reserve T for BinContinuous unital TopSpace-like non empty TopGrStr,
  x,y for Point of I[01],
  s,t for unital Point of T,
  f,g for Loop of t,
  c for constant Loop of t;

theorem Th7:
  for a,b being Point of I[01] holds
  for N being a_neighborhood of a*b
  ex N1 being a_neighborhood of a, N2 being a_neighborhood of b st
  for x,y being Point of I[01] st x in N1 & y in N2 holds x*y in N
  proof
    let a,b be Point of I;
    let N be a_neighborhood of a*b;
    set g = I[01]-TIMES;
    g.(a,b) = a*b by Th6;
    then consider H being a_neighborhood of [a,b] such that
A1: g.:H c= N by BORSUK_1:def 1;
    consider F being Subset-Family of [:I,I:] such that
A2: Int H = union F and
A3: for e being set st e in F ex X1, Y1 being Subset of I 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 I 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,y be Point of I such that
A10: x in A & y in B;
A11: Int H c= H by TOPS_1:16;
A12: g.(x,y) = x*y by Th6;
    [x,y] in M by A6,A10,ZFMISC_1:87;
    then [x,y] in Int H by A2,A5,TARSKI:def 4;
    then g.(x,y) in g.:H by A11,FUNCT_2:35;
    hence thesis by A1,A12;
  end;
