 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 Th25:
  G is commutative Group implies A * B = B * A
proof
  assume
A1: G is commutative Group;
  thus A * B c= B * A
  proof
    let x be object;
    assume x in A * B;
    then consider g,h such that
A2: x = g * h and
A3: g in A & h in B;
    x = h * g by A1,A2,Lm2;
    hence thesis by A3;
  end;
  let x be object;
  assume x in B * A;
  then consider g,h such that
A4: x = g * h and
A5: g in B & h in A;
  x = h * g by A1,A4,Lm2;
  hence thesis by A5;
end;
