reserve k,n,i,j for Nat;

theorem Th4:
  for p being Permutation of Seg 2 st p is being_transposition
  holds p = <*2,1*>
proof
  let p be Permutation of Seg 2;
  assume
A1: p is being_transposition;
  now
    set p0=<*1,2*>;
    assume
A2: p=<*1,2*>;
    consider i,j being Nat such that
A3: i in dom p and
A4: j in dom p & i<>j and
A5: p.i=j and
    p.j=i and
    for k being Nat st k <>i & k<>j & k in dom p holds p.k=k by A1;
    len p0=2 by FINSEQ_1:44;
    then
A6: dom p ={1,2} by A2,FINSEQ_1:2,def 3;
    then
A7: i=1 or i=2 by A3,TARSKI:def 2;
    now
      per cases by A4,A6,A7,TARSKI:def 2;
      case
        i=1 & j=2;
        hence contradiction by A2,A5;
      end;
      case
        i=2 & j=1;
        hence contradiction by A2,A5;
      end;
    end;
    hence contradiction;
  end;
  hence thesis by Th1;
end;
