reserve T,T1,T2,S for non empty TopSpace;

theorem Th12:
  for X,Y,Z being non empty TopSpace, B being Subset of Y, C being
  Subset of Z, f being Function of X,Y, h being Function of Y|B,Z|C st f is
  continuous & h is continuous & rng f c= B & B<>{} & C<>{} holds ex g being
  Function of X,Z st g is continuous & g=h*f
proof
  let X,Y,Z be non empty TopSpace, B be Subset of Y, C be Subset of Z, f be
  Function of X,Y, h be Function of Y|B,Z|C;
  assume that
A1: f is continuous and
A2: h is continuous and
A3: rng f c= B and
A4: B<>{} and
A5: C<>{};
A6: the carrier of X=dom f by FUNCT_2:def 1;
  the carrier of Y|B=[#](Y|B) .=B by PRE_TOPC:def 5;
  then reconsider u=f as Function of X,Y|B by A3,A6,FUNCT_2:2;
  reconsider V=B as non empty Subset of Y by A4;
  Y|V is non empty;
  then reconsider H=Y|B as non empty TopSpace;
  reconsider F=C as non empty Subset of Z by A5;
  reconsider k=u as Function of X,H;
  Z|F is non empty;
  then reconsider G=Z|C as non empty TopSpace;
  reconsider j=h as Function of H,G;
A7: the carrier of (Z|C)=[#](Z|C) .=C by PRE_TOPC:def 5;
  j*k is Function of X,G;
  then reconsider v=h*u as Function of X,Z by A7,FUNCT_2:7;
  u is continuous by A1,TOPMETR:6;
  then v is continuous by A2,A4,A5,PRE_TOPC:26;
  hence thesis;
end;
