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 Th36:
  for T being BinContinuous non empty TopSpace-like TopaddGrStr, a,
  b being Element of T, 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 BinContinuous non empty TopSpace-like TopaddGrStr, a, b be Element
  of T, W be a_neighborhood of a+b;
  the carrier of [:T,T:] = [:the carrier of T,the carrier of T:] by
BORSUK_1:def 2;
  then reconsider f = the addF of T as Function of [:T,T:], T;
  consider H being a_neighborhood of [a,b] such that
A1: f.:H c= W by Def8,BORSUK_1:def 1;
  consider F being Subset-Family of [:T,T:] such that
A2: Int H = union F and
A3: for e being set st e in F ex X1, Y1 being Subset of T st e = [:X1,Y1
  :] & X1 is open & Y1 is open by BORSUK_1:5;
  [a,b] in Int H by CONNSP_2:def 1;
  then consider M being set such that
A4: [a,b] in M and
A5: M in F by A2,TARSKI:def 4;
  consider A, B being Subset of T such that
A6: M = [:A,B:] and
A7: A is open and
A8: B is open by A3,A5;
  a in A by A4,A6,ZFMISC_1:87;
  then
A9: a in Int A by A7,TOPS_1:23;
  b in B by A4,A6,ZFMISC_1:87;
  then b in Int B by A8,TOPS_1:23;
  then reconsider B as open a_neighborhood of b by A8,CONNSP_2:def 1;
  reconsider A as open a_neighborhood of a by A7,A9,CONNSP_2:def 1;
  take A, B;
  let x be object;
  assume x in A+B;
  then consider g, h being Element of T such that
A10: x = g+h and
A11: g in A & h in B;
A12: Int H c= H by TOPS_1:16;
  [g,h] in M by A6,A11,ZFMISC_1:87;
  then [g,h] in Int H by A2,A5,TARSKI:def 4;
  then x in f.:H by A10,A12,FUNCT_2:35;
  hence thesis by A1;
end;
