 reserve x,X for set,
         n, m, i for Nat,
         p, q for Point of TOP-REAL n,
         A, B for Subset of TOP-REAL n,
         r, s for Real;
reserve N for non zero Nat,
        u,t for Point of TOP-REAL(N+1);

theorem Th1:
  for F be Element of N-tuples_on
      Funcs(the carrier of TOP-REAL(N+1),the carrier of R^1)
    st for i st i in dom F holds F.i = PROJ(N+1,i)
  holds
    (for t holds <:F:>.t = t|N) &
    for Sp,Sn be Subset of TOP-REAL(N+1) st
        Sp = {u: u.(N+1)>=0 & |.u.|=1} &
        Sn = {t: t.(N+1)<=0 & |.t.|=1}
      holds
        <:F:>.:Sp = cl_Ball(0.TOP-REAL N,1) &
        <:F:>.:Sn = cl_Ball(0.TOP-REAL N,1) &
        <:F:>.:(Sp/\Sn) = Sphere(0.TOP-REAL N,1) &
        (for H be Function of (TOP-REAL(N+1)) |Sp,Tdisk(0.TOP-REAL N,1) st
           H= <:F:>|Sp holds H is being_homeomorphism) &
        for H be Function of (TOP-REAL(N+1)) |Sn,Tdisk(0.TOP-REAL N,1) st
           H= <:F:>|Sn holds H is being_homeomorphism
proof
  set N1=N+1,Tn1=TOP-REAL N1,Tn=TOP-REAL N;
  N:N in NAT by ORDINAL1:def 12;
  set N0=(0*N1)+*(N1,-1);
A1: len N0 = N1 by CARD_1:def 7;
  rng N0 c= REAL;
  then N0 is FinSequence of REAL by FINSEQ_1:def 4;
  then reconsider N0 as Point of Tn1 by A1,TOPREAL7:17;
  set NF=N NormF,NNF=NF(#)NF;
  reconsider ONE=1 as Element of NAT;
  set TD=Tdisk(0.TOP-REAL N,1);
A2: [#]TD = cl_Ball(0.Tn,1) by PRE_TOPC:def 5;
  reconsider NNF as Function of Tn,R^1 by TOPMETR:17;
  reconsider mNNF=-NNF as Function of Tn,R^1 by TOPMETR:17;
  reconsider m1=1+mNNF as Function of Tn,R^1 by TOPMETR:17;
  1<= N1 by NAT_1:11;
  then
A3: N1 in Seg N1;
  dom (0*N1)=Seg N1;
  then
A4: N0.N1 = -1 by A3,FUNCT_7:31;
  let F be Element of N-tuples_on Funcs(the carrier of Tn1,the carrier of R^1);
  set FF=<:F:>;
A5:N < N1 by NAT_1:13;
  len F = N by CARD_1:def 7;
  then
A6: dom F=Seg N by FINSEQ_1:def 3;
  assume
A7: for i st i in dom F holds F.i = PROJ(N1,i);
  thus
A8: for t holds <:F:>.t = t|N
  proof
    let s be Point of Tn1;
    reconsider Fs=FF.s as Point of Tn;
A9: len Fs=N by CARD_1:def 7;
A10:len s = N1 by CARD_1:def 7;
  now let i;
      assume that
A12:      1<= i
        and
A13:      i <= N;
A14: Fs.i = F.i.s by TIETZE_2:1;
      i < N1 by A13, NAT_1:13;
      then
A15:    i in dom s by A10,A12,FINSEQ_3: 25;
      F.i = PROJ(N1,i) by A12,A13,FINSEQ_1:1,A7,A6;
      then
A16:    Fs.i = s/.i by A14,TOPREALC:def 6;
      (s|N).i=s.i by A12,A13,FINSEQ_1:1, FUNCT_1:49;
      hence Fs.i = (s|N).i by A16, A15,PARTFUN1:def 6;
    end;
    hence thesis by A9,A10,FINSEQ_1:59,A5;
  end;
  let Sp,Sn be Subset of Tn1;
  assume
 A17: Sp = {u: u.N1>=0 & |.u.|=1};
 A18: dom FF = the carrier of Tn1 by FUNCT_2:def 1;
 A19:cl_Ball(0.Tn,1) c= FF.:Sp
  proof
    let x be object;
    assume
A20:  x in cl_Ball(0.Tn,1);
    then reconsider s=x as Element of REAL N by EUCLID:22;
    set sq = sqrt(1-|.s.|^2), s1=s^<*sq*>;
    |.s.| <= 1 by TOPREAL9:11,A20;
    then |.s.|* |.s.| <= 1 * 1 by XREAL_1:66;
    then
A21:  1-|.s.|^2 >= |.s.|^2-|.s.|^2 by XREAL_1:13;
    then
A22:  sq^2 = 1-|.s.|^2 by SQUARE_1:def 2;
