
theorem Th12:
  for S being feasible ManySortedSign, T being Subsignature of S
holds the Arity of T = (the Arity of S)|the carrier' of T & the ResultSort of T
  = (the ResultSort of S)|the carrier' of T
proof
  let S be feasible ManySortedSign, T be Subsignature of S;
A1: the Arity of T c= the Arity of S by Th11;
  the carrier of T = {} implies the carrier' of T = {} by Def1;
  then
A2: dom the ResultSort of T = the carrier' of T by FUNCT_2:def 1;
  dom the Arity of T = the carrier' of T & the ResultSort of T c= the
  ResultSort of S by Th11,FUNCT_2:def 1;
  hence thesis by A2,A1,GRFUNC_1:23;
end;
