reserve k, n for Nat;

theorem
  for S being non void non empty ManySortedSign st S is monotonic
  holds InducedGraph S is well-founded
proof
  let S be non void non empty ManySortedSign;
  set G = InducedGraph S;
  assume S is monotonic;
  then reconsider S as monotonic non void non empty ManySortedSign;
  set A = the finite-yielding non-empty MSAlgebra over S;
  assume G is non 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;
  reconsider v as SortSymbol of S;
  consider s being finite non empty Subset of NAT such that
A2: s = the set of all
 depth t where t is Element of (the Sorts of FreeEnv A).v  and
  depth(v,A) = max s by CIRCUIT1:def 6;
  max s is Nat by TARSKI:1;
  then consider c being directed Chain of G such that
  c is non empty and
A3: (vertex-seq c).(len c +1) = v and
A4: len c>max s by A1;
  1<=len c by A4,NAT_1:14;
  then consider
  t being Element of (the Sorts of FreeMSA the Sorts of A).v such
  that
A5: depth t = len c by A3,Th2;
  reconsider t9 = t as Element of (the Sorts of FreeEnv A).v;
  (ex t99 being Element of (the Sorts of FreeMSA the Sorts of A ).v st t9
  = t99 & depth t9 = depth t99 )& depth t9 in s by A2,CIRCUIT1:def 5;
  hence contradiction by A4,A5,XXREAL_2:def 8;
end;
