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

theorem Th17:
  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) = (1-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
  ) = (1-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
      .= (1-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:  1-i > 0;
      then --(1-i) > -0;
      then -(1-i) < 0;
      then
A10:  (i-1)*(2*(1-i)*|.fx.|) <= 0 by A9;
      set t = 2*(1-i)*|.fx.|+r0;
      set c = (1-i)*r0 / t;
      i+c = i*t/t + ((1-i)*r0 / t) by A6,A9,XCMPLX_1:89
        .= (i*t + (1-i)*r0) / t by XCMPLX_1:62
        .= (i*2*(1-i)*|.fx.| + r0) / t;
      then i+c-1 = (i*2*(1-i)*|.fx.|+r0) / t - t / t by A6,A9,XCMPLX_1:60
        .= (i*2*(1-i)*|.fx.|+r0 - t) / t by XCMPLX_1:120;
      then
A11:  i+c-1+1 <= 0+1 by A6,A9,A10,XREAL_1:7;
      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/(1-i)) as Subset of TOP-REAL n by TOPREAL3:8;
      take W = [:B,X2:];
      0 < (1-i)*r0 by A6,A9,XREAL_1:129;
      then
A12:  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
A13:  i in X2 by XBOOLE_0:def 4;
      0 <= i by BORSUK_1:43;
      then
A14:  i+c is Point of I[01] by A6,A9,A11,BORSUK_1:43;
A15:  now
        per cases;
        suppose
A16:      0 <= i-c;
          X1 c= the carrier of I[01]
          proof
            let a be object;
            assume
A17:        a in X1;
            then reconsider a as Real;
            a < i+c by A17,XXREAL_1:4;
            then
A18:        a < 1 by A11,XXREAL_0:2;
            0 < a by A16,A17,XXREAL_1:4;
            hence thesis by A18,BORSUK_1:43;
          end;
          then reconsider X1 as Subset of I[01];
          now
            per cases;
            suppose
              i-c <= i+c;
              then i-c <= 1 by A11,XXREAL_0:2;
              then i-c is Point of I[01] by A16,BORSUK_1:43;
              hence X1 is open by A14,BORSUK_4:45;
            end;
            suppose
              i-c > i+c;
              then X1 = {}I[01] by XXREAL_1:28;
              hence X1 is open;
            end;
          end;
          hence X2 is open by A3;
        end;
        suppose
A19:      i-c < 0;
          X2 = [. 0, i+c .[
          proof
            hereby
              let a be object;
              assume
A20:          a in X2;
              then reconsider b = a as Real;
              a in X1 by A20,XBOOLE_0:def 4;
              then
A21:          b < i+c by XXREAL_1:4;
              0 <= b by A20,BORSUK_1:43;
              hence a in [. 0,i+c.[ by A21,XXREAL_1:3;
            end;
            let a be object;
            assume
A22:        a in [. 0,i+c .[;
            then reconsider b = a as Real;
A23:        0 <= b by A22,XXREAL_1:3;
A24:        b < i+c by A22,XXREAL_1:3;
            then b <= 1 by A11,XXREAL_0:2;
            then
A25:        a in I by A23,BORSUK_1:40,XXREAL_1:1;
            a in X1 by A19,A24,A23,XXREAL_1:4;
            hence thesis by A25,XBOOLE_0:def 4;
          end;
          hence X2 is open by A14,Th5;
        end;
      end;
      x in B by A8,A9,GOBOARD6:1,XREAL_1:139;
      hence p in W by A4,A13,ZFMISC_1:87;
      B is open by GOBOARD6:3;
      hence W is open by A15,BORSUK_1:6;
A26:  0 < 2*(1-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
A27:    z in dom F and
A28:    z in W and
A29:    F.z = m by FUNCT_1:def 6;
        reconsider z as Point of [:TOP-REAL n,I[01]:] by A27;
        consider y being Point of TOP-REAL n, j being Point of I[01] such that
A30:    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 = (1-i)*y, fy = y as Element of REAL n
        by EUCLID:22;
A31:    (1-i) * (r0/(1-i)/2) = r0/2 & r0/2/(1-i) = r0/(1-i)/2 by A9,
XCMPLX_1:48,97;
        fy in B by A28,A30,ZFMISC_1:87;
        then
A32:    dist(lx,ey) < r0/2/(1-i) by METRIC_1:11;
        j in X2 by A28,A30,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
A33:    |.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 A30
          .= (1-j)*y by A1;
        then fe-fz = (1-i)*yy -(1-j)*yy;
        then
A34:    |.fe-fz.| = |.(1-i-(1-j))*fy.| by Th8
          .= |.j-i.|*|.fy.| by EUCLID:11
          .= |.i-j.|*|.fy.| by UNIFORM1:11;
        reconsider gx = fx, gy=fy as Element of n-tuples_on REAL by
EUCLID:def 1;
A35:    dist(ep,ez) = |.fp-fz.| & |.fp-fz.| <= |.fp-fe.| + |.fe-fz.| by
