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 Th40:
  for T being TopologicaladdGroup, a, b being Element of T for W
  being a_neighborhood of a+ (-b) ex A being open a_neighborhood of a, B being
  open a_neighborhood of b st A + (-B) c= W
proof
  let T be TopologicaladdGroup, a, b be Element of T,
W be a_neighborhood of a+(-b);
  consider A being open a_neighborhood of a, B being open a_neighborhood of -b
  such that
A1: A+B c= W by Th36;
  consider C being open a_neighborhood of b such that
A2: -C c= B by Th38;
  take A, C;
  let x be object;
  assume x in A+(-C);
  then consider g, h being Element of T such that
A3: x = g+h and
A4: g in A and
A5: h in -C;
  consider k being Element of T such that
A6: h = -k and
  k in C by A5;
  g+(-k) in A+B by A2,A4,A5,A6;
  hence thesis by A1,A3,A6;
end;
