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

theorem Th37:
  for T being TopSpace-like non empty TopGrStr 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 TopGrStr 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 multF 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;
