theorem
  for S being non void Signature
  for X being with_missing_variables ManySortedSet of the carrier of S
  for t being Term of S, X (\/) ((the carrier of S)-->{0})
  st t in Union the Sorts of Free(S,X)
  holds t in (the Sorts of Free(S,X)).the_sort_of t
proof
  let S be non void Signature;
  let X be with_missing_variables ManySortedSet of the carrier of S;
  set V = X (\/) ((the carrier of S)-->{0});
  set A = Free(S, X);
  set U = the Sorts of A;
A1: U = S-Terms(X, V) by MSAFREE3:24;
  let t be Term of S, X (\/) ((the carrier of S)-->{0});
  assume t in Union U;
  then consider s being object such that
A2: s in dom U and
A3: t in U.s by CARD_5:2;
  reconsider s as SortSymbol of S by A2;
  U.s = {r where r is Term of S,V: the_sort_of r = s & variables_in r c= X}
  by A1,MSAFREE3:def 5;
  then
  ex r being Term of S,V st t = r & the_sort_of r = s & variables_in r c= X
  by A3;
  hence thesis by A3;
end;
