
theorem
  for S being ManySortedSign, T being feasible ManySortedSign for T9
being Subsignature of T, f,g being Function st f,g form_morphism_between S,T &
  rng f c= the carrier of T9 & rng g c= the carrier' of T9 holds f,g
  form_morphism_between S,T9
proof
  let S be ManySortedSign, T be feasible ManySortedSign;
  let T9 be Subsignature of T, f,g be Function;
  assume that
A1: dom f = the carrier of S and
A2: dom g = the carrier' of S and
  rng f c= the carrier of T and
  rng g c= the carrier' of T and
A3: f*the ResultSort of S = (the ResultSort of T)*g and
A4: for o being set, p being Function st o in the carrier' of S & p = (
  the Arity of S).o holds f*p = (the Arity of T).(g.o) and
A5: rng f c= the carrier of T9 and
A6: rng g c= the carrier' of T9;
  thus dom f = the carrier of S & dom g = the carrier' of S by A1,A2;
  thus rng f c= the carrier of T9 & rng g c= the carrier' of T9 by A5,A6;
  thus f*the ResultSort of S = (the ResultSort of T)*((id the carrier' of T9)*
  g) by A3,A6,RELAT_1:53
    .= (the ResultSort of T)*(id the carrier' of T9)*g by RELAT_1:36
    .= ((the ResultSort of T)|the carrier' of T9)*g by RELAT_1:65
    .= (the ResultSort of T9)*g by Th12;
  let o be set, p be Function;
  assume that
A7: o in the carrier' of S and
A8: p = (the Arity of S).o;
A9: the Arity of T9 c= the Arity of T & dom the Arity of T9 = the carrier'
  of T9 by Th11,FUNCT_2:def 1;
  g.o in rng g by A2,A7,FUNCT_1:def 3;
  then (the Arity of T9).(g.o) = (the Arity of T).(g.o) by A6,A9,GRFUNC_1:2;
  hence thesis by A4,A7,A8;
end;
