reserve x for set;
reserve a,b,d,ra,rb,r0,s1,s2 for Real;
reserve r,s,r1,r2,r3,rc for Real;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve X,Y,Z for non empty TopSpace;

theorem Th13:
  for P being non empty Subset of TOP-REAL 2, f being Function of
I[01], (TOP-REAL 2)|P st f is continuous ex g being Function of I[01],R^1 st g
  is continuous & for r,q st r in the carrier of I[01] & q= f.r holds q`2=g.r
proof
  reconsider rj=proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
  let P be non empty Subset of TOP-REAL 2, f be Function of I[01], (TOP-REAL 2
  )|P;
  assume
A1: f is continuous;
  reconsider f1=f as Function;
  set h=(proj2|P)*f;
A2: [#]((TOP-REAL 2)|P)=P by PRE_TOPC:def 5;
  then
A3: rng f1 c= P by RELAT_1:def 19;
  rj is continuous by Th11;
  then rj|((TOP-REAL 2)|P) is continuous Function of ((TOP-REAL 2)|P),R^1;
  then rj|P is continuous Function of ((TOP-REAL 2)|P),R^1 by A2,TMAP_1:def 3;
  then reconsider h2=h as continuous Function of I[01],R^1 by A1,Lm1;
  take h2;
  thus h2 is continuous;
  let r be Real,q be Point of TOP-REAL 2;
  assume that
A4: r in the carrier of I[01] and
A5: q= f.r;
A6: dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A7: r in dom f1 by A4,FUNCT_2:def 1;
  then f1.r in rng f1 by FUNCT_1:def 3;
  then
A8: q in dom proj2 /\ P by A5,A3,A6,XBOOLE_0:def 4;
  thus h2.r = (proj2|P).q by A5,A7,FUNCT_1:13
    .= proj2.q by A8,FUNCT_1:48
    .= q`2 by PSCOMP_1:def 6;
end;
