theorem Th2:
  i1 is Element of (n+k)
proof
  i1 is Element of Segm(n);
  then reconsider I=i1 as Nat;
  per cases;
  suppose n =0 & k=0;
    hence thesis;
  end;
  suppose A1: n =0 & k>0;
    then i1 is empty by SUBSET_1:def 1;
    then I in Segm(n+k) by A1,NAT_1:44;
    hence thesis;
  end;
  suppose n<>0;
    then I in Segm n;
    then I < n <= n+k by NAT_1:11,44;
    then I < n+k by  XXREAL_0:2;
    then I in Segm (n+k) by NAT_1:44;
    hence thesis;
  end;
end;
