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

theorem Th18:
  for F being Function of [:TOP-REAL n,I[01]:], TOP-REAL n st for
x being Point of TOP-REAL n, i being Point of I[01] holds F.(x,i) = i * x holds
  F is continuous
proof
  set I = the carrier of I[01];
  let F be Function of [:TOP-REAL n,I[01]:], TOP-REAL n such that
A1: for x being Point of TOP-REAL n, i being Point of I[01] holds F.(x,i
  ) = i * x;
A2: REAL n = n-tuples_on REAL by EUCLID:def 1;
A3: [#]I[01] = I;
  for p being Point of [:TOP-REAL n,I[01]:], V being Subset of TOP-REAL n
st F.p in V & V is open ex W being Subset of [:TOP-REAL n,I[01]:] st p in W & W
  is open & F.:W c= V
  proof
    let p be Point of [:TOP-REAL n,I[01]:], V be Subset of TOP-REAL n;
    reconsider ep = F.p as Point of Euclid n by TOPREAL3:8;
    consider x being Point of TOP-REAL n, i being Point of I[01] such that
A4: p = [x,i] by BORSUK_1:10;
A5: ep = F.(x,i) by A4
      .= i*x by A1;
    reconsider fx = x as Element of REAL n by EUCLID:22;
    reconsider lx = x as Point of Euclid n by TOPREAL3:8;
    assume F.p in V & V is open;
    then F.p in Int V by TOPS_1:23;
    then consider r0 being Real such that
A6: r0 > 0 and
A7: Ball(ep,r0) c= V by GOBOARD6:5;
A8: r0/2 > 0 by A6,XREAL_1:139;
    per cases;
    suppose
A9:  i > 0;
      set t = 2*i*|.fx.|+r0;
      set c = i*r0 / t;
      i <= 1 by BORSUK_1:43;
      then 1-1 >= i-1 by XREAL_1:9;
      then 2*i*|.fx.| * (i-1) <= 0 by A9;
      then
A10:  i*(2*i*|.fx.|) - 2*i*|.fx.| - r0 < r0 - r0 by A6;
A11:  i-c = i*t/t - (i*r0 / t) by A6,A9,XCMPLX_1:89
        .= (i*t - i*r0) / t by XCMPLX_1:120
        .= i*(2*i*|.fx.|) / t;
      then i-c-1 = i*(2*i*|.fx.|) / t - t / t by A6,A9,XCMPLX_1:60
        .= (i*(2*i*|.fx.|) - t) / t by XCMPLX_1:120;
      then i-c-1 < 0 by A6,A9,A10,XREAL_1:141;
      then i-c-1+1 < 0+1 by XREAL_1:8;
      then
A12:  i-c is Point of I[01] by A6,A9,A11,BORSUK_1:43;
      set X1 = ]. i-c, i+c .[;
      set X2 = X1 /\ I;
      reconsider X2 as Subset of I[01] by XBOOLE_1:17;
      reconsider B = Ball(lx,r0/2/i) as Subset of TOP-REAL n by TOPREAL3:8;
      take W = [:B,X2:];
      0 < i*r0 by A6,A9,XREAL_1:129;
      then
A13:  0 < c by A6,A9,XREAL_1:139;
      then |.i-i.| < c by ABSVALUE:def 1;
      then i in X1 by RCOMP_1:1;
      then
A14:  i in X2 by XBOOLE_0:def 4;
A15:  0 <= i-c by A6,A9,A11;
A16:  now
        per cases;
        suppose
A17:      i+c <= 1;
          X1 c= the carrier of I[01]
          proof
            let a be object;
            assume
A18:        a in X1;
            then reconsider a as Real;
            a < i+c by A18,XXREAL_1:4;
            then
A19:        a < 1 by A17,XXREAL_0:2;
            0 < a by A6,A9,A11,A18,XXREAL_1:4;
            hence thesis by A19,BORSUK_1:43;
          end;
          then reconsider X1 as Subset of I[01];
          i+c is Point of I[01] by A6,A9,A17,BORSUK_1:43;
          then X1 is open by A12,BORSUK_4:45;
          hence X2 is open by A3;
        end;
        suppose
A20:      i+c > 1;
          X2 = ]. i-c, 1 .]
