reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);

theorem Th1:
  X in 2Set Seg n iff ex i,j st i in Seg n & j in Seg n & i < j & X = {i,j}
proof
  thus X in 2Set Seg n implies ex i,j st i in Seg n & j in Seg n & i<j & X={i,
  j}
  proof
    assume X in 2Set Seg n;
    then consider x,y being object such that
A1: x in Seg n and
A2: y in Seg n and
A3: x<>y and
A4: X={x,y} by SGRAPH1:8;
    reconsider x,y as Element of NAT by A1,A2;
    x>y or y>x by A3,XXREAL_0:1;
    hence thesis by A1,A2,A4;
  end;
  assume ex i,j st i in Seg n & j in Seg n & i<j & X={i,j};
  then consider i,j such that
A5: i in Seg n and
A6: j in Seg n and
A7: i<j and
A8: X={i,j};
  {i,j} c= Seg n by A5,A6,ZFMISC_1:32;
  hence thesis by A5,A6,A7,A8,SGRAPH1:8;
end;
