theorem
  for f be linear-transformation of V1,V2, 
      g be linear-transformation of V2,V3 st 
    g|im f is one-to-one holds 
  rank (g*f) = rank f & nullity (g*f) = nullity f
proof
  let f be linear-transformation of V1,V2, g be linear-transformation of V2,V3
  such that
A1: g|im f is one-to-one;
  the carrier of im (g*f) = [#]im (g*f) .= (g*f).:[#]V1 by RANKNULL:def 2
    .= ((g|im f)*f).:([#]V1) by Lm8
    .= (g|im f).:(f.:[#]V1) by RELAT_1:126
    .= (g|im f).:([#]im f) by RANKNULL:def 2
    .= [#]im (g|im f) by RANKNULL:def 2
    .= the carrier of im(g|im f);
  then
A2: rank(g*f) = rank (g|im f) by VECTSP_4:29
    .= rank f by A1,RANKNULL:45;
  nullity(f) + rank (f) = dim V1 by RANKNULL:44
    .= nullity(g*f) + rank (g*f) by RANKNULL:44;
  hence thesis by A2;
end;
