reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th4:
  for C being Subcategory of A holds the Source of C = (the Source
of A)|the carrier' of C & the Target of C = (the Target of A)|the carrier' of C
& the Comp of C = (the Comp of A)||the carrier' of C
proof
  let C be Subcategory of A;
A1: dom the Source of A = the carrier' of A by FUNCT_2:def 1;
A2: now
    let x be object;
    assume x in dom the Source of C;
    then reconsider m = x as Morphism of C by FUNCT_2:def 1;
    reconsider m9=m as Morphism of A by CAT_2:8;
    thus (the Source of C).x = dom m .= dom m9 by CAT_2:9
      .= (the Source of A).x;
  end;
A3: now
    let x be object;
    assume x in dom the Target of C;
    then reconsider m = x as Morphism of C by FUNCT_2:def 1;
    reconsider m9=m as Morphism of A by CAT_2:8;
    thus (the Target of C).x = cod m .= cod m9 by CAT_2:9
      .= (the Target of A).x;
  end;
  dom the Source of C = the carrier' of C by FUNCT_2:def 1;
  then dom the Source of C = (dom the Source of A) /\ the carrier' of C by A1,
CAT_2:7,XBOOLE_1:28;
  hence the Source of C = (the Source of A)|the carrier' of C by A2,FUNCT_1:46;
A4: dom the Target of A = the carrier' of A by FUNCT_2:def 1;
  dom the Target of C = the carrier' of C by FUNCT_2:def 1;
  then dom the Target of C = (dom the Target of A) /\ the carrier' of C by A4,
CAT_2:7,XBOOLE_1:28;
  hence the Target of C = (the Target of A)|the carrier' of C by A3,FUNCT_1:46;
A5: dom the Comp of C = (dom the Comp of A) /\ [:the carrier' of C, the
  carrier' of C:]
  proof
    the Comp of C c= the Comp of A by CAT_2:def 4;
    then
A6: dom the Comp of C c= dom the Comp of A by RELAT_1:11;
    dom the Comp of C c= [:the carrier' of C, the carrier' of C:] by
RELAT_1:def 18;
    hence dom the Comp of C c= (dom the Comp of A) /\ [:the carrier' of C, the
    carrier' of C:] by A6,XBOOLE_1:19;
    let x be object;
    assume
A7: x in(dom the Comp of A) /\ [:the carrier' of C, the carrier' of C :];
    then x in [:the carrier' of C, the carrier' of C:] by XBOOLE_0:def 4;
    then reconsider f=x`1, g=x`2 as Morphism of C by MCART_1:10;
    reconsider f9=f, g9=g as Morphism of A by CAT_2:8;
    x in dom the Comp of A by A7,XBOOLE_0:def 4;
    then
A8: [f9,g9] in dom the Comp of A by MCART_1:21;
    dom f = dom f9 by CAT_2:9
      .= cod g9 by A8,CAT_1:15
      .= cod g by CAT_2:9;
    then [f,g] in dom the Comp of C by CAT_1:15;
    hence thesis by A7,MCART_1:21;
  end;
  the Comp of C c= the Comp of A by CAT_2:def 4;
  hence
  the Comp of C = (the Comp of A)| ((dom the Comp of A) /\ [:the carrier'
  of C, the carrier' of C:] qua set) by A5,GRFUNC_1:23
    .= ((the Comp of A) qua Relation)| (dom(the Comp of A) qua set)| ([:the
  carrier' of C, the carrier' of C:] qua set) by RELAT_1:71
    .= (the Comp of A)||the carrier' of C;
end;
