 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;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;

theorem Th37:
  1_G * A = A & A * 1_G = A
proof
  thus 1_G * A = A
  proof
    thus 1_G * A c= A
    proof
      let x be object;
      assume x in 1_G * A;
      then ex h st x = 1_G * h & h in A by Th27;
      hence thesis by GROUP_1:def 4;
    end;
    let x be object;
    assume
A1: x in A;
    then reconsider a = x as Element of G;
    1_G * a = a by GROUP_1:def 4;
    hence thesis by A1,Th27;
  end;
  thus A * 1_G c= A
  proof
    let x be object;
    assume x in A * 1_G;
    then ex h st x = h * 1_G & h in A by Th28;
    hence thesis by GROUP_1:def 4;
  end;
  let x be object;
  assume
A2: x in A;
  then reconsider a = x as Element of G;
  a * 1_G = a by GROUP_1:def 4;
  hence thesis by A2,Th28;
end;
