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;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem Th117:
  H is normal Subgroup of G iff for a holds a + H = H + a
proof
  thus H is normal Subgroup of G implies for a holds a + H = H + a
  proof
    assume
A1: H is normal Subgroup of G;
    let a;
A2: carr(H * a) = (-a) + H + a by ThB59;
    carr(H * a) = the carrier of the addMagma of H by A1,Def13
      .= carr H;
    hence a + H = a + ((-a) + H) + a by A2,ThA33
      .= a + (-a) + H + a by ThB105
      .= 0_G + H + a by Def5
      .= H + a by Th37;
  end;
  assume
A3: for a holds a + H = H + a;
  H is normal
  proof
    let a;
    the carrier of H * a = (-a) + H + a by ThB59
      .= H + (-a) + a by A3
      .= H + ((-a) + a) by ThA107
      .= H + 0_G by Def5
      .= the carrier of H by ThB109;
    hence thesis by Th59;
  end;
  hence thesis;
end;
