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 Th38:
  for T being UnContinuous TopaddGroup, a being Element of T, W being
  a_neighborhood of -a ex A being open a_neighborhood of a st -A c= W
proof
  let T be UnContinuous TopaddGroup, a be Element of T,
W be a_neighborhood of -a;
  reconsider f = add_inverse T as Function of T, T;
  f.a = -a & f is continuous by Def7,Def6;
  then consider H being a_neighborhood of a such that
A1: f.:H c= W by BORSUK_1:def 1;
  a in Int Int H by CONNSP_2:def 1;
  then reconsider A = Int H as open a_neighborhood of a by CONNSP_2:def 1;
  take A;
  let x be object;
  assume x in -A;
  then consider g being Element of T such that
A2: x = -g and
A3: g in A;
  Int H c= H & f.g = -g by Def6,TOPS_1:16;
  then -g in f.:H by A3,FUNCT_2:35;
  hence thesis by A1,A2;
end;
