reserve k, n for Nat;

theorem
  for S being void non empty ManySortedSign holds S is monotonic iff
  InducedGraph S is well-founded
proof
  let S be void non empty ManySortedSign;
  set G = InducedGraph S, OS = the carrier' of S, CA = the carrier of S, AR =
  the Arity of S;
  hereby
    assume S is monotonic;
    assume not G is well-founded;
    then consider v being Element of the carrier of G such that
A1: for n being Nat ex c being directed Chain of G st c is
    non empty & (vertex-seq c).(len c +1) = v & len c > n by MSSCYC_1:def 4;
    consider e being directed Chain of G such that
A2: e is non empty and
    (vertex-seq e).(len e +1) = v and
    len e>0 by A1;
    e is FinSequence of the carrier' of G by MSSCYC_1:def 1;
    then
A3: rng e c= the carrier' of G by FINSEQ_1:def 4;
    1 in dom e by A2,FINSEQ_5:6;
    then e.1 in rng e by FUNCT_1:def 3;
    then
    ex op, v being set st e.1 = [op, v] & op in OS & v in CA & ex n being
Nat, args being Element of CA* st AR.op=args & n in dom args & args.n = v by A3
,Def1;
    hence contradiction;
  end;
  assume G is well-founded;
  let A be finitely-generated non-empty MSAlgebra over S;
  thus the Sorts of A is finite-yielding by MSAFREE2:def 10;
end;
