
theorem :: PRE_COMP_8:
  for s being State of SCM, tl, tr being bin-term holds
  (tl+tr)@s = (tl@s)+(tr@s) & (tl-tr)@s = (tl@s)-(tr@s) &
  (tl*tr)@s = (tl@s)*(tr@s) & (tl div tr)@s = (tl@s) div (tr@s) &
  (tl mod tr)@s = (tl@s) mod (tr@s)
proof
  let s be State of SCM, tl, tr be bin-term;
  thus (tl+tr)@s = nt0-Meaning_on (tl@s, tr@s) by Th5
    .= tl@s + tr@s by Def8;
  thus (tl-tr)@s = nt1-Meaning_on (tl@s, tr@s) by Th5
    .= tl@s - tr@s by Def8;
  thus (tl * tr)@s = nt2-Meaning_on (tl@s, tr@s) by Th5
    .= tl@s * tr@s by Def8;
  thus (tl div tr)@s = nt3-Meaning_on (tl@s, tr@s) by Th5
    .= tl@s div tr@s by Def8;
  thus (tl mod tr)@s = nt4-Meaning_on (tl@s, tr@s) by Th5
    .= tl@s mod tr@s by Def8;
end;
