
theorem CMI:
  for a,b be non zero Integer, c be Integer holds c mod (a*b) mod a = c mod a
  proof
    let a,b be non zero Integer;
    let c be Integer;
    A1: c mod (a*b) = (a*b)*frac (c/(a*b)) by R3; then
    reconsider t = (a*b)*frac (c/(a*b)) as Integer;
    (c mod (a*b)) mod a = a* frac (a*(b*frac (c/(a*b)))/a) by A1,R3
    .= a*frac (b*frac (c/(a*b))) by XCMPLX_1:89
    .= a*frac (b*(c/(a*b))) by FR3
    .= a*frac ((b*c)/(a*b)) by XCMPLX_1:74
    .= a*(frac (c/a)) by XCMPLX_1:91;
    hence thesis by R3;
  end;
