reserve S, T, Y for non empty TopSpace,
  s, s1, s2, s3 for Point of S,
  t, t1, t2, t3 for Point of T,
  l1, l2 for Path of [s1,t1],[s2,t2],
  H for Homotopy of l1 ,l2;

theorem Th10:
  for f being continuous Function of Y,[:S,T:] holds pr2 f is continuous
proof
  let f be continuous Function of Y,[:S,T:];
  set g = pr2 f;
  for p being Point of Y, V being Subset of T st g.p in V & V is open
  holds ex W being Subset of Y st p in W & W is open & g.:W c= V
  proof
    let p be Point of Y, V be Subset of T such that
A1: g.p in V and
A2: V is open;
A3: [:[#]S,V:] is open by A2,BORSUK_1:6;
    the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
BORSUK_1:def 2;
    then consider s, t being object such that
A4: s in the carrier of S and
    t in the carrier of T and
A5: f.p = [s,t] by ZFMISC_1:def 2;
A6: dom f = the carrier of Y by FUNCT_2:def 1;
    then g.p = [s,t]`2 by A5,MCART_1:def 13
      .= t;
    then f.p in [:[#]S,V:] by A1,A4,A5,ZFMISC_1:87;
    then consider W being Subset of Y such that
A7: p in W & W is open and
A8: f.:W c= [:[#]S,V:] by A3,JGRAPH_2:10;
    take W;
    thus p in W & W is open by A7;
    let y be object;
    assume y in g.:W;
    then consider x being Point of Y such that
A9: x in W and
A10: y = g.x by FUNCT_2:65;
A11: g.x = (f.x)`2 by A6,MCART_1:def 13;
    f.x in f.:W by A6,A9,FUNCT_1:def 6;
    hence thesis by A8,A10,A11,MCART_1:10;
  end;
  hence thesis by JGRAPH_2:10;
end;
