
theorem Th21: :: Center3:
  for R being finite Skew-Field holds
  card the carrier of center R = card the carrier of R iff R is commutative
proof
  let R be finite Skew-Field;
  set X = the carrier of R;
  set Y = the carrier of center R;
  hereby
    assume
A1: card X = card Y;
A2: Y c= X by Th16;
    card (X \ Y) = card X - card X by A1,Th16,CARD_2:44;
    then X \ Y = {};
    then X c= Y by XBOOLE_1:37;
    then
A3: X = Y by A2,XBOOLE_0:def 10;
    for x being Element of X holds for s being Element of X holds x*s=s*x
    by A3,STRUCT_0:def 5,Th17;
    hence R is commutative;
  end;
  now
    assume
A4: R is commutative;
    for x being object st x in X holds x in Y
    proof
      let x be object such that
A5:   x in X;
      for x being Element of X holds x is Element of Y
      proof
        let x be Element of X;
        for y being Element of X holds x*y = y*x by A4;
        then x in center R by Th17;
        hence thesis;
      end;
      then x is Element of Y by A5;
      hence thesis;
    end;
    then
A6: X c= Y;
    Y c= X by Th16;
    hence card Y = card X by A6,XBOOLE_0:def 10;
  end;
  hence thesis;
end;
