reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem
  for S being non void non empty ManySortedSign, A being non-empty
  MSAlgebra over S, X being non-empty GeneratorSet of A st A is non
  finite-yielding holds FreeMSA X is non finite-yielding
proof
  let S be non void non empty ManySortedSign, A be non-empty MSAlgebra over S,
  X be non-empty GeneratorSet of A such that
A1: A is non finite-yielding and
A2: FreeMSA X is finite-yielding;
  the Sorts of A is non finite-yielding by A1,MSAFREE2:def 11;
  then consider i being object such that
A3: i in the carrier of S and
A4: (the Sorts of A).i is infinite;
  the Sorts of FreeMSA X is finite-yielding by A2,MSAFREE2:def 11;
  then
A5: (the Sorts of FreeMSA X).i is finite;
  reconsider FXi = (the Sorts of FreeMSA X).i as non empty set by A3;
  reconsider SAi = (the Sorts of A).i as non empty set by A3;
  consider F being ManySortedFunction of FreeMSA X, A such that
A6: F is_epimorphism FreeMSA X, A by Th21;
  reconsider i as Element of S by A3;
  reconsider Fi = F.i as Function of FXi, SAi;
  F is "onto" by A6;
  then rng Fi = SAi;
  hence contradiction by A4,A5;
end;
