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 Th41,STRUCT_0:def 5;
    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 + 0_G by Def4
      .= a + (-g + g) by Def5
      .= a + -g + g by RLVECT_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 Th41,STRUCT_0:def 5;
  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 = 0_G + a by Def4
    .= g + -g + a by Def5
    .= g + (-g + a) by RLVECT_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;
