reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th9:
  for p1,p2 being Point of TOP-REAL n st p1 <> p2 holds
  LSeg(p1,p2) is_an_arc_of p1,p2
proof
  let p1,p2 be Point of TOP-REAL n;
  set P = LSeg(p1,p2);
  reconsider PP = P as Subset of Euclid n by EUCLID:67;
  PP is non empty;
  then reconsider PP = P as non empty Subset of Euclid n;
  reconsider p19 = p1, p29 = p2 as Element of REAL n by EUCLID:22;
  defpred X[object,object] means
for x being Real st x = $1 holds $2 = (1-x)*p1 + x*
  p2;
A1: for a being object st a in [.0,1.] ex u being object st X[a,u]
  proof
    let a be object;
    assume a in [.0,1.];
    then reconsider x = a as Real;
    take (1-x)*p1 + x*p2;
    thus thesis;
  end;
  consider f being Function such that
A2: dom f = [.0,1.] and
A3: for a being object st a in [.0,1.] holds X[a,f.a] from CLASSES1:sch 1(
  A1);
A4: rng f c= the carrier of (TOP-REAL n)|P
  proof
    let y be object;
    assume y in rng f;
    then consider x being object such that
A5: x in dom f and
A6: y = f.x by FUNCT_1:def 3;
    x in {r: 0 <= r & r <= 1} by A2,A5,RCOMP_1:def 1;
    then consider r such that
A7: r = x and
A8: 0 <= r and
A9: r <= 1;
    y = (1-r)*p1 + r*p2 by A2,A3,A5,A6,A7;
    then y in { (1-lambda)*p1 + lambda*p2: 0 <= lambda & lambda <= 1 } by A8
,A9;
    then y in [#]((TOP-REAL n)|P) by PRE_TOPC:def 5;
    hence thesis;
  end;
  then reconsider f as Function of I[01], (TOP-REAL n)|P by A2,BORSUK_1:40
,FUNCT_2:def 1,RELSET_1:4;
A10: I[01] is compact by HEINE:4,TOPMETR:20;
  assume
A11: p1 <> p2;
  now
    let x1,x2 be object;
    assume that
A12: x1 in dom f and
A13: x2 in dom f and
A14: f.x1 = f.x2;
    x2 in {r: 0 <= r & r <= 1} by A2,A13,RCOMP_1:def 1;
    then consider r2 being Real such that
A15: r2 = x2 and
    0 <= r2 and
    r2 <= 1;
A16: f.x2 = (1-r2)*p1 + r2*p2 by A2,A3,A13,A15;
    x1 in {r: 0 <= r & r <= 1} by A2,A12,RCOMP_1:def 1;
    then consider r1 being Real such that
A17: r1 = x1 and
    0 <= r1 and
    r1 <= 1;
    f.x1 = (1-r1)*p1 + r1*p2 by A2,A3,A12,A17;
    then
    (1-r1)*p1 + r1*p2 + (-r1)*p2 = (1-r2)*p1 + (r2*p2 + (-r1)*p2) by A14,A16,
RLVECT_1:def 3;
    then (1-r1)*p1 + r1*p2 + (-r1)*p2 = (1-r2)*p1 + (r2+(-r1))*p2 by
RLVECT_1:def 6;
    then (1-r1)*p1 + (r1*p2 + (-r1)*p2) = (1-r2)*p1 + (r2-r1)*p2 by
RLVECT_1:def 3;
    then (1-r1)*p1 + (r1+(-r1))*p2 = (1-r2)*p1 + (r2-r1)*p2 by RLVECT_1:def 6;
    then (1-r1)*p1 + 0.TOP-REAL n = (1-r2)*p1 + (r2-r1)*p2 by RLVECT_1:10;
    then
    (-(1-r2))*p1 + (1-r1)*p1 = (-(1-r2))*p1 + ((1-r2)*p1 + (r2-r1)*p2) by
RLVECT_1:4;
    then
    (-(1-r2))*p1 + (1-r1)*p1 = ((-(1-r2))*p1 + (1-r2)*p1) + (r2-r1)*p2 by
RLVECT_1:def 3;
    then (-(1-r2))*p1 + (1-r1)*p1 = (-(1-r2)+ (1-r2))*p1 + (r2-r1)*p2 by
RLVECT_1:def 6;
    then (-(1-r2))*p1 + (1-r1)*p1 = 0.TOP-REAL n + (r2-r1)*p2 by RLVECT_1:10;
