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 Th13:
  for S being OrderSortedSign, X being non-empty ManySortedSet of
S, o be OperSymbol of S, x being FinSequence holds x in Args(o,ParsedTermsOSA(X
  )) iff x is FinSequence of TS(DTConOSA(X)) & OSSym(o,X) ==> roots x
proof
  let S be OrderSortedSign, X be non-empty ManySortedSet of S, o be OperSymbol
  of S, x be FinSequence;
  set PTA = ParsedTermsOSA(X);
  hereby
    assume
A1: x in Args(o,PTA);
    then
A2: x in ((the Sorts of PTA)# * the Arity of S).o by MSUALG_1:def 4;
    hence x is FinSequence of TS(DTConOSA(X)) by Th5;
    x in ((ParsedTerms X)# * (the Arity of S)).o by A1,MSUALG_1:def 4;
    then reconsider x1 = x as FinSequence of TS(DTConOSA(X)) by Th5;
    OSSym(o,X) ==> roots x1 by A2,Th7;
    hence OSSym(o,X) ==> roots x;
  end;
  assume that
A3: x is FinSequence of TS(DTConOSA(X)) and
A4: OSSym(o,X) ==> roots x;
  reconsider x1 = x as FinSequence of TS(DTConOSA(X)) by A3;
  x1 in ((ParsedTerms X)# * (the Arity of S)).o by A4,Th7;
  hence thesis by MSUALG_1:def 4;
end;
