reserve p, q, x, y for Real,
  n for Nat;

theorem Th15:
  for X being non empty TopSpace, f, g being Function of X,
  TOP-REAL n st f is continuous & for p being Point of X holds g.p = y * f.p
  holds g is continuous
proof
  let X being non empty TopSpace, f, g be Function of X,TOP-REAL n such that
A1: f is continuous and
A2: for p being Point of X holds g.p = y * f.p;
  for p being Point of X, V being Subset of TOP-REAL n st g.p in V & V is
  open ex W being Subset of X st p in W & W is open & g.:W c= V
  proof
    let p be Point of X, V be Subset of TOP-REAL n;
    reconsider r = g.p as Point of Euclid n by TOPREAL3:8;
    reconsider r1 = f.p as Point of Euclid n by TOPREAL3:8;
    assume g.p in V & V is open;
    then g.p in Int V by TOPS_1:23;
    then consider r0 being Real such that
A3: r0 > 0 and
A4: Ball(r,r0) c= V by GOBOARD6:5;
    reconsider G1 = Ball(r1,r0/|.y.|) as Subset of TOP-REAL n by TOPREAL3:8;
    per cases;
    suppose
A5:   y <> 0;
A6:   G1 is open by GOBOARD6:3;
A7:   0 < |.y.| by A5,COMPLEX1:47;
      then r1 in G1 by A3,GOBOARD6:1,XREAL_1:139;
      then consider W1 being Subset of X such that
A8:   p in W1 and
A9:  W1 is open and
A10:  f.:W1 c= G1 by A1,A6,JGRAPH_2:10;
      take W1;
      thus p in W1 by A8;
      thus W1 is open by A9;
      g.:W1 c= Ball(r,r0)
      proof
        let x be object;
        assume x in g.:W1;
        then consider z being object such that
A11:    z in dom g and
A12:    z in W1 and
A13:    g.z = x by FUNCT_1:def 6;
        reconsider z as Point of X by A11;
A14:    x = y * f.z by A2,A13;
        then reconsider e1x = x as Point of Euclid n by TOPREAL3:8;
        reconsider ea1 = f.z as Point of Euclid n by TOPREAL3:8;
        z in the carrier of X;
        then z in dom f by FUNCT_2:def 1;
        then f.z in f.:W1 by A12,FUNCT_1:def 6;
        then
A15:    dist(r1,ea1) < r0/|.y.| by A10,METRIC_1:11;
        r = y * f.p by A2;
        then dist(r,e1x) < |.y.|*(r0/|.y.|) by A5,A14,A15,Th13;
        then dist(r,e1x) < r0 by A7,XCMPLX_1:87;
        hence thesis by METRIC_1:11;
      end;
      hence thesis by A4;
    end;
    suppose
A16:  y = 0;
A17:  r = y * f.p by A2
        .= 0.TOP-REAL n by A16,RLVECT_1:10;
      take W = [#]X;
      thus p in W;
      thus W is open;
      let x be object;
      assume x in g.:W;
      then consider z being object such that
      z in dom g and
A18:  z in W and
A19:  g.z = x by FUNCT_1:def 6;
      reconsider z as Point of X by A18;
      x = y * f.z by A2,A19
        .= 0.TOP-REAL n by A16,RLVECT_1:10;
      then x in Ball(r,r0) by A3,A17,GOBOARD6:1;
      hence thesis by A4;
    end;
  end;
  hence thesis by JGRAPH_2:10;
end;
