
theorem Th28:
  for C1,C2 being Category, F being Functor of C1,C2 holds F-MSF(
  the carrier of CatSign the carrier of C1, the Sorts of MSAlg C1) is
ManySortedFunction of MSAlg C1, (MSAlg C2)|(CatSign the carrier of C1, Upsilon
  F, Psi F)
proof
  let C1,C2 be Category, F be Functor of C1,C2;
  set S1 = CatSign the carrier of C1, S2 = CatSign the carrier of C2;
  set A1 = MSAlg C1, A2 = MSAlg C2;
  set f = Upsilon F, g = Psi F, B1 = A2|(S1, f, g);
  set H = F-MSF(the carrier of S1, the Sorts of A1);
  let i be object;
  assume i in the carrier of S1;
  then reconsider s = i as SortSymbol of S1;
  consider a,b being Object of C1 such that
A1: s = homsym(a,b) by Th15;
  f, g form_morphism_between S1,S2 by Th24;
  then the Sorts of B1 = (the Sorts of A2)*f by INSTALG1:def 3;
  then
A2: (the Sorts of A2).(f.s) = (the Sorts of B1).s by FUNCT_2:15;
  f.s = homsym(F.a,F.b) by A1,Th21;
  then
A3: (the Sorts of A2).(f.s) = Hom(F.a,F.b) by Def13;
A4: (the Sorts of A1).s = Hom(a,b) by A1,Def13;
  H.s = F|((the Sorts of A1).s) by Def1;
  then H.s = hom(F,a,b) by A4;
  hence thesis by A2,A4,A3;
end;