          proof
            hereby
              let a be object;
              assume
A21:          a in X2;
              then reconsider b = a as Real;
              a in X1 by A21,XBOOLE_0:def 4;
              then
A22:          i-c < b by XXREAL_1:4;
              b <= 1 by A21,BORSUK_1:43;
              hence a in ]. i-c, 1 .] by A22,XXREAL_1:2;
            end;
            let a be object;
            assume
A23:        a in ]. i-c, 1 .];
            then reconsider b = a as Real;
A24:        i-c < b by A23,XXREAL_1:2;
A25:        b <= 1 by A23,XXREAL_1:2;
            then b < i+c by A20,XXREAL_0:2;
            then
A26:        a in X1 by A24,XXREAL_1:4;
            a in I by A15,A24,A25,BORSUK_1:40,XXREAL_1:1;
            hence thesis by A26,XBOOLE_0:def 4;
          end;
          hence X2 is open by A12,Th4;
        end;
      end;
      x in B by A8,A9,GOBOARD6:1,XREAL_1:139;
      hence p in W by A4,A14,ZFMISC_1:87;
      B is open by GOBOARD6:3;
      hence W is open by A16,BORSUK_1:6;
A27:  0 < 2*i by A9,XREAL_1:129;
      F.:W c= Ball(ep,r0)
      proof
        let m be object;
        assume m in F.:W;
        then consider z being object such that
A28:    z in dom F and
A29:    z in W and
A30:    F.z = m by FUNCT_1:def 6;
        reconsider z as Point of [:TOP-REAL n,I[01]:] by A28;
        consider y being Point of TOP-REAL n, j being Point of I[01] such that
A31:    z = [y,j] by BORSUK_1:10;
        reconsider ez = F.z, ey = y as Point of Euclid n by TOPREAL3:8;
        reconsider fp = ep, fz = ez, fe = i*y, fy = y as Element of REAL n by
EUCLID:22;
A32:    i * (r0/i/2) = r0/2 & r0/2/i = r0/i/2 by A9,XCMPLX_1:48,97;
        fy in B by A29,A31,ZFMISC_1:87;
        then
A33:    dist(lx,ey) < r0/2/i by METRIC_1:11;
        j in X2 by A29,A31,ZFMISC_1:87;
        then j in X1 by XBOOLE_0:def 4;
        then |.j-i.| < c by RCOMP_1:1;
        then |.i-j.| < c by UNIFORM1:11;
        then
A34:    |.i-j.|*|.fy.| <= c*|.fy.| by XREAL_1:64;
        reconsider yy=ey as Element of n-tuples_on REAL by A2,EUCLID:22;
        ez = F.(y,j) by A31
          .= j*y by A1;
        then fe-fz = i*yy -j*yy;
        then
A35:    |.fe-fz.| = |.(i-j)*fy.| by Th8
          .= |.i-j.|*|.fy.| by EUCLID:11;
        reconsider yy=y as Element of n-tuples_on REAL by A2,EUCLID:22;
A36:    dist(ep,ez) = |.fp-fz.| & |.fp-fz.| <= |.fp-fe.| + |.fe-fz.| by
EUCLID:19,SPPOL_1:5;
A37:    dist(lx,ey) = |.fx-fy.| by SPPOL_1:5;
        then i * |.fx-fy.| < i * (r0/2/i) by A9,A33,XREAL_1:68;
        then
A38:    |.i.| * |.fx-fy.| < r0/2 by A9,A32,ABSVALUE:def 1;
        |.fx-fy.| = |.fy-fx.| & |.fy.| - |.fx.| <= |.fy-fx.| by EUCLID:15,18;
        then |.fy.| - |.fx.| < r0/2/i by A33,A37,XXREAL_0:2;
        then |.fy.| < |.fx.| + r0/2/i by XREAL_1:19;
        then
A39:    c*|.fy.| < c * (|.fx.| + r0/2/i) by A13,XREAL_1:68;
        c * (|.fx.| + r0/2/i) = c * (|.fx.| + r0/(2*i)) by XCMPLX_1:78
          .= c * ((|.fx.|*(2*i))/(2*i)+r0/(2*i)) by A27,XCMPLX_1:89
          .= c * ((|.fx.|*(2*i)+r0)/(2*i)) by XCMPLX_1:62
          .= i*r0 / (2*i) by A6,A9,XCMPLX_1:98
          .= r0/2 by A9,XCMPLX_1:91;
        then
A40:    |.i-j.|*|.fy.| <= r0/2 by A34,A39,XXREAL_0:2;
        i*fx-fe = i*fx-i*yy