A23:len s1 = len s + 1 by FINSEQ_2:16;
A24:len s = N by CARD_1:def 7;
    s1 is FinSequence of REAL by RVSUM_1:145;
    then reconsider s1 as Element of REAL N1 by A23,A24,FINSEQ_2:92;
    (s1|N)^<*s1.N1*> = s1 by FINSEQ_3:55,A23, CARD_1:def 7;
    then
A25:  s1|N=s by FINSEQ_2:17;
    reconsider ss1=s1 as Point of Tn1 by EUCLID:22;
A26:  s1.N1 = sq by A24,FINSEQ_1:42;
    then |.s1.| ^2 = |.s.| ^2 + (sq)^2 by A25,REAL_NS1:10
                   .= 1 ^2 by A22;
    then |.s1.| = 1 or |.s1.| = -1 by SQUARE_1:40;
    then
A27:  ss1 in Sp by A17,A26,A21;
    FF.ss1 = s by A25, A8;
    hence thesis by A27, A18,FUNCT_1:def 6;
  end;
  assume
A28: Sn= {t: t.N1<=0 & |.t.| =1};
A29:cl_Ball(0.Tn,1) c= FF.:Sn
  proof
    let x be object;
    assume
A30:  x in cl_Ball(0.Tn,1);
    then reconsider s=x as Element of REAL N by EUCLID:22;
    |.s.| <= 1 by TOPREAL9:11,A30;
    then |.s.|* |.s.| <= 1 * 1 by XREAL_1:66;
    then
A31:  1-|.s.|^2 >= |.s.|^2-|.s.|^2 by XREAL_1:13;
    set sq = -sqrt(1-|.s.|^2), s1=s^<*sq*>;
A32:len s1 = len s + 1 by FINSEQ_2:16;
    sq^2 = (-sq)^2;
    then
A33:  sq^2 = 1-|.s.|^2 by SQUARE_1:def 2,A31;
A34:  len s = N by CARD_1:def 7;
    s1 is FinSequence of REAL by RVSUM_1:145;
    then reconsider s1 as Element of REAL N1 by A32,A34,FINSEQ_2:92;
    (s1|N)^<*s1.N1*> = s1 by FINSEQ_3:55,A32, CARD_1:def 7;
    then
A35:  s1|N=s by FINSEQ_2:17;
    reconsider ss1=s1 as Point of Tn1 by EUCLID:22;
A36:s1.N1 = sq by A34,FINSEQ_1:42;
    then |.s1.| ^2 = |.s.| ^2 + (sq)^2 by A35,REAL_NS1:10
                   .= 1 ^2 by A33;
    then |.s1.| = 1 or |.s1.| = -1 by SQUARE_1:40;
    then
A37:  ss1 in Sn by A28,A36,A31;
    FF.ss1 = s by A35,A8;
    hence thesis by A37, A18,FUNCT_1:def 6;
  end;
A38:Sphere(0.Tn,1) c= FF.:(Sn/\Sp)
  proof
    reconsider Z=0 as Element of REAL by XREAL_0:def 1;
    let x be object;
    assume
A39:  x in Sphere(0.Tn,1);
    then reconsider s=x as Element of REAL N by EUCLID:22;
A40: |.s.| = 1 by TOPREAL9:12,A39;
    set s1=s^<* Z *>;
A41:len s = N by CARD_1:def 7;
A42:len s1 = len s + 1 by FINSEQ_2:16;
    then reconsider s1 as Element of REAL N1 by A41,FINSEQ_2:92;
A43:s1.N1 = 0 by A41,FINSEQ_1:42;
    reconsider ss1=s1 as Point of Tn1 by EUCLID:22;
    (s1|N)^<*s1.N1*> = s1 by FINSEQ_3:55,A42, CARD_1:def 7;
    then
A44:  s1|N=s by FINSEQ_2:17;
    then |.s1.| ^2 = |.s.| ^2 + 0^2 by A43,REAL_NS1:10;
    then
A45:  |.s1.| = 1 or |.s1.| = -1 by A40,SQUARE_1:40;
    then
A46:  ss1 in Sn by A28,A43;
    ss1 in Sp by A45,A17,A43;
    then
A47:  ss1 in Sp/\Sn by A46,XBOOLE_0:def 4;
    FF.ss1 = s by A44,A8;
    hence thesis by A47, A18,FUNCT_1:def 6;
  end;
A48:FF.:Sp c= cl_Ball(0.Tn,1)
  proof
    let y be object;
    assume y in FF.:Sp;
    then consider x be object such that
        x in dom FF
      and
A49:    x in Sp
      and
A50:    FF.x=y by FUNCT_1:def 6;
    consider s be Point of Tn1 such that
A51:    s=x
      and
        s.N1>=0
      and
A52:    |.s.|=1 by A49,A17;
    reconsider ss=s as Element of REAL N1 by EUCLID:22;
