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
  for l being FinSequence of Group_of_Perm n st (len l) mod 2 = 0 & (for
i being Element of NAT st i in dom l ex q being Element of Permutations n st l.
i = q & q is being_transposition) holds Product l is even Permutation of Seg n
proof
  let l be FinSequence of Group_of_Perm n;
  Product l in the carrier of Group_of_Perm n;
  then Product l in Permutations n by MATRIX_1:def 13;
  then reconsider Pf = Product l as Permutation of Seg n by MATRIX_1:def 12;
  assume
A1: (len l) mod 2 = 0 & for i being Element of NAT st i in dom l ex q
  being Element of Permutations n st l.i = q & q is being_transposition;
  ex l be FinSequence of the carrier of Group_of_Perm n st (len l) mod 2 =
  0 & Pf = Product l & for i st i in dom l ex q being Element of Permutations n
  st l.i=q & q is being_transposition
  proof
    consider l be FinSequence of the carrier of Group_of_Perm n such that
A2: (len l) mod 2 = 0 & Pf = Product l and
A3: for i be Element of NAT st i in dom l ex q being Element of
    Permutations n st l.i=q & q is being_transposition by A1;
    take l;
    thus (len l) mod 2 = 0 & Pf = Product l by A2;
    let i;
    thus thesis by A3;
  end;
  hence thesis by MATRIX_1:def 15;
end;
