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 Th118:
  for H being Subgroup of G holds H is normal Subgroup of G iff
  for a holds a + H c= H + a
proof
  let H be Subgroup of G;
  thus H is normal Subgroup of G implies for a holds a + H c= H + a by Th117;
  assume
A1: for a holds a + H c= H + a;
  now
    let a;
    (-a) + H c= H + (-a) by A1;
    then a + ((-a) + H) c= a + (H + (-a)) by Th4;
    then a + (-a) + H c= a + (H + (-a)) by ThB105;
    then 0_G + H c= a + (H + (-a)) by Def5;
    then carr H c= a + (H + (-a)) by ThB109;
    then carr H c= a + H + (-a) by ThB106;
    then carr H + a c= a + H + (-a) + a by Th4;
    then H + a c= a + H + ((-a) + a) by ThB34;
    then H + a c= a + H + 0_G by Def5;
    hence H + a c= a + H by Th37;
  end;
  then for a holds a + H = H + a by A1;
  hence thesis by Th117;
end;
