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 Th15:
  for S being OrderSortedSign, X being non-empty ManySortedSet of
  S, s be Element of S, x being set st x in (the Sorts of ParsedTermsOSA(X)).s
  holds x is Element of TS DTConOSA(X)
proof
  let S being OrderSortedSign, X being non-empty ManySortedSet of S, s be
  Element of S, x being set such that
A1: x in (the Sorts of ParsedTermsOSA(X)).s;
  set PTA = ParsedTermsOSA(X), SPTA = the Sorts of PTA;
  s in the carrier of S;
  then s in dom SPTA by PARTFUN1:def 2;
  then SPTA.s in rng SPTA by FUNCT_1:def 3;
  then x in union rng SPTA by A1,TARSKI:def 4;
  then reconsider x1=x as Element of Union SPTA by CARD_3:def 4;
  x1 is Element of PTA;
  hence thesis by Th14;
end;
