
theorem Th30:
  for C1,C2 being Category, F being Functor of C1,C2 for a,b,c
being Object of C1 for f,g being Morphism of C1 st f in Hom(a,b) & g in Hom(b,c
  ) for x being Element of Args(compsym(a,b,c),MSAlg C1) st x = <*g,f*> for H
  being ManySortedFunction of MSAlg C1, (MSAlg C2)|(CatSign the carrier of C1,
  Upsilon F, Psi F) st H = F-MSF(the carrier of CatSign the carrier of C1, the
  Sorts of MSAlg C1) holds H#x = <*F.g,F.f*>
proof
  let C1,C2 be Category, F be Functor of C1,C2;
  set CS1 = CatSign the carrier of C1, CS2 = CatSign the carrier of C2;
  set A1 = MSAlg C1, A2 = MSAlg C2;
  set u = Upsilon F, p = Psi F;
  set B = A2|(CS1, u, p);
  let a,b,c be Object of C1;
  set o = compsym(a,b,c);
  let f,g be Morphism of C1 such that
A1: f in Hom(a,b) and
A2: g in Hom(b,c);
  let x be Element of Args(o, A1) such that
A3: x = <*g,f*>;
  F.g in Hom(F.b,F.c) by A2,CAT_1:81;
  then
A4: dom (F.g) = F.b & cod (F.g) = F.c by CAT_1:1;
  F.f in Hom(F.a,F.b) by A1,CAT_1:81;
  then dom (F.f) = F.a & cod (F.f) = F.b by CAT_1:1;
  then
A5: <*F.g,F.f*> in Args(compsym(F.a,F.b,F.c), A2) by A4,Th29;
A6: dom g = b & cod g = c by A2,CAT_1:1;
  dom f = a & cod f = b by A1,CAT_1:1;
  then
A7: x in Args(o, A1) by A3,A6,Th29;
  let H be ManySortedFunction of A1, B such that
A8: H = F-MSF(the carrier of CS1, the Sorts of A1);
  (the Sorts of A1).homsym(b,c) = Hom(b,c) by Def13;
  then H.homsym(b,c) = F|Hom(b,c) by A8,Def1;
  then
A9: (H.homsym(b,c)).g = F.g by A2,FUNCT_1:49;
A10: dom <*g,f*> = Seg 2 by FINSEQ_1:89;
  then
A11: 1 in dom <*g,f*> by FINSEQ_1:2,TARSKI:def 2;
  (the Sorts of A1).homsym(a,b) = Hom(a,b) by Def13;
  then H.homsym(a,b) = F|Hom(a,b) by A8,Def1;
  then
A12: (H.homsym(a,b)).f = F.f by A1,FUNCT_1:49;
A13: 2 in dom <*g,f*> by A10,FINSEQ_1:2,TARSKI:def 2;
  u,p form_morphism_between CS1, CS2 by Th24;
  then
A14: Args(o, B) = Args(p.o, A2) by INSTALG1:24
    .= Args(compsym(F.a,F.b,F.c), A2) by Th23;
  then consider g9,f9 being Morphism of C2 such that
A15: H#x = <*g9,f9*> and
  dom f9 = F.a and
  cod f9 = F.b and
  dom g9 = F.b and
  cod g9 = F. c by A5,Th29;
A17: the_arity_of o = <*homsym(b,c),homsym(a,b)*> by Def3;
  then <*g,f*>.1 = g & (the_arity_of o)/.1 = homsym(b,c) by FINSEQ_4:17;
  then
A18: <*g9,f9*>.1 = F.g by A3,A7,A5,A14,A15,A9,A11,MSUALG_3:24;
  <*g,f*>.2 = f & (the_arity_of o)/.2 = homsym(a,b ) by A17,FINSEQ_4:17;
  then <*g9,f9*>.2 = F.f by A3,A7,A5,A14,A15,A12,A13,MSUALG_3:24;
  hence thesis by A15,A18;
end;