          .= i*(fx-fy) by Th7;
        then
A41:    |.i*fx-fe.| < r0/2 by A38,EUCLID:11;
        i*fx-fe = fp-fe by A5;
        then |.fp-fe.| + |.fe-fz.| < r0/2 + r0/2 by A35,A40,A41,XREAL_1:8;
        then dist(ep,ez) < r0 by A36,XXREAL_0:2;
        hence thesis by A30,METRIC_1:11;
      end;
      hence thesis by A7;
    end;
    suppose
A42:  i <= 0;
      set t = |.fx.|+r0;
      reconsider B = Ball(lx,r0) as Subset of TOP-REAL n by TOPREAL3:8;
      set c = r0 / t;
      set X1 = [. 0, c .[;
A43:  0 < c by A6,XREAL_1:139;
      0+r0 <= t by XREAL_1:7;
      then
A44:  c <= 1 by A6,XREAL_1:185;
A45:  X1 c= I
      proof
        let s be object;
        assume
A46:    s in X1;
        then reconsider s as Real;
        s < c by A46,XXREAL_1:3;
        then
A47:    s <= 1 by A44,XXREAL_0:2;
        0 <= s by A46,XXREAL_1:3;
        hence thesis by A47,BORSUK_1:43;
      end;
A48:  B is open by GOBOARD6:3;
      reconsider X1 as Subset of I[01] by A45;
      take W = [:B,X1:];
A49:  x in B by A6,GOBOARD6:1;
A50:  i = 0 by A42,BORSUK_1:43;
      then i in X1 by A43,XXREAL_1:3;
      hence p in W by A4,A49,ZFMISC_1:87;
      c is Point of I[01] by A6,A44,BORSUK_1:43;
      then X1 is open by Th5;
      hence W is open by A48,BORSUK_1:6;
      F.:W c= Ball(ep,r0)
      proof
        let m be object;
        assume m in F.:W;
        then consider z being object such that
A51:    z in dom F and
A52:    z in W and
A53:    F.z = m by FUNCT_1:def 6;
        reconsider z as Point of [:TOP-REAL n,I[01]:] by A51;
        consider y being Point of TOP-REAL n, j being Point of I[01] such that
A54:    z = [y,j] by BORSUK_1:10;
        reconsider ez = F.z, ey = y as Point of Euclid n by TOPREAL3:8;
        reconsider fp = ep, fz = ez, fy = y as Element of REAL n by EUCLID:22;
        fy in B by A52,A54,ZFMISC_1:87;
        then
A55:    dist(lx,ey) < r0 by METRIC_1:11;
A56:    ez = F.(y,j) by A54
          .= j*y by A1;
        fp = i*x by A5
          .= 0.TOP-REAL n by A50,RLVECT_1:10;
        then
A57:    fz-fp = F.z - 0.TOP-REAL n
          .= fz by RLVECT_1:13;
A58:    |.fy.| - |.fx.| <= |.fy-fx.| by EUCLID:15;
        dist(lx,ey) = |.fx-fy.| & |.fx-fy.| = |.fy-fx.| by EUCLID:18
,SPPOL_1:5;
        then |.fy.| - |.fx.| < r0 by A55,A58,XXREAL_0:2;
        then
A59:    |.fy.| < t by XREAL_1:19;
A60:    now
          per cases;
          suppose
A61:        0 < j;
            j in X1 by A52,A54,ZFMISC_1:87;
            then j < c by XXREAL_1:3;
            then r0/j > r0/c by A6,A61,XREAL_1:76;
            then t < r0/j by A43,XCMPLX_1:54;
            then |.fy.| < r0/j by A59,XXREAL_0:2;
            then j*|.fy.| < j*(r0/j) by A61,XREAL_1:68;
            hence j*|.fy.| < r0 by A61,XCMPLX_1:87;
          end;
          suppose
            j <= 0;
            hence j*|.fy.| < r0 by A6;
          end;
        end;
A62:    0 <= j by BORSUK_1:43;
        dist(ep,ez) = |.fz-fp.| by SPPOL_1:5
          .= |.fz.| by A57
          .= |.j*fy.| by A56
          .= |.j.|*|.fy.| by EUCLID:11
          .= j*|.fy.| by A62,ABSVALUE:def 1;
        hence thesis by A53,A60,METRIC_1:11;
      end;
      hence thesis by A7;
    end;
  end;
  hence thesis by JGRAPH_2:10;
end;
