reserve C for category,
  o1, o2, o3 for Object of C;

theorem Th36:
  for B being non empty subcategory of C for A being non empty
  subcategory of B holds A is non empty subcategory of C
proof
  let B be non empty subcategory of C, A be non empty subcategory of B;
  reconsider D = A as with_units non empty SubCatStr of C by ALTCAT_2:21;
  now
    let o be Object of D, o1 be Object of C such that
A1: o = o1;
    o in the carrier of D & the carrier of D c= the carrier of B by
ALTCAT_2:def 11;
    then reconsider oo = o as Object of B;
    idm o = idm oo by ALTCAT_2:34
      .= idm o1 by A1,ALTCAT_2:34;
    hence idm o1 in <^o,o^>;
  end;
  hence thesis by ALTCAT_2:def 14;
end;
