
theorem Th24: :: Central02a:
  for R be Skew-Field, s, a be Element of R
  holds a in the carrier of centralizer s iff a*s = s*a
proof
  let R be Skew-Field, s, a be Element of R;
  set cs = centralizer s;
  set ccs = the carrier of cs;
A1: ccs = {x where x is Element of R: x*s = s*x} by Def5;
  hereby
    assume a in the carrier of centralizer s;
    then ex x being Element of R st a=x & x*s=s*x by A1;
    hence a*s = s*a;
  end;
  assume a*s = s*a;
  hence thesis by A1;
end;
