
theorem
  for F being Field holds for a,b being Element of F
holds for c,d being Element of NonZero F holds omf(F).(omf(F).(a,revf(F).c),omf
  (F).(b,revf(F).d)) = omf(F).(omf(F).(a,b),revf(F).(omf(F).(c,d)))
proof
  let F be Field;
  let a,b be Element of F;
  let c,d be Element of NonZero F;
  reconsider revfc = revf(F).c, revfd = revf(F).d as Element of NonZero F;
  omf(F).(c,d) is Element of NonZero F by REALSET2:24;
  then reconsider revfcd = revf(F).(c*d) as Element of F by REALSET2:24;
  thus omf(F).(omf(F).(a,revf(F).c),omf(F).(b,revf(F).d)) = a*revfc*(b*revfd)
    .= a*(revfc*(b*revfd)) by REALSET2:19
    .= a*(b*(revfc*revfd)) by REALSET2:19
    .= omf(F).(a,omf(F).(b,revf(F).(omf(F).(c,d)))) by Th3
    .= a*(b*revfcd)
    .= a*b*revfcd by REALSET2:19
    .= omf(F).(omf(F).(a,b),revf(F).(omf(F).(c,d)));
end;
