
theorem Th26: :: Central02:
  for R be Skew-Field, s, a, b be Element of R
  st a in the carrier of center R & b in the carrier of centralizer s
  holds a*b in the carrier of centralizer s
proof
  let R be Skew-Field, s, a, b be Element of R such that
A1: a in the carrier of center R and
A2: b in the carrier of centralizer s;
  set cs = centralizer s;
  set ccs = the carrier of cs;
A3: ccs = {x where x is Element of R: x*s = s*x} by Def5;
A4: a in center R by A1;
  a*b*s=a*(b*s) by GROUP_1:def 3
    .= (b*s)*a by A4,Th17
    .= (s*b)*a by A2,Th24
    .= s*(b*a) by GROUP_1:def 3
    .= s*(a*b) by A4,Th17;
  hence thesis by A3;
end;
