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 Th2:
  Seg 1 = { 1 } & Seg 2 = { 1,2 }
proof
  now
    let x be object;
    thus x in Seg 1 implies x in { 1 }
    proof
      assume x in Seg 1;
      then consider k being Nat such that
A1:   x = k and
A2:   1 <= k & k <= 1;
      k = 1 by A2,XXREAL_0:1;
      hence thesis by A1,TARSKI:def 1;
    end;
    assume x in { 1 };
    then x = 1 by TARSKI:def 1;
    hence x in Seg 1;
  end;
  hence Seg 1 = { 1 } by TARSKI:2;
  now
    let x be object;
    thus x in Seg 2 implies x in { 1,2 }
    proof
      assume x in Seg 2;
      then consider k being Nat such that
A3:   x = k and
A4:   1 <= k and
A5:   k <= 2;
      k <= 1 + 1 by A5;
      then k = 1 or k = 2 by A4,NAT_1:9;
      hence thesis by A3,TARSKI:def 2;
    end;
    assume x in { 1,2 };
    then x = 1 or x = 2 by TARSKI:def 2;
    hence x in Seg 2;
  end;
  hence thesis by TARSKI:2;
end;
