reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th15:
  for C,D being Categorial Category st the carrier' of C c= the carrier' of D
  holds C is Subcategory of D
proof
  let C,D be Categorial Category;
  assume
A1: the carrier' of C c= the carrier' of D;
  thus the carrier of C c= the carrier of D
  proof
    let x be object;
    assume x in the carrier of C;
    then reconsider a = x as Object of C;
    reconsider i = id a as Morphism of D by A1;
A2: dom i = i`11 by Th13;
    dom id a = i`11 by Th13;
    hence thesis by A2;
  end;
  hereby
    let a,b be Object of C, a9,b9 be Object of D;
    assume that
A3: a = a9 and
A4: b = b9;
    thus Hom(a,b) c= Hom(a9,b9)
    proof
      let x be object;
      assume x in Hom(a,b);
      then consider f being Morphism of C such that
A5:   x = f and
A6:   dom f = a and
A7:   cod f = b;
      reconsider A = a, B = b as Category by Th12;
A8:   ex F being Functor of A,B st ( f = [[A,B], F]) by A6,A7,Def6;
      reconsider f as Morphism of D by A1;
A9:  dom f = f`11 by Th13;
A10:  cod f = f`12 by Th13;
A11:  f`11 = A by A8,MCART_1:85;
      f`12 = B by A8,MCART_1:85;
      hence thesis by A3,A4,A5,A9,A10,A11;
    end;
  end;
A12: dom the Comp of C c= dom the Comp of D
  proof
    let x be object;
    assume
A13: x in dom the Comp of C;
    then reconsider g = x`1, f = x`2 as Morphism of C by MCART_1:10;
    reconsider g9 = g, f9 = f as Morphism of D by A1;
A14: x = [g,f] by A13,MCART_1:21;
    then
A15: dom g = cod f by A13,CAT_1:15;
A16: dom g = g`11 by Th13;
A17: dom g9 = g `11 by Th13;
A18: cod f = f`12 by Th13;
    cod f9 = f`12 by Th13;
    hence thesis by A14,A15,A16,A17,A18,CAT_1:15;
  end;
  now
    let x be object;
    assume
A19: x in dom the Comp of C;
    then reconsider g = x`1, f = x`2 as Morphism of C by MCART_1:10;
    reconsider g9 = g, f9 = f as Morphism of D by A1;
A20: x = [g,f] by A19,MCART_1:21;
A21: dom g = g`11 by Th13;
    cod g = g`12 by Th13;
    then consider G being Functor of g`11, g`12 such that
A22: g = [[g`11,g`12],G] by A21,Def6;
A23: dom f = f`11 by Th13;
    cod f = dom g by A19,A20,CAT_1:15;
    then consider F being Functor of f`11, g`11 such that
A24: f = [[f`11,g`11],F] by A21,A23,Def6;
    thus (the Comp of C).x = (the Comp of C).(g,f) by A19,MCART_1:21
      .= g(*)f by A19,A20,CAT_1:def 1
      .= [[f`11,g`12],G*F] by A22,A24,Def6
      .= g9(*)f9 by A22,A24,Def6
      .= (the Comp of D).(g,f) by A12,A19,A20,CAT_1:def 1
      .= (the Comp of D).x by A19,MCART_1:21;
  end;
  hence the Comp of C c= the Comp of D by A12,GRFUNC_1:2;
  let a be Object of C, a9 be Object of D;
  assume
A25: a = a9;
  reconsider A = a as Category by Th12;
  thus id a = [[A,A], id A] by Def6
    .= id a9 by A25,Def6;
end;
