reserve x,y for set;

theorem Th27:
  for A being category, C being non empty subcategory of A for a,b
being Object of C st <^a,b^> <> {} for f being Morphism of a,b holds (incl C).f
  = f
proof
  let A be category, C be non empty subcategory of A;
  let a,b be Object of C such that
A1: <^a,b^> <> {};
  let f be Morphism of a,b;
A2: the MorphMap of incl C = id the Arrows of C & [a,b] in [:the carrier of
  C, the carrier of C:] by FUNCTOR0:def 28,ZFMISC_1:def 2;
  <^(incl C).a, (incl C).b^> <> {} by A1,FUNCTOR0:28,XBOOLE_1:3;
  hence (incl C).f = Morph-Map(incl C, a, b).f by A1,FUNCTOR0:def 15
    .= (id ((the Arrows of C).(a,b))).f by A2,MSUALG_3:def 1
    .= f;
end;
