reserve S for non void non empty ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S;
reserve A for MSAlgebra over S,
  t for Term of S,V;

theorem
  S-Terms V = Union FreeSort V
proof
  set T = S-Terms V, X = V;
A1: dom FreeSort X = the carrier of S by PARTFUN1:def 2;
  hereby
    let x be object;
    assume x in T;
    then reconsider t = x as Term of S,V;
    consider s being SortSymbol of S such that
A2: t in FreeSort (X, s) by Th11;
    FreeSort (X,s) = (FreeSort X).s by MSAFREE:def 11;
    hence x in Union FreeSort X by A1,A2,CARD_5:2;
  end;
  let x be object;
  assume x in Union FreeSort X;
  then consider y being object such that
A3: y in dom FreeSort X and
A4: x in (FreeSort X).y by CARD_5:2;
  reconsider y as SortSymbol of S by A3,PARTFUN1:def 2;
  x in FreeSort(X,y) by A4,MSAFREE:def 11;
  hence thesis;
end;
