reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;

theorem
  for k being natural Number holds Seg k = Seg(k + 1) \ {k + 1}
proof
  let k be natural Number;
A1: Seg(k + 1) = Seg k \/ {k + 1} by Th9;
  Seg k misses {k + 1}
  proof
    assume
A2: not thesis;
    set x = the Element of Seg k /\ {k + 1};
A3: x in Seg k by A2,XBOOLE_0:def 4;
    x in {k + 1} by A2,XBOOLE_0:def 4;
    then x = k + 1 by TARSKI:def 1;
    hence thesis by A3,Th1,XREAL_1:29;
  end;
  hence thesis by A1,XBOOLE_1:88;
end;
