reserve x,y,z for set;

theorem Th7:
  for S being non void Signature for X being non-empty
ManySortedSet of the carrier of S for s being SortSymbol of S for x being Term
  of S, X holds x in (the Sorts of FreeMSA X).s iff the_sort_of x = s
proof
  let S be non void Signature;
  let X be non-empty ManySortedSet of the carrier of S;
  let s be SortSymbol of S;
  FreeMSA X = MSAlgebra(# FreeSort X, FreeOper X #) by MSAFREE:def 14;
  then (the Sorts of FreeMSA X).s = FreeSort(X, s) by MSAFREE:def 11;
  hence thesis by MSATERM:def 5;
end;