    then (-(1-r2))*p1 + (1-r1)*p1 = (r2-r1)*p2 by RLVECT_1:4;
    then ((r2-1)+(1-r1))*p1 = (r2-r1)*p2 by RLVECT_1:def 6;
    then r2-r1 = 0 by A11,RLVECT_1:36;
    hence x1 = x2 by A17,A15;
  end;
  then
A18: f is one-to-one;
  [#]((TOP-REAL n)|P) c= rng f
  proof
    let a be object;
    assume a in [#]((TOP-REAL n)|P);
    then a in P by PRE_TOPC:def 5;
    then consider lambda such that
A19: a = (1-lambda)*p1 + lambda*p2 and
A20: 0 <= lambda and
A21: lambda <= 1;
    lambda in { r: 0 <= r & r <= 1} by A20,A21;
    then
A22: lambda in dom f by A2,RCOMP_1:def 1;
    then a = f.lambda by A2,A3,A19;
    hence thesis by A22,FUNCT_1:def 3;
  end;
  then
A23: rng f = [#]((TOP-REAL n)|P) by A4;
A24: TopSpaceMetr(Euclid n) is T_2 by PCOMPS_1:34;
A25: (TOP-REAL n)|P = TopSpaceMetr((Euclid n)|PP) by EUCLID:63;
  for W being Point of I[01], G being a_neighborhood of f.W ex H being
  a_neighborhood of W st f.:H c= G
  proof
    reconsider p11=p1,p22=p2 as Point of Euclid n by EUCLID:67;
    let W be Point of I[01], G be a_neighborhood of f.W;
    reconsider W9 = W as Point of Closed-Interval-MSpace(0,1) by BORSUK_1:40
,TOPMETR:10;
A26: (Pitag_dist n).(p19,p29) = dist(p11,p22) by METRIC_1:def 1;
    [#]((TOP-REAL n)|P) = P by PRE_TOPC:def 5;
    then reconsider Y = f.W as Point of (Euclid n)|PP by TOPMETR:def 2;
A27: dist(p11,p22) >= 0 by METRIC_1:5;
    reconsider W1 = W as Real;
    P = the carrier of (Euclid n)|PP by TOPMETR:def 2;
    then reconsider WW9 = Y as Point of Euclid n by TARSKI:def 3;
    f.W in Int G by CONNSP_2:def 1;
    then consider Q being Subset of (TOP-REAL n)|P such that
A28: Q is open and
A29: Q c= G and
A30: f.W in Q by TOPS_1:22;
    consider r being Real such that
A31: r > 0 and
A32: Ball(Y,r) c= Q by A25,A28,A30,TOPMETR:15;
    reconsider r as Element of REAL by XREAL_0:def 1;
    set delta = r/(Pitag_dist n).(p19,p29);
    reconsider H = Ball(W9,delta) as Subset of I[01] by BORSUK_1:40,TOPMETR:10;
    Closed-Interval-TSpace(0,1) = TopSpaceMetr(Closed-Interval-MSpace(0,1
    )) by TOPMETR:def 7;
    then
A33: H is open by TOPMETR:14,20;
A34: dist(p11,p22) <> 0 by A11,METRIC_1:2;
    then W in H by A31,A26,A27,TBSP_1:11,XREAL_1:139;
    then W in Int H by A33,TOPS_1:23;
    then reconsider H as a_neighborhood of W by CONNSP_2:def 1;
    take H;
A35: delta > 0 by A31,A26,A27,A34,XREAL_1:139;
    f.:H c= Ball(Y,r)
    proof
      reconsider WW1 = WW9 as Element of REAL n;
      let a be object;
A36:  rng f c= the carrier of (TOP-REAL n)|P by RELAT_1:def 19;
      assume a in f.:H;
      then consider u being object such that
A37:  u in dom f and
A38:  u in H and
A39:  a = f.u by FUNCT_1:def 6;
      reconsider u1 = u as Real by A2,A37;
A40:  [#] ((TOP-REAL n)|P) = P by PRE_TOPC:def 5;
      reconsider u9 = u as Point of Closed-Interval-MSpace(0,1) by A38;
      reconsider W99 = W9, u99 = u9 as Point of RealSpace by TOPMETR:8;
