 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 Th22:
  for G being Group, A being Subset of G holds
  (for g1,g2 being Element of G st g1 in A & g2 in A holds g1 * g2 in A) &
  (for g being Element of G st g in A holds g" in A) implies A * A = A
proof
  let G be Group, A be Subset of G such that
A1: for g1,g2 being Element of G st g1 in A & g2 in A holds g1 * g2 in A and
A2: for g being Element of G st g in A holds g" in A;
  thus A * A c= A
  proof
    let x be object;
    assume x in A * A;
    then ex g1,g2 being Element of G st x = g1 * g2 & g1 in A & g2 in A;
    hence thesis by A1;
  end;
  let x be object;
  assume
A3: x in A;
  then reconsider a = x as Element of G;
  a" in A by A2,A3;
  then a" * a in A by A1,A3;
  then
A4: 1_G in A by GROUP_1:def 5;
  1_G * a = a by GROUP_1:def 4;
  hence thesis by A3,A4;
end;
