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