reserve i for object;
reserve S for non empty ManySortedSign;
reserve D for non empty set,
  n for Nat;

theorem Th8:
  for MS being segmental non void 1-element ManySortedSign
, A being non-empty MSAlgebra over MS holds the Charact of A is FinSequence of
  PFuncs((the_sort_of A)*,the_sort_of A)
proof
  let MS be segmental non void 1-element ManySortedSign, A be
  non-empty MSAlgebra over MS;
A1: dom the Charact of A = the carrier' of MS by PARTFUN1:def 2;
  ex n being Element of NAT st the carrier' of MS = Seg n
  proof
    consider n such that
A2: the carrier' of MS = Seg n by Def7;
    n in NAT by ORDINAL1:def 12;
    hence thesis by A2;
  end;
  then reconsider f = the Charact of A as FinSequence by A1,FINSEQ_1:def 2;
  f is FinSequence of PFuncs((the_sort_of A)*,the_sort_of A)
  proof
    let x be object;
    assume x in rng f;
    then consider i being object such that
A3: i in the carrier' of MS and
A4: f.i = x by A1,FUNCT_1:def 3;
    reconsider i as OperSymbol of MS by A3;
A5: (the Sorts of A).((the ResultSort of MS).i) is Component of the Sorts
    of A by PBOOLE:139;
    dom(the ResultSort of MS) = the carrier' of MS by FUNCT_2:def 1;
    then ((the Sorts of A)*the ResultSort of MS).i = (the Sorts of A).((the
    ResultSort of MS).i) by FUNCT_1:13
      .= the_sort_of A by A5,Def12;
    then
A6: x is Function of Args(i,A),the_sort_of A by A4,PBOOLE:def 15;
    Args(i,A) c= (the_sort_of A)* by Th7;
    then x is PartFunc of (the_sort_of A)*,the_sort_of A by A6,RELSET_1:7;
    hence thesis by PARTFUN1:45;
  end;
  hence thesis;
end;
