
theorem
  for F being Field holds for a,b,c being Element of F
  holds osf(F).(a,osf(F).(b,c)) = (the addF of F).(osf(F).(a,b),c)
proof
  let F be Field;
  let a,b,c be Element of F;
  thus osf(F).(a,osf(F).(b,c)) = osf(F).(a,(the addF of F).(b,(comp F).c)) by
Def1
    .= a+(comp F).(b+(comp F).c) by Def1
    .= a+(comp F).(b+-c) by VECTSP_1:def 13
    .= a+-(b+-c) by VECTSP_1:def 13
    .= a+(-b+--c) by RLVECT_1:31
    .= a+((comp F).b+--c) by VECTSP_1:def 13
    .= a+((comp F).b+(comp F).(-c)) by VECTSP_1:def 13
    .= a+((comp F).b+(comp F).((comp F).c)) by VECTSP_1:def 13
    .= a+((comp F).b+c) by REALSET2:9
    .= a+((comp F).b)+c by RLVECT_1:def 3
    .= (the addF of F).(osf(F).(a,b),c) by Def1;
end;
