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 ThW37:
  for T being TopSpace-like non empty TopaddGrStr 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 BinContinuous
proof
  let T be TopSpace-like non empty TopaddGrStr 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;
  let f be Function of [:T,T:], T such that
A2: f = the addF of T;
  for W being Point of [:T,T:], G being a_neighborhood of f.W ex H being
  a_neighborhood of W st f.:H c= G
  proof
    let W be Point of [:T,T:], G be a_neighborhood of f.W;
    consider a, b being Point of T such that
A3: W = [a,b] by BORSUK_1:10;
    f.W = a+b by A2,A3;
    then consider
    A being a_neighborhood of a, B being a_neighborhood of b such that
A4: A+B c= G by A1;
    reconsider H = [:A,B:] as a_neighborhood of W by A3;
    take H;
    thus thesis by A2,A4,Th14;
  end;
  hence thesis by BORSUK_1:def 1;
end;
