
theorem
  for F being Field holds for a,b being Element of F
holds for c,d being Element of NonZero F holds (the addF of F).(omf(F).(a,revf(
F).c),omf(F).(b,revf(F).d)) = omf(F).((the addF of F).(omf(F).(a,d),omf(F).(b,c
  )),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 revfd = revf(F).d as Element of F by XBOOLE_0:def 5;
A1: a = a*1.F by REALSET2:21
    .= omf(F).(a,1.F)
    .= a*(d*revfd) by REALSET2:def 6
    .= a*d*revfd by REALSET2:19;
  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;
  b = b*1.F by REALSET2:21
    .= omf(F).(b,1.F)
    .= b*(c*revfc) by REALSET2:def 6
    .= b*c*revfc by REALSET2:19;
  then
A2: omf(F).(b,revf(F).d) = b*c*revfc*revfd .= b*c*(revfc*revfd) by REALSET2:19
    .= omf(F).(omf(F).(b,c),revf(F).(omf(F).(c,d))) by Th3;
  omf(F).(a,revf(F).c) = a*d*revfd*revfc by A1
    .= a*d*(revfd*revfc) by REALSET2:19
    .= omf(F).(omf(F).(a,d),revfc*revfd)
    .= omf(F).(omf(F).(a,d),revf(F).(omf(F).(c,d))) by Th3;
  hence (the addF of F).(omf(F).(a,revf(F).c),omf(F).(b,revf(F).d)) = a*d*
  revfcd+b*c*revfcd by A2
    .= (a*d+b*c)*revfcd by VECTSP_1:def 7
    .= omf(F).((the addF of F). (omf(F).(a,d),omf(F).(b,c)),revf(F).(omf(F).
  (c,d)));
end;
