ { x } & card C = k
proof
  let D be finite set,k be Nat;
  assume
A1: card D = k+1;
  then D <> {};
  then consider x being object such that
A2: x in D by XBOOLE_0:def 1;
  reconsider C=D \ { x } as Subset of D;
  reconsider x as Element of D by A2;
  take x,C;
  x in {x} by TARSKI:def 1;
  then
A3: not x in C by XBOOLE_0:def 5;
  {x} c= D by A2,ZFMISC_1:31;
  hence D=C \/ {x} by XBOOLE_1:45;
  then card D = card C + 1 by A3,CARD_2:41;
  hence thesis by A1;
end;
