 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;

theorem Th7:
  A <> {} iff A" <> {}
proof
  set x = the Element of A";
  thus A <> {} implies A" <> {}
  proof
    set x = the Element of A;
    assume
A1: A <> {};
    then reconsider x as Element of G by Lm1;
    x" in A" by A1;
    hence thesis;
  end;
  assume A" <> {};
  then ex a st x = a" & a in A by Th2;
  hence thesis;
end;
