reserve k, n for Nat;

theorem
  for S being non void non empty ManySortedSign st S is
  finitely_operated & InducedGraph S is well-founded holds S is monotonic
proof
  let S be non void non empty ManySortedSign;
  set G = InducedGraph S;
  assume that
A1: S is finitely_operated and
A2: G is well-founded;
  given A being finitely-generated non-empty MSAlgebra over S such that
A3: A is non finite-yielding;
  set GS = the non-empty finite-yielding GeneratorSet of A;
  reconsider gs = GS as non-empty finite-yielding ManySortedSet of the carrier
  of S;
  FreeMSA gs is non finite-yielding by A3,MSSCYC_1:23;
  then the Sorts of FreeMSA gs is non finite-yielding;
  then consider v being object such that
A4: v in the carrier of S and
A5: (the Sorts of FreeMSA gs).v is non finite by FINSET_1:def 5;
  reconsider v as SortSymbol of S by A4;
  consider n being Nat such that
A6: for c being directed Chain of G st c is non empty & (vertex-seq c).(
  len c +1) = v holds len c <= n by A2,MSSCYC_1:def 4;
  set V = (the Sorts of FreeMSA gs).v;
  set Vn = {t where t is Element of V : depth t<=n};
  Vn is finite by A1,Th5;
  then not V c= Vn by A5;
  then consider t being object such that
A7: t in V and
A8: not t in Vn;
  reconsider t as Element of V by A7;
A9: not depth t<=n by A8;
  then 1 <=depth t by NAT_1:14;
  then ex d being directed Chain of InducedGraph S st len d = depth t & (
  vertex-seq d).(len d +1) = v by Th2;
  hence contradiction by A6,A9,CARD_1:27;
end;
