theorem Th20:
  for tr be Element of Permutations(k) st
  tr is being_transposition holds tr*tr = idseq k & tr = tr"
proof
  set I=idseq k;
  let tr be Element of Permutations(k);
  assume tr is being_transposition;
  then consider i,j such that
  i in dom tr and
  j in dom tr and
  i<>j and
A1: tr.i=j and
A2: tr.j=i and
A3: for m st m <>i & m<>j & m in dom tr holds tr.m=m;
  reconsider TR=tr as Permutation of Seg k by MATRIX_1:def 12;
  set TT=TR*TR;
A4: dom TT=Seg k by FUNCT_2:52;
A5: dom TR=Seg k by FUNCT_2:52;
A6: for x being object st x in dom TT holds TT.x = I.x
  proof
    let x being object such that
A7: x in dom TT;
    reconsider m=x as Nat by A4,A7;
    now
      per cases;
      suppose
        m=i or m=j;
        hence TT.m=m by A1,A2,A7,FUNCT_1:12;
      end;
      suppose
        m<>i & m<>j;
        then tr.m=m by A3,A4,A5,A7;
        hence TT.m=m by A7,FUNCT_1:12;
      end;
    end;
    hence thesis by A4,A7,FUNCT_1:18;
  end;
A8: dom I=Seg k;
  hence tr*tr = idseq k by A6,FUNCT_1:2,FUNCT_2:52;
  rng TR=Seg k by FUNCT_2:def 3;
  hence thesis by A4,A8,A5,A6,FUNCT_1:2,42;
end;
