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;
reserve G for addGroup-like non empty addMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for addGroup;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

theorem Th119:
  a in H iff H + a = carr(H)
proof
  thus a in H implies H + a = carr(H)
  proof
    assume
A1: a in H;
    thus H + a c= carr(H)
    proof
      let x be object;
      assume x in H + a;
      then consider g such that
A2:   x = g + a and
A3:   g in H by Th104;
      g + a in H by A1,A3,Th50;
      hence thesis by A2;
    end;
    let x be object;
    assume
A4: x in carr(H);
    then
A5: x in H;
    reconsider b = x as Element of G by A4;
A6: (b + -a) + a = b + (-a + a) by RLVECT_1:def 3
      .= b + 0_G by Def5
      .= x by Def4;
    -a in H by A1,Th51;
    hence thesis by A5,A6,Th50,Th104;
  end;
  assume
A7: H + a = carr(H);
  0_G + a = a & 0_G in H by Th46,Def4;
  hence thesis by A7,Th104;
end;
