
theorem Th30: :: Central2:
  for R being finite Skew-Field, s being Element of R,
  t being Element of MultGroup R st t = s
  holds card Centralizer t = card (the carrier of centralizer s) - 1
proof
  let R be finite Skew-Field, s be Element of R,
  t be Element of MultGroup R such that
A1: t = s;
  set ct = Centralizer t, cs = centralizer s;
  set cct = the carrier of ct, ccs = the carrier of cs;
  the carrier of MultGroup R = NonZero R by UNIROOTS:def 1;
  then not 0.R in the carrier of MultGroup R by ZFMISC_1:56;
  then not 0.R in MultGroup R;
  then not 0.R in ct by Th7;
  then
A2: not 0.R in cct;
  cct \/ {0.R} = ccs by A1,Th29;
  then card ccs = card cct +1 by A2,CARD_2:41;
  hence thesis;
end;
