
theorem Th33: :: CardCentral:
  for R being finite Skew-Field, s being Element of R
  holds card the carrier of (centralizer s) =
  (card (the carrier of center R)) |^ (dim VectSp_over_center s)
proof
  let R be finite Skew-Field, s be Element of R;
  set vR = VectSp_over_center s;
A1: the addLoopStr of vR = the addLoopStr of centralizer s by Def7;
  set B = the Basis of vR;
  B is finite by A1;
  then vR is finite-dimensional by MATRLIN:def 1;
  hence thesis by A1,Th15;
end;
