theorem
  for r being Element of T,s holds
  Hom(T,x1,x2).s.r = ((canonical_homomorphism T)**Hom(Free(S,X),x1,x2)).s.r
  proof let r be Element of T,s;
    the Sorts of T is MSSubset of Free(S,X) by MSAFREE4:def 6;
    then
A1: (the Sorts of T).s c= (the Sorts of Free(S,X)).s
    by PBOOLE:def 2,def 18;
    ((canonical_homomorphism T)**Hom(Free(S,X),x1,x2)).s.r =
    (Hom(T,x1,x2)**canonical_homomorphism T).s.r =
    ((Hom(T,x1,x2).s)*((canonical_homomorphism T).s)).r =
    (Hom(T,x1,x2).s).((canonical_homomorphism T).s.r) =
    Hom(T,x1,x2).s.r by A1,Th66,MSUALG_3:2,FUNCT_2:15,MSAFREE4:47;
    hence thesis;
  end;
