
theorem
  for F being Field holds for a being Element of F holds
for b,c being Element of NonZero F holds ovf(F).(a,omf(F).(b,c)) = ovf(F).(ovf(
  F).(a,b),c)
proof
  let F be Field;
  let a be Element of F;
  let b,c be Element of NonZero F;
  reconsider revfb = revf(F).b, revfc = revf(F).c as Element of F by
XBOOLE_0:def 5;
  omf(F).(b,c) is Element of NonZero F by REALSET2:24;
  hence ovf(F).(a,omf(F).(b,c)) = omf(F).(a,revf(F).(omf(F).(b,c))) by Def2
    .= a*(revfb*revfc) by Th3
    .= a*revfb*revfc by REALSET2:19
    .= omf(F).(ovf(F).(a,b),revf(F).c) by Def2
    .= ovf(F).(ovf(F).(a,b),c) by Def2;
end;
