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

theorem Th20:
  for S being non void non empty ManySortedSign, A being MSAlgebra
  over S, G being GeneratorSet of A, B being MSSubset of A st G c= B holds B is
  GeneratorSet of A
proof
  let S be non void non empty ManySortedSign, A be MSAlgebra over S, G be
  GeneratorSet of A, B be MSSubset of A;
  B is MSSubset of GenMSAlg B by MSUALG_2:def 17;
  then
A1: B c= the Sorts of GenMSAlg B by PBOOLE:def 18;
  assume G c= B;
  then G c= the Sorts of GenMSAlg B by A1,PBOOLE:13;
  then G is MSSubset of GenMSAlg B by PBOOLE:def 18;
  then GenMSAlg G is MSSubAlgebra of GenMSAlg B by MSUALG_2:def 17;
  then the Sorts of GenMSAlg G is MSSubset of GenMSAlg B by MSUALG_2:def 9;
  then
A2: the Sorts of GenMSAlg G c= the Sorts of GenMSAlg B by PBOOLE:def 18;
A3: the Sorts of GenMSAlg(G) = the Sorts of A by MSAFREE:def 4;
  then the Sorts of GenMSAlg B is MSSubset of GenMSAlg G by MSUALG_2:def 9;
  then the Sorts of GenMSAlg B c= the Sorts of GenMSAlg G by PBOOLE:def 18;
  hence the Sorts of GenMSAlg(B) = the Sorts of A by A3,A2,PBOOLE:146;
end;
