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;
reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty addMagma,
   P, Q, P1, Q1 for Subset of H,
   h for Element of H;
 reserve a for Element of G;

theorem
  for T being TopaddGroup st for a, b being Element of T, W being
a_neighborhood of a+(-b) ex A being a_neighborhood of a, B being a_neighborhood
  of b st A+(-B) c= W holds T is TopologicaladdGroup
proof
  let T be TopaddGroup such that
A1: for a, b being Element of T, W being a_neighborhood of a+(-b) ex A
  being a_neighborhood of a, B being a_neighborhood of b st A+(-B) c= W;
A2: for a being Element of T, W being a_neighborhood of -a ex A being
  a_neighborhood of a st -A c= W
  proof
    let a be Element of T, W be a_neighborhood of -a;
    W is a_neighborhood of 0_T+(-a) by Def4;
    then consider
    A being a_neighborhood of 0_T, B being a_neighborhood of a such
    that
A3: A+(-B) c= W by A1;
    take B;
    let x be object;
    assume
A4: x in -B;
    then consider g being Element of T such that
A5: x = -g and
    g in B;
    0_T in A by CONNSP_2:4;
    then 0_T + (-g) in A+(-B) by A4,A5;
    then -g in A+(-B) by Def4;
    hence thesis by A3,A5;
  end;
  for a, b being Element of T, W being a_neighborhood of a+b ex A being
  a_neighborhood of a, B being a_neighborhood of b st A+B c= W
  proof
    let a, b be Element of T, W be a_neighborhood of a+b;
    W is a_neighborhood of a+(- -b);
    then consider
    A being a_neighborhood of a, B being a_neighborhood of -b such
    that
A6: A+(-B) c= W by A1;
    consider B1 being a_neighborhood of b such that
A7: -B1 c= B by A2;
    take A, B1;
    let x be object;
    assume x in A + B1;
    then consider g, h being Element of T such that
A8: x = g + h & g in A and
A9: h in B1;
    -h in -B1 by A9;
    then h in -B by A7,Th7;
    then x in A+(-B) by A8;
    hence thesis by A6;
  end;
  hence thesis by A2,ThW37,Th39;
end;
