reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  K for non empty doubleLoopStr,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D,
  F for add-associative right_zeroed
  right_complementable Abelian non empty doubleLoopStr;
reserve A,B for Matrix of n,K;
reserve A,A9,B,B9,C for Matrix of n,F;
reserve i,j,n for Nat,
  K for Field,
  a,b for Element of K;
reserve x,y,x1,x2,y1,y2 for set,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Field,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of K;
reserve p,q for Element of Permutations(n);

theorem
  id Seg n is even
proof
  set l=<*>the carrier of Group_of_Perm n;
  0=2 * 0 + 0;
  then
A1: (len l) mod 2=0 by NAT_D:def 2;
  Product <*> the carrier of Group_of_Perm(n)=1_Group_of_Perm(n) by GROUP_4:8;
  then
A2: idseq n=Product l by Th15;
  for i st i in dom l ex q st l.i=q & q is being_transposition;
  hence thesis by A1,A2;
end;
