reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem Th38:
    for X,Y be Algebraic_Set of n,R holds X c< Y implies Ideal_Y c< Ideal_X
    proof
      let X,Y be Algebraic_Set of n,R;
      assume
A1:   X c< Y;
      Ideal_X <> Ideal_Y
      proof
        assume Ideal_X = Ideal_Y; then
        X = Zero_(Ideal_Y) by Th36 .= Y by Th36;
        hence contradiction by A1;
      end;
      hence thesis by A1,Th29;
    end;
