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 Th9:
  for a being natural Number holds Seg a \/ { a+1 } = Seg (a+1)
proof
  let a be natural Number;
  thus Seg a \/ { a+1 } c= Seg (a+1)
  proof
    a+0<=a+1 by XREAL_1:7;
    then
A1: Seg a c= Seg(a+1) by Th5;
    let x be object;
    assume x in Seg a \/ { a+1 };
    then x in Seg a or x in { a+1 } by XBOOLE_0:def 3;
    then x in Seg (a+1) or x = a+1 by A1,TARSKI:def 1;
    hence thesis by Th3;
  end;
  let x be object;
  assume
A2: x in Seg (a+1);
  then reconsider x as Element of NAT;
A3: x<=a+1 by A2,Th1;
A4: 1<=x by A2,Th1;
  x<=a or x=a+1 by A3,NAT_1:8;
  then x in Seg a or x in {a+1} by A4,TARSKI:def 1;
  hence thesis by XBOOLE_0:def 3;
end;
