reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;

theorem Th9:
  A <> {} & B <> {} iff A + B <> {}
proof
  thus A <> {} & B <> {} implies A + B <> {}
  proof
    assume
A1: A <> {};
    then reconsider x = the Element of A as Element of G by TARSKI:def 3;
    assume
A2: B <> {};
    then reconsider y = the Element of B as Element of G by TARSKI:def 3;
    x + y in A + B by A1,A2;
    hence thesis;
  end;
  set x = the Element of A + B;
  assume A + B <> {};
  then ex a,b st x = a + b & a in A & b in B by ThX8;
  hence thesis;
end;
