reserve k,n,i,j for Nat;

theorem Th3:
  Permutations(2)={<*1,2*>,<*2,1*>}
proof
  now
    let x be object;
    assume
A1: x in {idseq 2,<*2,1*>};
    now
      per cases by A1,TARSKI:def 2;
      case
        x=idseq 2;
        hence x in Permutations(2) by MATRIX_1:def 12;
      end;
      case
A2:     x=<*2,1*>;
        <*2,1*> is Permutation of Seg 2 by Th2;
        hence x in Permutations(2) by A2,MATRIX_1:def 12;
      end;
    end;
    hence x in Permutations(2);
  end;
  then
A3: {idseq 2,<*2,1*>} c=Permutations(2);
  now
    let p be object;
    assume p in Permutations(2);
    then reconsider q=p as Permutation of Seg 2 by MATRIX_1:def 12;
    q=<*1,2*> or q=<*2,1*> by Th1;
    hence p in {idseq 2,<*2,1*>} by FINSEQ_2:52,TARSKI:def 2;
  end;
  then Permutations(2) c={idseq 2,<*2,1*>};
  hence thesis by A3,FINSEQ_2:52;
end;