    len ss = N1 by CARD_1:def 7;
    then len (ss|N)=N by A5,FINSEQ_1:59;
    then reconsider sN=ss|N as Point of Tn by TOPREAL7:17;
    |.ss.| ^2 = |.sN .| ^2 + (s.N1)^2 by REAL_NS1:10;
    then |.ss.| ^2 >= |.sN .| ^2+0 by XREAL_1:6;
    then
A53:  1 >= |. sN .| by SQUARE_1:16,A52;
    sN -0.Tn = sN by RLVECT_1:13;
    then sN in cl_Ball(0.Tn,1) by A53;
    hence thesis by A8,A51,A50;
  end;
  hence FF.:Sp = cl_Ball(0.Tn,1) by A19;
A54:FF.:Sn c= cl_Ball(0.Tn,1)
  proof
    let y be object;
    assume y in FF.:Sn;
    then consider x be object such that
        x in dom FF and
A55:    x in Sn
      and
A56:    FF.x=y by FUNCT_1:def 6;
    consider s be Point of Tn1 such that
A57:    s=x
      and
        s.N1<=0
      and
A58:    |.s.|=1 by A55,A28;
    reconsider ss=s as Element of REAL N1 by EUCLID:22;
    len ss = N1 by CARD_1:def 7;
    then len (ss|N)=N by A5,FINSEQ_1:59;
    then reconsider sN=ss|N as Point of Tn by TOPREAL7:17;
    |.ss.| ^2 = |.sN .| ^2 + (s.N1)^2 by REAL_NS1:10;
    then |.ss.| ^2 >= |.sN .| ^2+0 by XREAL_1:6;
    then
A59:  1 >= |. sN .| by SQUARE_1:16,A58;
    sN -0.Tn = sN by RLVECT_1:13;
    then sN in cl_Ball(0.Tn,1) by A59;
    hence thesis by A8,A57,A56;
  end;
  hence FF.:Sn = cl_Ball(0.Tn,1) by A29;
  FF.:(Sn/\Sp) c= Sphere(0.Tn,1)
  proof
    let y be object;
    assume y in FF.:(Sn/\Sp);
    then consider x be object such that
        x in dom FF
      and
A60:    x in (Sn/\Sp)
      and
A61:    FF.x=y by FUNCT_1:def 6;
    x in Sn by A60,XBOOLE_0:def 4;
    then consider s be Point of Tn1 such that
A62:    s=x
      and
A63:    s.N1 <= 0
      and
A64:    |.s.|=1 by A28;
    reconsider ss=s as Element of REAL N1 by EUCLID:22;
    len ss = N1 by CARD_1:def 7;
    then len (ss|N)=N by A5,FINSEQ_1:59;
    then reconsider sN=ss|N as Point of Tn by TOPREAL7:17;
A65:  |.ss.| ^2 = |.sN .| ^2 + (s.N1)^2 by REAL_NS1:10;
    x in Sp by A60,XBOOLE_0:def 4;
    then ex s1 be Point of Tn1 st s1=x & s1.N1 >= 0 & |.s1.|=1 by A17;
    then (s.N1)^2 =0 by A62,A63;
    then
A66:  |.ss.| = |.sN .| or |.ss.| = -|.sN .| by A65,SQUARE_1:40;
    sN -0.Tn = sN by RLVECT_1:13;
    then sN in Sphere(0.Tn,1) by A66, A64;
    hence thesis by A8, A62,A61;
  end;
  hence FF.:(Sp/\Sn) = Sphere(0.Tn,1) by A38;
  thus for H be Function of (TOP-REAL(N+1)) |Sp,Tdisk(0.TOP-REAL N,1) st
    H=<:F:>|Sp holds H is being_homeomorphism
  proof
    set N0=(0*N1)+*(N1,1);
    rng N0 c= REAL;
    then
W:    N0 is FinSequence of REAL by FINSEQ_1:def 4;
    len N0 = N1 by CARD_1:def 7;
    then reconsider N0 as Point of Tn1 by W,TOPREAL7:17;
    reconsider ONE=1 as Element of NAT;
    set NF=N NormF,NNF=NF(#)NF;
A67:[#]TD = cl_Ball(0.Tn,1) by PRE_TOPC:def 5;
    reconsider NNF as Function of Tn,R^1 by TOPMETR:17;
    reconsider mNNF=-NNF as Function of Tn,R^1 by TOPMETR:17;
    reconsider m1=1+mNNF as Function of Tn,R^1 by TOPMETR:17;
    1<= N1 by NAT_1:11;
    then
A68:  N1 in Seg N1;
    then
A69:  |.N0.| = |. 1 .| by TOPREALC:13
            .= 1 by ABSVALUE:def 1;
    let H be Function of Tn1 |Sp,TD such that
