theorem Th137:
  union Left_Cosets H = the carrier of G &
  union Right_Cosets H = the carrier of G
proof
  thus union Left_Cosets H = the carrier of G
  proof
    set h = the Element of H;
    reconsider g = h as Element of G by Th42;
    thus union Left_Cosets H c= the carrier of G;
    let x be object;
    assume x in the carrier of G;
    then reconsider a = x as Element of G;
A1: a = a * 1_G by GROUP_1:def 4
      .= a * (g" * g) by GROUP_1:def 5
      .= a * g" * g by GROUP_1:def 3;
A2: a * g" * H in Left_Cosets H by Def15;
    h in H;
    then a in a * g" * H by A1,Th103;
    hence thesis by A2,TARSKI:def 4;
  end;
  set h = the Element of H;
  reconsider g = h as Element of G by Th42;
  thus union Right_Cosets H c= the carrier of G;
  let x be object;
  assume x in the carrier of G;
  then reconsider a = x as Element of G;
A3: a = 1_G * a by GROUP_1:def 4
    .= g * g" * a by GROUP_1:def 5
    .= g * (g" * a) by GROUP_1:def 3;
A4: H * (g" * a) in Right_Cosets H by Def16;
  h in H;
  then a in H * (g" * a) by A3,Th104;
  hence thesis by A4,TARSKI:def 4;
end;
