reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th13:
  for X being ManySortedSet of S
  for o being OperSymbol of S
  for p being FinSequence st p in Args(o,Free(S,X)) holds
  Den(o,Free(S,X)).p = [o,the carrier of S]-tree p
  proof
    let X be ManySortedSet of S;
    let o be OperSymbol of S;
    let p be FinSequence such that
A1: p in Args(o,Free(S,X));
    set Y = X (\/) ((the carrier of S) --> {0});
    consider A being MSSubset of FreeMSA Y such that
A2: Free(S,X) = GenMSAlg A & A = (Reverse (Y))""X by MSAFREE3:def 1;
A3: Free(S,Y) = FreeMSA Y by MSAFREE3:31;
    then Args(o,Free(S,X)) c= Args(o,Free(S,Y)) by A2,MSAFREE3:37;
    then Den(o,Free(S,Y)).p = [o,the carrier of S]-tree p by A1,A3,INSTALG1:3;
    hence Den(o,Free(S,X)).p = [o,the carrier of S]-tree p
    by A1,A2,A3,EQUATION:19;
  end;
