
theorem Th29: :: Central2a:
  for R being Skew-Field, s being Element of R, t being Element of MultGroup R
  st t = s
  holds the carrier of centralizer s = (the carrier of Centralizer t) \/ {0.R}
proof
  let R be Skew-Field, s be Element of R,
  t be Element of MultGroup R such that
A1: t = s;
  set ct = Centralizer t, cs = centralizer s;
  set cct = the carrier of ct, ccs = the carrier of cs;
A2: the carrier of MultGroup R = NonZero R by UNIROOTS:def 1;
A3: cct = { b where b is Element of MultGroup R : t*b = b*t } by Def1;
A4: ccs = {x where x is Element of R: x*s = s*x} by Def5;
  now
    let x be object;
    hereby
      assume x in ccs;
      then consider c being Element of R such that
A5:   c = x and
A6:   c*s=s*c by A4;
      per cases;
      suppose c = 0.R;
        then c in {0.R} by TARSKI:def 1;
        hence x in cct \/ {0.R} by A5,XBOOLE_0:def 3;
      end;
      suppose c <> 0.R;
        then not c in {0.R} by TARSKI:def 1;
        then reconsider c1 = c as Element of MultGroup R by A2,XBOOLE_0:def 5;
        t*c1 = s*c by A1,UNIROOTS:16
          .= c1*t by A1,A6,UNIROOTS:16;
        then c in cct by A3;
        hence x in cct \/ {0.R} by A5,XBOOLE_0:def 3;
      end;
    end;
    assume
A7: x in cct \/ {0.R};
    per cases by A7,XBOOLE_0:def 3;
    suppose x in cct;
      then consider b being Element of MultGroup R such that
A8:   x = b and
A9:   t*b = b*t by A3;
      reconsider b1 = b as Element of R by UNIROOTS:19;
      b1*s = t*b by A1,A9,UNIROOTS:16
        .= s*b1 by A1,UNIROOTS:16;
      hence x in ccs by A4,A8;
    end;
    suppose x in {0.R};
      then
A10:  x = 0.R by TARSKI:def 1;
      0.R*s = s*0.R;
      hence x in ccs by A4,A10;
    end;
  end;
  hence thesis by TARSKI:2;
end;
