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 Th8:
  for F,G being Function of [:I[01],I[01]:],T
  for M,N being Subset of [:I[01],I[01]:] holds
  HomotopyMlt(F,G).:(M/\N) c= F.:M * G.:N
  proof
    let F,G be Function of II,T;
    let M,N be Subset of II;
    let y be object;
    assume y in HomotopyMlt(F,G).:(M/\N);
    then consider x being Point of II such that
A1: x in M /\ N and
A2: HomotopyMlt(F,G).x = y by FUNCT_2:65;
    consider a,b being Point of I such that
A3: x = [a,b] by BORSUK_1:10;
A4: HomotopyMlt(F,G).(a,b) = F.(a,b) * G.(a,b) by Def7;
    [a,b] in M & [a,b] in N by A1,A3,XBOOLE_0:def 4;
    then F.(a,b) in F.:M & G.(a,b) in G.:N by FUNCT_2:35;
    hence thesis by A2,A3,A4;
  end;
