
theorem Th35: :: Skew1:
  for R being finite Skew-Field, r being Element of R
  st r is Element of MultGroup R holds
  ((card (the carrier of center R)) |^ (dim VectSp_over_center r) - 1) divides
  ((card (the carrier of center R)) |^ (dim VectSp_over_center R) - 1)
proof
  let R be finite Skew-Field, r be Element of R such that
A1: r is Element of MultGroup R;
  set G = MultGroup R;
  set q = card (the carrier of center R);
  set qr= card (the carrier of centralizer r);
  set n = dim VectSp_over_center R;
  reconsider s=r as Element of MultGroup R by A1;
  card R = q |^ n by Th31;
  then card G = (q |^ n) - 1 by UNIROOTS:18;
  then q |^ n - 1 = card con_class s * card Centralizer s by Th13;
  then card Centralizer s divides (q |^ n - 1) by INT_1:def 3;
  then (qr - 1) divides (q |^ n -1) by Th30;
  hence thesis by Th33;
end;
