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;
reserve p,q,r,s,t for FinSequence;
reserve D for set;
reserve a, b, c, d, e, f for object;

theorem
  Seg n = (n+1) \ {0}
proof
A1: n+1 = {m: m < n+1} by AXIOMS:4;
  thus Seg n c= (n+1) \ {0}
  proof
    let x be object;
    assume x in Seg n;
    then consider k being Nat such that
A2: x = k and
A3: 1 <= k and
A4: k <= n;
    k < n+1 by A4,NAT_1:13;
    then
A5: x in n+1 by A1,A2;
    not x in {0} by A2,A3,TARSKI:def 1;
    hence thesis by A5,XBOOLE_0:def 5;
  end;
  let x be object;
  assume
A6: x in (n+1) \ {0};
  then
A7: x in n+1;
A8: not x in {0} by A6,XBOOLE_0:def 5;
  consider m such that
A9: x = m and
A10: m < n+1 by A1,A7;
A11: x <> 0 by A8,TARSKI:def 1;
  0+1 = 1;
  then
A12: 1 <= m by A9,A11,NAT_1:13;
  m <= n by A10,NAT_1:13;
  hence thesis by A9,A12;
end;
