
theorem Th34: :: DimCentral:
  for R being finite Skew-Field, s being Element of R
  holds 0 < dim VectSp_over_center s
proof
  let R be finite Skew-Field, s be Element of R;
  set q = card the carrier of center R;
  set ns= dim VectSp_over_center s;
  now
    assume
A1: ns = 0;
    q |^ ns = q #Z ns by PREPOWER:36;
    then q |^ ns = 1 by A1,PREPOWER:34;
    then card the carrier of centralizer s = 1 by Th33;
    hence contradiction by Th28;
  end;
  hence thesis;
end;
