reserve k,n,i,j for Nat;

theorem Th1:
  for f being Permutation of Seg 2 holds f=<*1,2*> or f=<*2,1*>
proof
  let f be Permutation of Seg 2;
A1: rng f = Seg 2 by FUNCT_2:def 3;
  2 in {1,2} by TARSKI:def 2;
  then
A2: f.2 in Seg 2 by A1,FINSEQ_1:2,FUNCT_2:4;
A3: dom f=Seg 2 by FUNCT_2:52;
  then reconsider p=f as FinSequence by FINSEQ_1:def 2;
A4: len p= 2 by A3,FINSEQ_1:def 3;
A5: 1 in dom f & 2 in dom f by A3;
  1 in {1,2} by TARSKI:def 2;
  then
A6: f.1 in Seg 2 by A1,FINSEQ_1:2,FUNCT_2:4;
  now
    per cases by A6,A2,FINSEQ_1:2,TARSKI:def 2;
    case
      f.1=1 & f.2=1;
      hence contradiction by A5,FUNCT_1:def 4;
    end;
    case
      f.1=1 & f.2=2;
      hence thesis by A4,FINSEQ_1:44;
    end;
    case
      f.1=2 & f.2=1;
      hence thesis by A4,FINSEQ_1:44;
    end;
    case
      f.1=2 & f.2=2;
      hence contradiction by A5,FUNCT_1:def 4;
    end;
  end;
  hence thesis;
end;