A70:  H = FF|Sp;
A71:dom H = [#](Tn1 |Sp) by FUNCT_2:def 1;
A72:[#](Tn1 |Sp) = Sp by PRE_TOPC:def 5;
    then
A73:    rng H = (FF|Sp).:Sp by A71,A70,RELAT_1:113
             .= cl_Ball(0.Tn,1) by A19,A48,RELAT_1:129;
    then
A74:  rng H = [#]TD by PRE_TOPC:def 5;
A75:for x1,x2 be object st x1 in dom H & x2 in dom H & H.x1 =H.x2 holds x1=x2
    proof
      let x1,x2 be object;
      assume
A76:    x1 in dom H;
      then consider s1 be Point of Tn1 such that
A77:      x1=s1
        and
A78:      s1.N1>=0
        and
A79:      |.s1.|=1 by A17,A71,A72;
      assume
A80:    x2 in dom H;
      then consider s2 be Point of Tn1 such that
A81:      x2=s2
        and
A82:      s2.N1>=0
        and
A83:      |.s2.|=1 by A17,A71,A72;
      reconsider s1,s2 as Element of REAL N1 by EUCLID:22;
A84:  1 ^2 = |.s1|N .| ^2 + (s1.N1)^2 by REAL_NS1:10,A79;
A85:  H.x2 = FF.x2 by A70,FUNCT_1:47,A80;
      assume
A86:    H.x1 = H.x2;
      H.x1 = FF.x1 by A76,A70,FUNCT_1:47;
      then
A87:    s1|N = FF.s2 by A85,A86,A8,A77, A81
            .= s2|N by A8;
      then 1 ^2 = |.s1|N .| ^2 + (s2.N1)^2 by REAL_NS1:10,A83;
      then
A88:    s1.N1 = s2.N1 or s1.N1 = -s2.N1 by A84, SQUARE_1:40;
A89:  s1.N1 >0 or s1.N1=0 by A78;
A90:  len s2 = N1 by CARD_1:def 7;
      len s1= N1 by CARD_1:def 7;
      hence x1 = (s1|N)^<*s1.N1*> by FINSEQ_3:55, A77
              .= x2 by FINSEQ_3:55, A88,A89,A82,A90,A87, A81;
    end;
    then
A91:  H is one-to-one;
    set TD=Tdisk(0.Tn,1);
    set M=m1 | TD;
A92:dom M = the carrier of TD by FUNCT_2:def 1;
    reconsider MM=M as continuous Function of TD,REAL
      by TOPMETR:17,JORDAN5A:27;
A93:M= m1 | the carrier of TD by TMAP_1:def 4;
A94: for p be Point of Tn st p in the carrier of TD holds
      MM.p = 1-(|.p.|*|.p.|)
    proof
      let x be Point of Tn;
      NF.x = |.x.| by JGRAPH_4:def 1;
      then NNF.x = |.x.| * |.x.| by VALUED_1:5;
      then mNNF.x = -|.x.| * |.x.| by VALUED_1:8;
      then m1.x = 1+-|.x.| * |.x.| by VALUED_1:2;
      hence thesis by A93,FUNCT_1:49;
    end;
A95: the carrier of TD = cl_Ball(0.Tn,1) by N,BROUWER:3;
A96: now
      let r be Real;
      assume
      r in rng MM;
      then consider x be object such that
A97:      x in dom MM
        and
A98:      MM.x =r by FUNCT_1:def 3;
      reconsider x as Point of Tn by A92,A95,A97;
      |.x.| <= 1 by A97,A95,TOPREAL9:11;
      then |.x.| * |.x.| <= 1 *1 by XREAL_1:66;
      then 1-|.x.| * |.x.| >= 1-1*1 by XREAL_1:10;
      hence r>=0 by A94,A97,A98;
    end;
    then MM is nonnegative-yielding by PARTFUN3:def 4;
    then reconsider SQR = |[sqrt MM]| as continuous Function of TD,TOP-REAL 1;
A99:dom (sqrt MM)=dom MM by PARTFUN3:def 5;
A100: rng id TD = the carrier of TD;
    dom id TD = the carrier of TD;
    then reconsider ID=id TD as continuous Function of TD,Tn
      by A100,FUNCT_2:2,A95,PRE_TOPC:26;
    reconsider SQR as continuous Function of TD,TOP-REAL ONE;
    reconsider II=ID^^SQR as continuous Function of TD,TOP-REAL (N+ONE);
    reconsider II as continuous Function of TD,Tn1;
