
theorem
  for S, T be TopSpace, Y being non empty TopSpace, A being Subset of S,
f being Function of [:S,T:],Y, g being Function of [:S|A,T:],Y st g = f | [:A,
  the carrier of T:] & f is continuous holds g is continuous
proof
  let S, T be TopSpace, Y be non empty TopSpace;
  let A be Subset of S;
  let f be Function of [:S,T:],Y;
  let g be Function of [:S|A,T:],Y;
  assume
A1: g = f | [:A,the carrier of T:] & f is continuous;
  set TT = the TopStruct of T;
A2: [:S|A,TT:] = [:S|A,TT| [#]TT:] by TSEP_1:3
    .= [:S,TT:]| [:A,[#]TT:] by BORSUK_3:22;
  the TopStruct of [:S,T:] = [:the TopStruct of S,the TopStruct of T:] by Th13;
  then
A3: the TopStruct of [:S,TT:] = the TopStruct of [:S,T:] by Th13;
  the TopStruct of [:S|A,TT:] = the TopStruct of [:S|A,T:] by Th13;
  hence thesis by A1,A3,A2,TOPMETR:7;
end;
