
theorem Th22: :: Center4:
  for R being Skew-Field
  holds the carrier of center R = (the carrier of center MultGroup R) \/ {0.R}
proof
  let R being Skew-Field;
A1: the carrier of center MultGroup R c= the carrier of MultGroup R
  by GROUP_2:def 5;
A2: the carrier of MultGroup R = NonZero R by UNIROOTS:def 1;
A3: (the carrier of MultGroup R)\/ {0.R} = the carrier of R by UNIROOTS:15;
A4: the carrier of center R c= the carrier of R by Th16;
A5: (the carrier of center MultGroup R) \/ {0.R} c= the carrier of center R
  proof
    let x be object;
    assume
A6: x in (the carrier of center MultGroup R) \/ {0.R};
    per cases by A6,XBOOLE_0:def 3;
    suppose
A7:   x in the carrier of center MultGroup R;
      then reconsider a = x as Element of MultGroup R by A1;
A8:   a in center MultGroup R by A7;
      reconsider a1 = a as Element of R by UNIROOTS:19;
      now
        let b be Element of R;
        thus a1*b = b*a1
        proof
          per cases by A3,XBOOLE_0:def 3;
          suppose b in (the carrier of MultGroup R);
            then reconsider b1 = b as Element of MultGroup R;
            thus a1*b = a*b1 by UNIROOTS:16
              .= b1*a by A8,GROUP_5:77
              .= b*a1 by UNIROOTS:16;
          end;
          suppose b in {0.R};
            then
A9:         b = 0.R by TARSKI:def 1;
            hence a1*b = 0.R
              .= b*a1 by A9;
          end;
        end;
      end;
      then a1 in center R by Th17;
      hence thesis;
    end;
    suppose x in {0.R};
      then x = 0.R by TARSKI:def 1;
      then x in center R by Th18;
      hence thesis;
    end;
  end;
  the carrier of center R c= (the carrier of center MultGroup R) \/ {0.R}
  proof
    let x be object;
    assume
A10: x in the carrier of center R;
    then reconsider a = x as Element of center R;
    reconsider a1 = a as Element of R by A4;
    per cases;
    suppose a1 = 0.R;
      then a1 in {0.R} by TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose a1 <> 0.R;
      then not a1 in {0.R} by TARSKI:def 1;
      then reconsider a2 = a1 as Element of MultGroup R by A2,XBOOLE_0:def 5;
      now
        let b be Element of MultGroup R;
        b in the carrier of MultGroup R;
        then reconsider b1 = b as Element of R by A2;
A11:    x in center R by A10;
        thus a2*b=a1*b1 by UNIROOTS:16
          .= b1*a1 by A11,Th17
          .= b*a2 by UNIROOTS:16;
      end;
      then a1 in center MultGroup R by GROUP_5:77;
      then a1 in the carrier of center MultGroup R;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  hence thesis by A5,XBOOLE_0:def 10;
end;