A101: dom II = the carrier of TD by FUNCT_2:def 1;
A102: dom SQR = (dom sqrt MM) by Def1;
    for y,x be object holds y in rng H & x = II.y iff x in dom H & y = H.x
    proof
      let y,x be object;
      hereby
        assume that
A103:       y in rng H
          and
A104:       x = II.y;
        reconsider p=y as Point of Tn by A103, A73;
        set p1 = 1-(|.p.|*|.p.|),sp =sqrt p1,ssp=|[ sp ]|;
A105:   ID.p=p by A103,FUNCT_1:17;
A106:   MM.p = 1-(|.p.|*|.p.|) by A103,A94;
        then (sqrt MM).p = sp by A103,A92,PARTFUN3:def 5,A99;
        then SQR.p = ssp by Def1,A99,A102, A103,A92;
        then
A107:     II.p = p^ssp by A103, A101,A105,PRE_POLY:def 4;
        II.p in rng II by A103, A101,FUNCT_1:def 3;
        then reconsider IIp=II.p as Point of Tn1;
        MM.p in rng MM by A103,A92,FUNCT_1:def 3;
        then
A109:     MM.p >=0 by A96;
A110:   sqr ssp = <*sp^2*> by RVSUM_1:55
               .= <*MM.p*> by A109,SQUARE_1:def 2,A106;
        sqr IIp = (sqr p) ^ (sqr ssp) by RVSUM_1:144,A107;
        then Sum sqr IIp = Sum (sqr p) + MM.p by A110,RVSUM_1:74
                        .= |.p.|^2 + MM.p by TOPREAL9:5
                        .=1 by A106;
        then
A111:     |.IIp.| =1;
A112:   len p = N by CARD_1:def 7;
        then IIp.N1 = sp by A107, FINSEQ_1:42;
        then
A113:     IIp in Sp by A106,A109,A17,A111;
        hence x in dom H by A104,A71, PRE_TOPC:def 5;
        FF.IIp = H.IIp by A113,A71,A72,A70,FUNCT_1:47;
        hence H.x = (p^ssp) |N by A8,A107,A104
                 .= (p^ssp) | (dom p) by A112,FINSEQ_1:def 3
                 .=y by FINSEQ_1:21;
      end;
      assume that
A114:     x in dom H
        and
A115:     y = H.x;
      consider p be Point of Tn1 such that
A116:     x=p
        and
A117:     p.N1>=0
        and
A118:     |.p.|=1 by A114,A17,A71,A72;
A119: p|N is FinSequence of REAL by RVSUM_1:145;
      len p = N1 by CARD_1:def 7;
      then len (p|N) = N by NAT_1:11,FINSEQ_1:59;
      then reconsider pN=p|N as Point of Tn by A119,TOPREAL7:17;
A121: p =pN ^<*p.N1*> by CARD_1:def 7,FINSEQ_3:55;
      set p1 = 1-(|.pN.|*|.pN.|),sp =sqrt p1,ssp=|[ sp ]|;
A122: sqr <*p.N1*> = <*(p.N1)^2*> by RVSUM_1:55;
A123: pN-0.Tn = pN by VECTSP_1:18;
      sqr p =sqr pN ^ sqr <*p.N1*> by A121,RVSUM_1:144;
      then Sum sqr p = Sum sqr pN + (p.N1)^2 by RVSUM_1:74,A122
                     .= |.pN.|^2 + (p.N1)^2 by TOPREAL9:5;
      then
A124:   |.p.|^2 = |.pN.|^2 + (p.N1)^2 by TOPREAL9:5;
      then |.p.|^2 >= |.pN.|^2 by XREAL_1:38;
      then |.pN.| <= 1 by SQUARE_1:47,A118;
      then
A125:   pN in rng H by A123, A73;
      then MM.pN = 1-(|.pN.|*|.pN.|) by A94;
      then (sqrt MM).pN = sp by A125,A92,PARTFUN3:def 5,A99;
      then
A126:   SQR.pN = ssp by Def1,A99,A102, A125,A92;
A127: FF.p = p|N by A8;
      hence y in rng H by A125,A115,A116, A114,A70,FUNCT_1:47;
      ID.pN=pN by A125,FUNCT_1:17;
      then II.pN = pN^ssp by A125, A101, A126,PRE_POLY:def 4;
      then II.pN = p by A118,A124,A117,SQUARE_1:def 2,A121;
      hence x = II.y by A115,A116, A127, A114,A70,FUNCT_1:47;
    end;
