reserve e for set;

theorem Th10:
  for C being Category, i,j,k being Object of C holds [:Hom(j,k),
  Hom(i,j):] c= dom the Comp of C
proof
  let C be Category, i,j,k be Object of C;
  let e be object;
  assume
A1: e in [:Hom(j,k),Hom(i,j):];
  then reconsider y = e`1, x = e`2 as Morphism of C by MCART_1:10;
A2: e`2 in Hom(i,j) by A1,MCART_1:10;
A3: e = [y,x] by A1,MCART_1:21;
  e`1 in Hom(j,k) by A1,MCART_1:10;
  then dom y = j by CAT_1:1
    .= cod x by A2,CAT_1:1;
  hence thesis by A3,CAT_1:15;
end;
