reserve e for set;

theorem Th11:
  for C being Category, i,j,k being Object of C holds (the Comp of
  C).:[:Hom(j,k),Hom(i,j):] c= Hom(i,k)
proof
  let C be Category, i,j,k be Object of C;
  let e be object;
  assume e in (the Comp of C).:[:Hom(j,k),Hom(i,j):];
  then consider x being object such that
A1: x in dom the Comp of C and
A2: x in [:Hom(j,k),Hom(i,j):] and
A3: e = (the Comp of C).x by FUNCT_1:def 6;
  reconsider y = x`1, z = x`2 as Morphism of C by A2,MCART_1:10;
A4: x = [y,z] & e = (the Comp of C).(y,z) by A2,A3,MCART_1:21;
A5: x`2 in Hom(i,j) by A2,MCART_1:10;
  then
A6: z is Morphism of i,j by CAT_1:def 5;
A7: x`1 in Hom(j,k) by A2,MCART_1:10;
  then y is Morphism of j,k by CAT_1:def 5;
  then y(*)z in Hom(i,k) by A7,A5,A6,CAT_1:23;
  hence thesis by A1,A4,CAT_1:def 1;
end;