    then
A128: (H qua Function)"=II by A91, A101,A74,FUNCT_1:32;
    dom (0*N1)=Seg N1;
    then N0.N1 = 1 by A68,FUNCT_7:31;
    then
A129: N0 in Sp by A69,A17;
    for P being Subset of Tn1 |Sp holds P is open iff H.:P is open
    proof
      let P being Subset of Tn1 |Sp;
      for i st i in dom F for h be Function of Tn1,R^1 st h = F.i holds
        h is continuous
      proof
        let i such that
A130:     i in dom F;
        i <= N by A130,A6,FINSEQ_1:1;
        then
A131:     i <= N1 by NAT_1:13;
        let h be Function of Tn1,R^1 such that
A132:     h = F.i;
A133:   h = PROJ(N1,i) by A130,A132,A7;
        1 <= i by A130,A6,FINSEQ_1:1;
        then i in Seg N1 by A131;
        hence thesis by A133,TOPREALC:57;
      end;
      then
A134:   FF is continuous by TIETZE_2:21;
      FF| (Tn1|Sp) = FF|Sp by TMAP_1:def 4,A72;
      then
A135:   H is continuous by A134, A129,A70,PRE_TOPC:27;
      hereby
        rng II = dom H by A128,A91,FUNCT_1:33;
        then reconsider ii=II as Function of TD,Tn1 |Sp by FUNCT_2:2,A101;
        assume
A136:     P is open;
A137:   ii is continuous by PRE_TOPC:27;
        dom H is non empty by A73,A71;
        then ii"P is open by A137,A71,A136,TOPS_2:43;
        hence H.:P is open by A75,FUNCT_1:def 4,FUNCT_1:84, A128;
      end;
      assume
A138:   H.:P is open;
      H"(H.:P)=P by A71, A75,FUNCT_1:def 4,FUNCT_1:94;
      hence P is open by A138, TOPS_2:43,A135,A67;
    end;
    hence thesis by A71,A74,A91, A129,TOPGRP_1:25;
  end;
  let H be Function of Tn1 |Sn,TD such that
A139: H = FF|Sn;
A140: dom H = [#](Tn1 |Sn) by FUNCT_2:def 1;
A141: [#](Tn1 |Sn) = Sn by PRE_TOPC:def 5;
  then
A142: rng H = (FF|Sn).:Sn by A140,A139,RELAT_1:113
           .= cl_Ball(0.Tn,1) by A29,A54,RELAT_1:129;
  then A143: rng H = [#]TD by PRE_TOPC:def 5;
  A144: for x1,x2 be object st x1 in dom H & x2 in dom H & H.x1 =H.x2
  holds x1=x2
  proof
    let x1,x2 be object;
    assume
A145: x1 in dom H;
    then consider s1 be Point of Tn1 such that
A146:   x1=s1
      and
A147:   s1.N1<=0
      and
A148:   |.s1.|=1 by A28,A140,A141;
    assume
A149: x2 in dom H;
    then consider s2 be Point of Tn1 such that
A150:   x2=s2
      and
A151:   s2.N1<=0
      and
A152:   |.s2.|=1 by A28,A140,A141;
    reconsider s1,s2 as Element of REAL N1 by EUCLID:22;
A153: 1 ^2 = |.s1|N .| ^2 + (s1.N1)^2 by REAL_NS1:10,A148;
A154: H.x2 = FF.x2 by A139,FUNCT_1:47,A149;
    assume
A155: H.x1 = H.x2;
    H.x1 = FF.x1 by A145,A139,FUNCT_1:47;
    then
A156: s1|N = FF.s2 by A154,A155,A8,A146, A150
          .= s2|N by A8;
    then 1 ^2 = |.s1|N .| ^2 + (s2.N1)^2 by REAL_NS1:10,A152;
    then
A157: s1.N1 = s2.N1 or s1.N1 = -s2.N1 by A153, SQUARE_1:40;
A158: s1.N1 <0 or s1.N1=0 by A147;
A159: len s2 = N1 by CARD_1:def 7;
    len s1= N1 by CARD_1:def 7;
    hence x1 = (s1|N)^<*s1.N1*> by FINSEQ_3:55, A146
            .= x2 by FINSEQ_3:55, A157,A158,A151,A159,A156, A150;
  end;
  then
A160: H is one-to-one;
  set TD=Tdisk(0.Tn,1);
  set M=m1 | TD;
A161: dom M = the carrier of TD by FUNCT_2:def 1;
