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

theorem Th1:
  Segm n \/ { n } = Segm(n+1)
proof
   n in Segm(n+1) by NAT_1:45;
   then
A1:{n} c= Segm(n+1) by ZFMISC_1:31;
   Segm n c= Segm(n+1) by NAT_1:39,11;
  hence Segm n \/ { n } c= Segm(n+1) by A1,XBOOLE_1:8;
  let x be object;
  assume
A2: x in Segm(n+1);
    then reconsider x as Nat;
  now
   x < n+1 by A2,NAT_1:44;
   then per cases by NAT_1:22;
   case x < n;
    hence x in Segm n by NAT_1:44;
   end;
   case x = n;
    hence x in {n} by TARSKI:def 1;
   end;
  end;
 hence thesis by XBOOLE_0:def 3;
end;
