reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th29:
  for K0 being Subset of TOP-REAL 2, f being Function of (TOP-REAL
  2)|K0,R^1 st (for p being Point of (TOP-REAL 2)|K0 holds f.p=proj1.p) holds f
  is continuous
proof
  reconsider g=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
  let K0 be Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K0,R^1;
A1: dom f= the carrier of (TOP-REAL 2)|K0 & (the carrier of TOP-REAL 2)/\K0=
  K0 by FUNCT_2:def 1,XBOOLE_1:28;
A2: g is continuous by JORDAN5A:27;
  assume for p being Point of (TOP-REAL 2)|K0 holds f.p=proj1.p;
  then
A3: for x being object st x in dom f holds f.x=proj1.x;
  the carrier of (TOP-REAL 2)|K0 =[#]((TOP-REAL 2)|K0) .=K0 by PRE_TOPC:def 5;
  then f=g|K0 by A1,A3,Th6,FUNCT_1:46;
  hence thesis by A2,TOPMETR:7;
end;
