
theorem Th7:
  for S being non void non empty ManySortedSign for X being
non-empty ManySortedSet of the carrier of S, v be SortSymbol of S, e be Element
  of (the Sorts of FreeMSA X).v holds e is finite DecoratedTree
proof
  let S be non void non empty ManySortedSign, X be non-empty ManySortedSet of
the carrier of S, v be SortSymbol of S, e be Element of (the Sorts of FreeMSA X
  ).v;
  FreeMSA X = MSAlgebra (# FreeSort(X), FreeOper(X) #) by MSAFREE:def 14;
  then (the Sorts of FreeMSA X).v = FreeSort (X, v) by MSAFREE:def 11;
  then
A1: e in TS(DTConMSA(X)) by TARSKI:def 3;
  then reconsider e9 = e as DecoratedTree;
  dom e9 is finite by A1;
  hence thesis by FINSET_1:10;
end;
