reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem
  for J being non empty non void Signature
  for T being MSAlgebra over J
  for X being VariableSet of T holds X c= Union the Sorts of T
  proof
    let J be non empty non void Signature;
    let T be MSAlgebra over J;
    let X be VariableSet of T;
    consider G being GeneratorSet of T such that
A1: X = Union G by Def8;
    let x; assume x in X; then
    consider y being object such that
A2: y in dom G & x in G.y by A1,CARD_5:2;
    y in the carrier of J by A2; then
A3: y in dom the Sorts of T by PARTFUN1:def 2;
    G c= the Sorts of T by PBOOLE:def 18; then
    G.y c= (the Sorts of T).y by A2;
    hence thesis by A2,A3,CARD_5:2;
  end;
