reserve C for CategoryStr;
reserve f,f1,f2,f3 for morphism of C;
reserve g1,g2 for morphism of C opp;
reserve C,D,E for with_identities CategoryStr;
reserve F for Functor of C,D;
reserve G for Functor of D,E;
reserve f for morphism of C;

theorem Th44:
  for C being non empty category, a1,a2 being (Morphism of Alter(C)),
  f1,f2 being morphism of C st a1 = f1 & a2 = f2 & f1 |> f2
  holds a1(*)a2 = f1(*)f2
  proof
    let C be non empty category;
    let a1,a2 be Morphism of Alter(C);
    let f1,f2 be morphism of C;
    assume
A1: a1 = f1 & a2 = f2;
    assume
A2: f1 |> f2;
    thus a1(*)a2 = (the Comp of Alter(C)).(a1,a2) by A1,A2,CAT_1:def 1
    .= f1(*)f2 by A1,A2,Def3;
  end;
