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

theorem
  for C1,C2 being full Categorial Category st
  the carrier of C1 = the carrier of C2 holds
  the CatStr of C1 = the CatStr of C2
proof
  let C1, C2 be full Categorial Category;
  assume
A1: the carrier of C1 = the carrier of C2;
  reconsider A = the carrier of C1 as categorial non empty set by Def6;
  set B = the set of all m`2 where m is Morphism of C1;
  set m = the Morphism of C1;
  m`2 in B;
  then reconsider B as non empty set;
  reconsider D1 = the CatStr of C1, D2 = the CatStr of C2 as strict Category
  by Th1;
A2: D1 is Categorial by Th10;
A3: D2 is Categorial by Th10;
A4: now
    let a,b be Element of A, F be Functor of a,b;
    reconsider m = [[a,b], F] as Morphism of C1 by Def8;
    m`2 = F;
    hence [[a,b],F] is Morphism of the CatStr of C1 iff F in B;
  end;
  now
    let a,b be Element of A, F be Functor of a,b;
    reconsider a9 = a, b9 = b as Object of C2 by A1;
A5: cat a9 = a by Th16;
    cat b9 = b by Th16;
    then reconsider m2 = [[a,b], F] as Morphism of C2 by A5,Def8;
    reconsider m = [[a,b], F] as Morphism of C1 by Def8;
A6: m`2 = F;
    m2 = m2;
    hence [[a,b],F] is Morphism of the CatStr of C2 iff F in B by A6;
  end;
  hence thesis by A1,A2,A3,A4,Th18;
end;
