 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;
reserve G for non empty multMagma,
  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 Th8;
  hence thesis;
end;
