reserve x,y,z for set;

theorem Th4:
  for S being non void Signature for X being ManySortedSet of the
carrier of S for s being SortSymbol of S st x in X.s holds root-tree [x,s] in (
  the Sorts of Free(S, X)).s
proof
  let S be non void Signature;
  let X be ManySortedSet of the carrier of S;
  let s be SortSymbol of S such that
A1: x in X.s;
  set Y = X (\/) ((the carrier of S) --> {0});
  consider A being MSSubset of FreeMSA Y such that
A2: Free(S, X) = GenMSAlg A and
A3: A = (Reverse Y)""X by Def1;
  A is MSSubset of Free(S,X) by A2,MSUALG_2:def 17;
  then A c= the Sorts of Free(S,X) by PBOOLE:def 18;
  then
A4: A.s c= (the Sorts of Free(S,X)).s;
  X c= Y by PBOOLE:14;
  then X.s c= Y.s;
  then root-tree [x,s] in A.s by A1,A3,Th3;
  hence thesis by A4;
end;
