
theorem Th36:
  for f1,f2 being Function of [:I[01],I[01]:],I[01] st f1 is
continuous & f2 is continuous & (for p being Point of [:I[01],I[01]:] holds f1.
p * f2.p is Point of I[01]) holds ex g being Function of [:I[01],I[01]:],I[01]
  st (for p being Point of [:I[01],I[01]:],r1,r2 being Real st f1.p=r1 &
  f2.p=r2 holds g.p=r1*r2) & g is continuous
proof
  reconsider A = [.0,1.] as non empty Subset of R^1 by TOPMETR:17,XXREAL_1:1;
  set X = [:I[01],I[01]:];
  let f1,f2 be Function of [:I[01],I[01]:],I[01];
  assume that
A1: f1 is continuous & f2 is continuous and
A2: for p being Point of [:I[01],I[01]:] holds f1.p * f2.p is Point of I[01];
  reconsider f19 = f1, f29 = f2 as Function of X, R^1 by BORSUK_1:40,FUNCT_2:7
,TOPMETR:17;
  f19 is continuous & f29 is continuous by A1,PRE_TOPC:26;
  then consider g being Function of X,R^1 such that
A3: for p being Point of X,r1,r2 being Real st f19.p=r1 & f29.p=
  r2 holds g.p=r1*r2 and
A4: g is continuous by JGRAPH_2:25;
A5: rng g c= [.0,1.]
  proof
    let x be object;
    assume x in rng g;
    then consider y being object such that
A6: y in dom g and
A7: x = g.y by FUNCT_1:def 3;
    reconsider y as Point of X by A6;
    g.y = f1.y * f2.y by A3;
    then g.y is Point of I[01] by A2;
    hence thesis by A7,BORSUK_1:40;
  end;
  [.0,1.] = the carrier of R^1|A & dom g = the carrier of X by FUNCT_2:def 1
,PRE_TOPC:8;
  then reconsider g as Function of X, R^1|A by A5,FUNCT_2:2;
  R^1 | A = I[01] by BORSUK_1:def 13,TOPMETR:def 6;
  then reconsider g as continuous Function of X, I[01] by A4,JGRAPH_1:45;
  take g;
  thus thesis by A3;
end;
