 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
  for G being commutative Group, A,B being Subset of G holds (A * B)" = A" * B"
proof
  let G be commutative Group, A,B be Subset of G;
  thus (A * B)" c= A" * B"
  proof
    let x be object;
    assume x in (A * B)";
    then consider g being Element of G such that
A1: x = g" and
A2: g in A * B;
    consider g1,g2 being Element of G such that
A3: g = g1 * g2 and
A4: g1 in A & g2 in B by A2;
A5: g1" in A" & g2" in B" by A4;
    x = g1" * g2" by A1,A3,GROUP_1:47;
    hence thesis by A5;
  end;
  let x be object;
  assume x in A" * B";
  then consider g1,g2 being Element of G such that
A6: x = g1 * g2 and
A7: g1 in A" and
A8: g2 in B";
  consider b being Element of G such that
A9: g2 = b" and
A10: b in B by A8;
  consider a being Element of G such that
A11: g1 = a" and
A12: a in A by A7;
A13: a * b in A * B by A12,A10;
  x = (a * b)" by A6,A11,A9,GROUP_1:47;
  hence thesis by A13;
end;
