reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;

theorem Th69:
  for g being Function of I[01],TOP-REAL n st g is continuous
holds ex f being Function of I[01],R^1 st (for t being Point of I[01] holds f.t
  =|.g.t.|) & f is continuous
proof
  let g be Function of I[01],TOP-REAL n;
  consider h being Function of TOP-REAL n,R^1 such that
A1: for q holds h.q=|.q.| and
A2: h is continuous by Th68;
  set f1=h*g;
A3: for t being Point of I[01] holds f1.t=|.g.t.|
  proof
    let t be Point of I[01];
    reconsider q=g.t as Point of TOP-REAL n;
    dom g = the carrier of I[01] by FUNCT_2:def 1;
    then f1.t=h.(g.t) by FUNCT_1:13
      .= |.q.| by A1;
    hence thesis;
  end;
  assume g is continuous;
  hence thesis by A2,A3;
end;