EUCLID:19,SPPOL_1:5;
A36:    (1-i)*(fx-fy) = (1-i)*(gx-gy) .= (1-i)*gx-(1-i)*gy by Th7
          .= (1-i)*fx-(1-i)*fy
          .= (1-i)*fx-fe;
A37:    dist(lx,ey) = |.fx-fy.| by SPPOL_1:5;
        then (1-i) * |.fx-fy.| < (1-i) * (r0/2/(1-i)) by A9,A32,XREAL_1:68;
        then |.1-i.| * |.fx-fy.| < r0/2 by A9,A31,ABSVALUE:def 1;
        then
A38:    |.(1-i)*fx-fe.| < r0/2 by A36,EUCLID:11;
        |.fx-fy.| = |.fy-fx.| & |.fy.| - |.fx.| <= |.fy-fx.| by EUCLID:15,18;
        then |.fy.| - |.fx.| < r0/2/(1-i) by A32,A37,XXREAL_0:2;
        then |.fy.| < |.fx.| + r0/2/(1-i) by XREAL_1:19;
        then
A39:    c*|.fy.| < c * (|.fx.| + r0/2/(1-i)) by A12,XREAL_1:68;
        c * (|.fx.| + r0/2/(1-i)) = c * (|.fx.| + r0/(2*(1-i))) by XCMPLX_1:78
          .= c * ((|.fx.|*(2*(1-i)))/(2*(1-i))+r0/(2*(1-i))) by A26,XCMPLX_1:89
          .= c * ((|.fx.|*(2*(1-i))+r0)/(2*(1-i))) by XCMPLX_1:62
          .= (1-i)*r0 / (2*(1-i)) by A6,A9,XCMPLX_1:98
          .= r0/2 by A9,XCMPLX_1:91;
        then
A40:    |.i-j.|*|.fy.| <= r0/2 by A33,A39,XXREAL_0:2;
        fp = (1-i)*x by A5
          .= (1-i)*fx;
        then |.fp-fe.| + |.fe-fz.| < r0/2 + r0/2 by A34,A40,A38,XREAL_1:8;
        then dist(ep,ez) < r0 by A35,XXREAL_0:2;
        hence thesis by A29,METRIC_1:11;
      end;
      hence thesis by A7;
    end;
    suppose
A41:  1-i <= 0;
A42:  i <= 1 by BORSUK_1:43;
      1-i+i <= 0+i by A41,XREAL_1:6;
      then
A43:  i = 1 by A42,XXREAL_0:1;
      set t = |.fx.|+r0;
      reconsider B = Ball(lx,r0) as Subset of TOP-REAL n by TOPREAL3:8;
      set c = r0 / t;
      set X1 = ]. 1-c, 1 .];
A44:  x in B by A6,GOBOARD6:1;
      0+r0 <= t by XREAL_1:7;
      then
A45:  c <= 1 by A6,XREAL_1:185;
      then
A46:  c-c <= 1-c by XREAL_1:9;
      X1 c= I
      proof
        let s be object;
        assume
A47:    s in X1;
        then reconsider s as Real;
        s <= 1 & 1-c < s by A47,XXREAL_1:2;
        hence thesis by A46,BORSUK_1:43;
      end;
      then reconsider X1 as Subset of I[01];
      c is Point of I[01] by A6,A45,BORSUK_1:43;
      then 1-c is Point of I[01] by JORDAN5B:4;
      then
A48:  X1 is open by Th4;
      take W = [:B,X1:];
A49:  0 < c by A6,XREAL_1:139;
      then 1-c < 1-0 by XREAL_1:15;
      then i in X1 by A43,XXREAL_1:2;
      hence p in W by A4,A44,ZFMISC_1:87;
      B is open by GOBOARD6:3;
      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
A50:    z in dom F and
A51:    z in W and
A52:    F.z = m by FUNCT_1:def 6;
        reconsider z as Point of [:TOP-REAL n,I[01]:] by A50;
        consider y being Point of TOP-REAL n, j being Point of I[01] such that
A53:    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 A51,A53,ZFMISC_1:87;
        then
A54:    dist(lx,ey) < r0 by METRIC_1:11;
A55:    ez = F.(y,j) by A53
          .= (1-j)*y by A1;
        fp = (1-i)*x by A5
          .= 0.TOP-REAL n by A43,RLVECT_1:10;
        then
A56:    fz-fp = F.z - 0.TOP-REAL n
          .= fz by RLVECT_1:13;
A57:    |.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 A54,A57,XXREAL_0:2;
        then
A58:    |.fy.| < t by XREAL_1:19;
A59:    now
          per cases;
          suppose
            j < 1;
            then
A60:        j-j < 1-j by XREAL_1:14;
            j in X1 by A51,A53,ZFMISC_1:87;
            then 1-c < j by XXREAL_1:2;
            then 1-c+c < j+c by XREAL_1:8;
            then 1-j < j+c-j by XREAL_1:14;
            then r0/(1-j) > r0/c by A6,A60,XREAL_1:76;
            then t < r0/(1-j) by A49,XCMPLX_1:54;
            then |.fy.| < r0/(1-j) by A58,XXREAL_0:2;
            then (1-j)*|.fy.| < (1-j)*(r0/(1-j)) by A60,XREAL_1:68;
            hence (1-j)*|.fy.| < r0 by A60,XCMPLX_1:87;
          end;
          suppose
A61:        j >= 1;
            j <= 1 by BORSUK_1:43;
            then j = 1 by A61,XXREAL_0:1;
            hence (1-j)*|.fy.| < r0 by A6;
          end;
        end;
        1-j is Point of I[01] by JORDAN5B:4;
        then
A62:    0 <= 1-j by BORSUK_1:43;
        dist(ep,ez) = |.fz-fp.| by SPPOL_1:5
          .= |.fz.| by A56
          .= |.(1-j)*fy.| by A55
          .= |.1-j.|*|.fy.| by EUCLID:11
          .= (1-j)*|.fy.| by A62,ABSVALUE:def 1;
        hence thesis by A52,A59,METRIC_1:11;
      end;
      hence thesis by A7;
    end;
  end;
  hence thesis by JGRAPH_2:10;
end;
