 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 Th16:
  {}(the carrier of G) * A = {} & A * {}(the carrier of G) = {}
proof
A1: now
    set x = the Element of A * {}(the carrier of G);
    assume A * {}(the carrier of G) <> {};
    then ex g1,g2 st x = g1 * g2 & g1 in A & g2 in {}(the carrier of G) by Th8;
    hence contradiction;
  end;
  now
    set x = the Element of {}(the carrier of G) * A;
    assume {}(the carrier of G) * A <> {};
    then ex g1,g2 st x = g1 * g2 & g1 in {}(the carrier of G) & g2 in A by Th8;
    hence contradiction;
  end;
  hence thesis by A1;
end;
