reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem Th14:
  for S being OrderSortedSign, X being non-empty ManySortedSet of
  S, x being set holds x is Element of ParsedTermsOSA(X) iff x is Element of TS
  DTConOSA(X)
proof
  let S being OrderSortedSign, X being non-empty ManySortedSet of S, x being
  set;
  TS DTConOSA X = union rng (ParsedTerms X) by Th8
    .= Union (the Sorts of ParsedTermsOSA(X)) by CARD_3:def 4;
  hence thesis;
end;
