
theorem Th36: :: Skew2:
  for R being finite Skew-Field, s being Element of R
  st s is Element of MultGroup R
  holds (dim VectSp_over_center s) divides (dim VectSp_over_center R)
proof
  let R be finite Skew-Field, s be Element of R such that
A1: s is Element of MultGroup R;
  set n = dim VectSp_over_center R;
  set ns= dim VectSp_over_center s;
  set q = card the carrier of center R;
A2: n in NAT by ORDINAL1:def 12;
A3: ns in NAT by ORDINAL1:def 12;
A4: 0 < ns by Th34;
A5: 1 < q by Th20;
  0 < q |^ ns by PREPOWER:6;
  then 0+1 < q |^ ns + 1 by XREAL_1:8;
  then
A6: 1 <= q |^ ns by NAT_1:13;
  0 < q |^ n by PREPOWER:6;
  then 0+1 < q |^ n + 1 by XREAL_1:8;
  then 1 <= q |^ n by NAT_1:13;
  then
A7: (q |^ n - 1) = (q |^ n -' 1) by XREAL_1:233;
  (q |^ ns - 1) = (q |^ ns -' 1) by A6,XREAL_1:233;
  hence thesis by A1,A2,A3,A4,A5,A7,Th3,Th35;
end;