  reconsider MM=M as continuous Function of TD,REAL by TOPMETR:17,JORDAN5A:27;
  reconsider Msqr=-sqrt MM as Function of TD,REAL;
A162: M= m1 | the carrier of TD by TMAP_1:def 4;
A163: for p be Point of Tn st p in the carrier of TD holds MM.p = 1-(|.p.|*
  |.p.|)
  proof
    let x be Point of Tn;
    NF.x = |.x.| by JGRAPH_4:def 1;
    then NNF.x = |.x.| * |.x.| by VALUED_1:5;
    then mNNF.x = -|.x.| * |.x.| by VALUED_1:8;
    then m1.x = 1+-|.x.| * |.x.| by VALUED_1:2;
    hence thesis by A162,FUNCT_1:49;
  end;
A164: the carrier of TD = cl_Ball(0.Tn,1) by N,BROUWER:3;
A165:now
    let r be Real;
    assume r in rng MM;
    then consider x be object such that
A166:   x in dom MM
      and
A167:   MM.x =r by FUNCT_1:def 3;
    reconsider x as Point of Tn by A161,A164,A166;
    |.x.| <= 1 by A166,A164,TOPREAL9:11;
    then |.x.| * |.x.| <= 1 *1 by XREAL_1:66;
    then 1-|.x.| * |.x.| >= 1-1*1 by XREAL_1:10;
    hence r>=0 by A163,A166,A167;
  end;
  then MM is nonnegative-yielding by PARTFUN3:def 4;
  then reconsider SQR = |[Msqr]| as continuous Function of TD,TOP-REAL 1;
A168: dom (sqrt MM)=dom MM by PARTFUN3:def 5;
A169: rng id TD = the carrier of TD;
  dom id TD = the carrier of TD;
  then reconsider ID=id TD as continuous Function of TD,Tn
    by A169,FUNCT_2:2,A164,PRE_TOPC:26;
  reconsider SQR as continuous Function of TD,TOP-REAL ONE;
  reconsider II=ID^^SQR as continuous Function of TD,TOP-REAL (N+ONE);
  reconsider II as continuous Function of TD,Tn1;
A170: dom II = the carrier of TD by FUNCT_2:def 1;
  dom (-sqrt MM)=dom sqrt MM by VALUED_1:8;
  then
A171: dom SQR = (dom sqrt MM) by Def1;
  for y,x be object holds y in rng H & x = II.y iff x in dom H & y = H.x
  proof
    let y,x be object;
    hereby
      assume that
A172:     y in rng H
        and
A173:     x = II.y;
      reconsider p=y as Point of Tn by A172, A142;
      set p1 = 1-(|.p.|*|.p.|),sp =sqrt p1,ssp=|[ -sp ]|;
A174: ID.p=p by A172,FUNCT_1:17;
A175: MM.p = 1-(|.p.|*|.p.|) by A172,A163;
      then (sqrt MM).p = sp by A172,A161,PARTFUN3:def 5,A168;
      then Msqr.p = -sp by VALUED_1:8;
      then SQR.p = ssp by Def1,A168,A171, A172,A161;
      then
A176:   II.p = p^ssp by A172, A170,A174,PRE_POLY:def 4;
      II.p in rng II by A172, A170,FUNCT_1:def 3;
      then reconsider IIp=II.p as Point of Tn1;
      MM.p in rng MM by A172,A161,FUNCT_1:def 3;
      then
A178:   MM.p >=0 by A165;
A179:   sqr IIp = (sqr p) ^ (sqr ssp) by RVSUM_1:144,A176;
      sp^2 = (-sp)^2;
      then sqr ssp = <*sp^2*> by RVSUM_1:55
                   .= <*MM.p*> by A178,SQUARE_1:def 2,A175;
      then Sum sqr IIp = Sum (sqr p) + MM.p by A179,RVSUM_1:74
                       .= |.p.|^2 + MM.p by TOPREAL9:5
                       .= 1 by A175;
      then
A180:   |.IIp.| =1;
A181: len p = N by CARD_1:def 7;
      then IIp.N1 = -sp by A176, FINSEQ_1:42;
      then
A182:   IIp in Sn by A175,A178,A28,A180;
      hence x in dom H by A173,A140, PRE_TOPC:def 5;
      FF.IIp = H.IIp by A182,A140,A141,A139,FUNCT_1:47;
      hence H.x = (p^ssp) |N by A8,A176,A173
