
theorem Th5:
  for F being Field holds for a,c being Element of F
holds for b,d being Element of NonZero F holds omf(F).(a,revf(F).b) = omf(F).(c
  ,revf(F).d) iff omf(F).(a,d) = omf(F).(b,c)
proof
  let F be Field;
  let a,c be Element of F;
  let b,d be Element of NonZero F;
A1: omf(F).(a,revf(F).b) = omf(F).(c,revf(F).d) implies omf(F).(a,d) = omf(F
  ).(b,c)
  proof
    reconsider revfb = revf(F).b, revfd = revf(F).d as Element of NonZero F;
    reconsider crevfd = c*revfd as Element of F;
    assume
A2: omf(F).(a,revf(F).b) = omf(F).(c,revf(F).d);
A3: c = c*1.F by REALSET2:21
      .= omf(F).(c,d*revfd) by REALSET2:def 6
      .= c*(revfd*d)
      .= c*revfd*d by REALSET2:19;
    a = a*1.F by REALSET2:21
      .= omf(F).(a,b*revfb) by REALSET2:def 6
      .= a*(revfb*b)
      .= a*revfb*b by REALSET2:19
      .= b*crevfd by A2
      .= omf(F).(b,omf(F).(c,revf(F).d));
    hence omf(F).(a,d) = b*(c*revfd)*d .= b*(c*revfd*d) by REALSET2:19
      .= omf(F).(b,c) by A3;
  end;
  omf(F).(a,d) = omf(F).(b,c) implies omf(F).(a,revf(F).b) = omf(F).(c,
  revf(F).d)
  proof
    reconsider revfd = revf(F).d, revfb = revf(F).b as Element of NonZero F;
    reconsider crevfd = omf(F).(c,revfd) as Element of F;
    assume
A4: omf(F).(a,d) = omf(F).(b,c);
    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
      .= b*c*revfd by A4
      .= b*(c*revfd) by REALSET2:19
      .= crevfd*b;
    hence omf(F).(a,revf(F).b) = crevfd*b*revfb
      .= crevfd*(b*revfb) by REALSET2:19
      .= omf(F).(c,revfd)*1.F by REALSET2:def 6
      .= omf(F).(c,revf(F).d) by REALSET2:21;
  end;
  hence thesis by A1;
end;
