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
  for N1,N2 being strict normal Subgroup of G ex N being strict normal
  Subgroup of G st the carrier of N = carr N1 + carr N2
proof
  let N1,N2 be strict normal Subgroup of G;
  set A = carr N1 + carr N2;
  set B = carr N1;
  set C = carr N2;
  carr N1 + carr N2 = carr N2 + carr N1 by Lm5;
  then consider H being strict Subgroup of G such that
A1: the carrier of H = A by Th78;
  now
    let a;
    thus a + H = a + N1 + C by A1,ThB29
      .= N1 + a + C by Th117
      .= B + (a + N2) by ThA30
      .= B + (N2 + a) by Th117
      .= H + a by A1,ThB31;
  end;
  then H is normal Subgroup of G by Th117;
  hence thesis by A1;
end;
