
theorem Th37: :: MGC1:
  for R being finite Skew-Field holds
  card the carrier of center MultGroup R = card (the carrier of center R) - 1
proof
  let R be finite Skew-Field;
A1: the carrier of MultGroup R = NonZero R by UNIROOTS:def 1;
  the carrier of center MultGroup R c= the carrier of MultGroup R
  by GROUP_2:def 5;
  then
A2: not 0.R in the carrier of center MultGroup R by A1,ZFMISC_1:56;
  the carrier of center R = (the carrier of center MultGroup R) \/ {0.R}
  by Th22;
  then card the carrier of center R = card (the carrier of center MultGroup
  R) + 1 by A2,CARD_2:41;
  hence thesis;
end;
