reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;

theorem
  n+1 = {0} \/ Seg n
proof
  thus n+1 c= {0} \/ Seg n
  proof
   let x be object;
    assume x in n+1;
    then x in {j where j is Nat: j<n+1} by AXIOMS:4;
    then consider j being Nat such that
A1: j=x and
A2: j<n+1;
    j=0 or 1<j+1 & j<=n by A2,NAT_1:13,XREAL_1:29;
    then j=0 or 1<=j & j<=n by NAT_1:13;
    then x in {0} or x in Seg n by A1,FINSEQ_1:1,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
A3: Segm 1 c= Segm(n+1) by NAT_1:39,11;
  Seg(n) c= Segm(n+1) by Th2;
  hence thesis by A3,CARD_1:49,XBOOLE_1:8;
end;
