reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th30:
  for p being Permutation of Seg 3 st p = <* 1,3,2 *> holds p is
  being_transposition
proof
  let p be Permutation of Seg 3;
  assume
A1: p = <*1,3,2*>;
  then
A2: dom p = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  ex i,j st i in dom p & j in dom p & i <> j & p.i = j & p.j = i & for k
  st k <> i & k <> j & k in dom p holds p.k = k
  proof
    take i = 2, j = 3;
    thus i in dom p & j in dom p by A2,ENUMSET1:def 1;
    for k st k <> i & k <> j & k in dom p holds p.k = k
    proof
      let k;
      assume k <> i & k <> j & k in dom p;
      then k = 1 by A2,ENUMSET1:def 1;
      hence thesis by A1;
    end;
    hence thesis by A1;
  end;
  hence thesis by MATRIX_1:def 14;
end;
