reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;

theorem ThX17:
  for G being addGroup, 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 addGroup, 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 RLVECT_1:def 3
      .= a + 0_G by Def5
      .= a by Def4;
    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 RLVECT_1:def 3
    .= 0_G + a by Def5
    .= a by Def4;
  hence thesis by A1;
end;
