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

theorem Th10:
  for C,D being Category st the CatStr of C = the CatStr of D holds
  C is Categorial implies D is Categorial
proof
  let C,D be Category;
  assume
A1: the CatStr of C = the CatStr of D;
  assume that
A2: the carrier of C is categorial and
A3: for a being Object of C, A being Category st a = A holds
  id a = [[A,A], id A] and
A4: for m being Morphism of C for A,B being Category st A = dom m & B = cod m
  ex F being Functor of A,B st m = [[A,B], F] and
A5: for m1,m2 being Morphism of C for A,B,C being Category
  for F being Functor of A,B
  for G being Functor of B,C st m1 = [[A,B],F] & m2 = [[B,C],G]
  holds m2(*)m1 = [[A,C],G*F];
  thus the carrier of D is categorial by A1,A2;
  thus for a being Object of D, A being Category st a = A
      holds id a = [[A,A], id A]
  proof
    let a be Object of D, A be Category;
    reconsider b = a as Object of C by A1;
    assume a = A;
    then [[A,A], id A] = id b by A3;
    hence id a = [[A,A], id A] by A1,Lm1;
  end;
  hereby
    let m be Morphism of D;
    reconsider m9 = m as Morphism of C by A1;
    let A,B be Category;
    assume that
A6: A = dom m and
A7: B = cod m;
A8: A = dom m9 by A1,A6;
    B = cod m9 by A1,A7;
    hence ex F being Functor of A,B st m = [[A,B], F] by A4,A8;
  end;
  let m1,m2 be Morphism of D;
  reconsider f1 = m1, f2 = m2 as Morphism of C by A1;
  let a,b,c be Category;
  let F be Functor of a,b, G be Functor of b,c;
  assume that
A9: m1 = [[a,b],F] and
A10: m2 = [[b,c],G];
  reconsider a1 = dom f1, b1 = cod f1, a2 = dom f2, b2 = cod f2 as Category
  by A2;
A11: ex F1 being Functor of a1,b1 st ( f1 = [[a1,b1], F1]) by A4;
  ex F2 being Functor of a2,b2 st ( f2 = [[a2,b2], F2]) by A4;
  then
A12: dom f2 = m2`11 by MCART_1:85;
A13: m2`11 = b by A10,MCART_1:85;
A14: cod f1 = m1`12 by A11,MCART_1:85;
  m1`12 = b by A9,MCART_1:85;
  then
A15: [f2,f1] in dom the Comp of C by A12,A13,A14,CAT_1:15;
  hence m2(*)m1 = (the Comp of D).(m2,m1) by A1,CAT_1:def 1
    .= f2(*)f1 by A1,A15,CAT_1:def 1
    .= [[a,c],G*F] by A5,A9,A10;
end;
