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

theorem Th14:
  for C1, C2 being Categorial Category st
  the carrier of C1 = the carrier of C2 &
  the carrier' of C1 = the carrier' of C2
  holds the CatStr of C1 = the CatStr of C2
proof
  let C1, C2 be Categorial Category;
  assume that
A1: the carrier of C1 = the carrier of C2 and
A2: the carrier' of C1 = the carrier' of C2;
A3: dom the Source of C1 = the carrier' of C1 by FUNCT_2:def 1;
A4: dom the Source of C2 = the carrier' of C2 by FUNCT_2:def 1;
A5: dom the Target of C1 = the carrier' of C1 by FUNCT_2:def 1;
A6: dom the Target of C2 = the carrier' of C2 by FUNCT_2:def 1;
  now
    let x be object;
    assume x in the carrier' of C1;
    then reconsider m1 = x as Morphism of C1;
    reconsider m2 = m1 as Morphism of C2 by A2;
    thus (the Source of C1).x = dom m1 .= m1`11 by Th13
      .= dom m2 by Th13
      .= (the Source of C2).x;
  end;
  then
A7: the Source of C1 = the Source of C2 by A2,A3,A4;
A8: now
    let x be object;
    assume x in the carrier' of C1;
    then reconsider m1 = x as Morphism of C1;
    reconsider m2 = m1 as Morphism of C2 by A2;
    thus (the Target of C1).x = cod m1 .= m1`12 by Th13
      .= cod m2 by Th13
      .= (the Target of C2).x;
  end;
  then
A9: the Target of C1 = the Target of C2 by A2,A5,A6;
A10: dom the Comp of C1 = dom the Comp of C2
  proof
    hereby
      let x be object;
      assume
A11:  x in dom the Comp of C1;
      then reconsider xx = x as
      Element of [:the carrier' of C1, the carrier' of C1:];
      reconsider y = xx as
      Element of [:the carrier' of C2, the carrier' of C2:] by A2;
A12:  y = [xx`1,xx`2] by MCART_1:21;
      then dom(xx`1) = cod(xx`2) by A11,CAT_1:def 6;
      then dom(y`1) = cod(y`2) by A8,A7;
      hence x in dom the Comp of C2 by A12,CAT_1:def 6;
    end;
    let x be object;
    assume
A13: x in dom the Comp of C2;
    then reconsider xx = x as
    Element of [:the carrier' of C1, the carrier' of C1:] by A2;
    reconsider y = xx as
    Element of [:the carrier' of C2, the carrier' of C2:] by A2;
A14: xx = [y`1,y`2] by MCART_1:21;
    then dom(y`1) = cod(y`2) by A13,CAT_1:def 6;
    then dom(xx`1) = cod(xx`2) by A8,A7;
    hence thesis by A14,CAT_1:def 6;
  end;
  now
    let x,y be object;
    assume
A15: [x,y] in dom the Comp of C1;
    then reconsider g1 = x, h1 = y as Morphism of C1 by ZFMISC_1:87;
    reconsider g2 = g1, h2 = h1 as Morphism of C2 by A2;
    reconsider a1 = dom g1, b1 = cod g1 as Category by Th12;
    consider f1 being Functor of a1,b1 such that
A16: g1 = [[a1,b1],f1] by Def6;
    reconsider c1 = dom h1, d1 = cod h1 as Category by Th12;
    consider i1 being Functor of c1,d1 such that
A17: h1 = [[c1,d1],i1] by Def6;
A18: a1 = d1 by A15,CAT_1:15;
    thus (the Comp of C1).(x,y) = g1(*)h1 by A15,CAT_1:def 1
      .= [[c1,b1],f1*i1] by A16,A17,A18,Def6
      .= g2(*)h2 by A16,A17,A18,Def6
      .= (the Comp of C2).(x,y) by A10,A15,CAT_1:def 1;
  end;
  then the Comp of C1 = the Comp of C2 by A2,A10,BINOP_1:20;
  hence thesis by A1,A2,A7,A9;
end;