A41:  dist(W9,u9) = (the distance of Closed-Interval-MSpace(0,1)).(W9,u9
      ) by METRIC_1:def 1
        .= (the distance of RealSpace).(W99,u99) by TOPMETR:def 1
        .= dist(W99,u99) by METRIC_1:def 1;
      dist(W9,u9) < delta by A38,METRIC_1:11;
      then |.W1-u1.| < delta by A41,TOPMETR:11;
      then |.-(u1-W1).| < delta;
      then
A42:  |.u1-W1.| < delta by COMPLEX1:52;
A43:  delta <> 0 by A31,A26,A27,A34,XREAL_1:139;
      then (Pitag_dist n).(p19,p29) = r/delta by XCMPLX_1:54;
      then
A44:  |.p19 - p29.| = r/delta by EUCLID:def 6;
      f.u in rng f by A37,FUNCT_1:def 3;
      then reconsider aa = a as Point of (Euclid n)|PP by A39,A36,A40,
TOPMETR:def 2;
      P = the carrier of (Euclid n)|PP by TOPMETR:def 2;
      then reconsider aa9 = aa as Point of Euclid n by TARSKI:def 3;
      reconsider aa1 = aa9 as Element of REAL n;
A45:  p19 - p29 = p1 - p2 by EUCLID:69;
A46:  WW1 = (1-W1)*p1 + W1*p2 by A3,BORSUK_1:40;
      aa1 = ((1-u1)*p1 + u1*p2) by A2,A3,A37,A39;
      then WW1 - aa1 = (1-W1)*p1 + W1*p2 - ((1-u1)*p1 + u1*p2) by A46,EUCLID:69
        .= W1*p2 + (1-W1)*p1 - (1-u1)*p1 - u1*p2 by RLVECT_1:27
        .= W1*p2 + ((1-W1)*p1 - (1-u1)*p1) - u1*p2 by RLVECT_1:def 3
        .= W1*p2 + ((1-W1)-(1-u1))*p1 - u1*p2 by RLVECT_1:35
        .= (u1-W1)*p1 + (W1*p2 - u1*p2) by RLVECT_1:def 3
        .= (u1-W1)*p1 + (W1-u1)*p2 by RLVECT_1:35
        .= (u1-W1)*p1 + (-(u1-W1))*p2
        .= (u1-W1)*p1 + -(u1-W1)*p2 by RLVECT_1:79
        .= (u1-W1)*p1 - (u1-W1)*p2
        .= (u1-W1)*(p1 - p2) by RLVECT_1:34
        .= (u1-W1)*(p19 - p29) by A45,EUCLID:65;
      then
A47:  |. WW1 -aa1 .| = |.u1-W1.|*|.p19 - p29.| by EUCLID:11;
      r/delta > 0 by A31,A35,XREAL_1:139;
      then |. WW1 - aa1.| < delta*(r/delta) by A47,A42,A44,XREAL_1:68;
      then |. WW1 - aa1 .| < r by A43,XCMPLX_1:87;
      then (the distance of Euclid n).(WW9,aa9) < r by EUCLID:def 6;
      then (the distance of (Euclid n)|PP).(Y,aa) < r by TOPMETR:def 1;
      then dist(Y,aa) < r by METRIC_1:def 1;
      hence thesis by METRIC_1:11;
    end;
    then f.:H c= Q by A32;
    hence thesis by A29;
  end;
  then
A48: f is continuous by BORSUK_1:def 1;
  take f;
A49: the TopStruct of TOP-REAL n = TopSpaceMetr(Euclid n) by EUCLID:def 8;
  then reconsider PP = P as Subset of TopSpaceMetr(Euclid n);
  (TopSpaceMetr(Euclid n))|PP = (TOP-REAL n)|P by A49,PRE_TOPC:36;
  then (TOP-REAL n)|P is T_2 by A24,TOPMETR:2;
  hence f is being_homeomorphism by A23,A18,A48,A10,COMPTS_1:17;
  0 in [.0,1.] by XXREAL_1:1;
  hence f.0 = (1-0)*p1 + 0 * p2 by A3
    .= p1 + 0 * p2 by RLVECT_1:def 8
    .= p1 + 0.TOP-REAL n by RLVECT_1:10
    .= p1 by RLVECT_1:4;
  1 in [.0,1.] by XXREAL_1:1;
  hence f.1 = (1-1)*p1 + 1*p2 by A3
    .= 0.TOP-REAL n + 1*p2 by RLVECT_1:10
    .= 1*p2 by RLVECT_1:4
    .= p2 by RLVECT_1:def 8;
end;
