 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 Th17:
  for G being Group, A being Subset of G holds A <> {} implies
  [#](the carrier of G) * A = the carrier of G &
  A * [#](the carrier of G) = the carrier of G
proof
  let G be Group, A be Subset of G;
  set y = the Element of A;
  assume
A1: A <> {};
  then reconsider y as Element of G by Lm1;
  thus [#](the carrier of G) * A = the carrier of G
  proof
    set y = the Element of A;
    reconsider y as Element of G by A1,Lm1;
    thus [#](the carrier of G) * A c= the carrier of G;
    let x be object;
    assume x in the carrier of G;
    then reconsider a = x as Element of G;
    (a * y") * y = a * (y" * y) by GROUP_1:def 3
      .= a * 1_G by GROUP_1:def 5
      .= a by GROUP_1:def 4;
    hence thesis by A1;
  end;
  thus A * [#](the carrier of G) c= the carrier of G;
  let x be object;
  assume x in the carrier of G;
  then reconsider a = x as Element of G;
  y * (y" * a) = (y * y") * a by GROUP_1:def 3
    .= 1_G * a by GROUP_1:def 5
    .= a by GROUP_1:def 4;
  hence thesis by A1;
end;
