
theorem
  for S being non empty non void ManySortedSign
  for X being non-empty ManySortedSet of the carrier of S
  for T being all_vars_including inheriting_operations (X,S)-terms
  MSAlgebra over S
  for G being basic GeneratorSet of T
  for s being SortSymbol of S
  for a being set holds a is pure Element of G.s iff a in (FreeGen T).s
  proof
    let S be non empty non void ManySortedSign;
    let X be non-empty ManySortedSet of the carrier of S;
    let T be all_vars_including inheriting_operations (X,S)-terms
    MSAlgebra over S;
    let G be basic GeneratorSet of T;
    let s be SortSymbol of S;
    let a be set;
    (FreeGen T).s c= G.s by Def3,PBOOLE:def 2;
    hence thesis by Def4;
  end;