               .= (p^ssp) | (dom p) by A181,FINSEQ_1:def 3
               .= y by FINSEQ_1:21;
    end;
    assume that
A183:   x in dom H
      and
A184:   y = H.x;
    consider p be Point of Tn1 such that
A185:   x=p
      and
A186:   p.N1<=0
      and
A187:   |.p.|=1 by A183,A28,A140,A141;
A188: p|N is FinSequence of REAL by RVSUM_1:145;
    len p = N1 by CARD_1:def 7;
    then len (p|N) = N by NAT_1:11,FINSEQ_1:59;
    then reconsider pN=p|N as Point of Tn by A188,TOPREAL7:17;
A190: p =pN ^<*p.N1*> by CARD_1:def 7,FINSEQ_3:55;
A191: sqr <*p.N1*> = <*(p.N1)^2*> by RVSUM_1:55;
    sqr p =sqr pN ^ sqr <*p.N1*> by A190,RVSUM_1:144;
    then
A192: Sum sqr p = Sum sqr pN + (p.N1)^2 by RVSUM_1:74,A191
               .= |.pN.|^2 + (p.N1)^2 by TOPREAL9:5;
    then |.p.|^2 = |.pN.|^2 + (p.N1)^2 by TOPREAL9:5;
    then |.p.|^2 >= |.pN.|^2 by XREAL_1:38;
    then
A193: |.pN.| <= 1 by SQUARE_1:47,A187;
    set p1 = 1-(|.pN.|*|.pN.|),sp =sqrt p1,ssp=|[ -sp ]|;
    pN-0.Tn = pN by VECTSP_1:18;
    then
A194: pN in rng H by A193, A142;
    then MM.pN = 1-(|.pN.|*|.pN.|) by A163;
    then (sqrt MM).pN = sp by A194,A161,PARTFUN3:def 5,A168;
    then (Msqr).pN = -sp by VALUED_1:8;
    then
A195: SQR.pN = ssp by Def1,A168,A171, A194,A161;
    1^2 = |.pN.|^2 + (p.N1)^2 by A187, A192,TOPREAL9:5;
    then
A196: -p.N1 = sqrt (1 - |.pN.|^2) by A186,SQUARE_1:23;
A197: FF.p = p|N by A8;
    hence y in rng H by A194,A184,A185, A183,A139,FUNCT_1:47;
    ID.pN=pN by A194,FUNCT_1:17;
    then II.pN = pN^ssp by A194, A170, A195,PRE_POLY:def 4;
    hence x = II.y by A196,A190,A184,A185, A197, A183,A139,FUNCT_1:47;
  end;
  then
A198: (H qua Function)"=II by A160, A170,A143,FUNCT_1:32;
  |.N0.| = |. -1 .| by TOPREALC:13,A3
        .= --1 by ABSVALUE:def 1;
  then
A199:N0 in Sn by A28,A4;
  for P being Subset of Tn1 |Sn holds P is open iff H.:P is open
  proof
    let P being Subset of Tn1 |Sn;
    for i st i in dom F for h be Function of Tn1,R^1 st h = F.i
      holds h is continuous
    proof
      let i such that
A200:   i in dom F;
      i <= N by A200,A6,FINSEQ_1:1;
      then
A201:   i <= N1 by NAT_1:13;
      let h be Function of Tn1,R^1 such that
A202:   h = F.i;
A203: h = PROJ(N1,i) by A200,A202,A7;
      1 <= i by A200,A6,FINSEQ_1:1;
      then i in Seg N1 by A201;
      hence thesis by A203,TOPREALC:57;
    end;
    then
A204: FF is continuous by TIETZE_2:21;
    FF| (Tn1|Sn) = FF|Sn by TMAP_1:def 4,A141;
    then
A205:H is continuous by A204, A199,A139,PRE_TOPC:27;
    hereby
      rng II = dom H by A198,A160,FUNCT_1:33;
      then reconsider ii=II as Function of TD,Tn1 |Sn by FUNCT_2:2,A170;
      assume
A206:   P is open;
A207: ii is continuous by PRE_TOPC:27;
      dom H is non empty by A142,A140;
      then ii"P is open by A207,A140,A206,TOPS_2:43;
      hence H.:P is open by A144,FUNCT_1:def 4,FUNCT_1:84, A198;
    end;
    assume
A208: H.:P is open;
    H"(H.:P)=P by A140, A144,FUNCT_1:def 4,FUNCT_1:94;
    hence P is open by A208,TOPS_2:43,A205,A2;
  end;
  hence thesis by A140,A143,A160, A199,TOPGRP_1:25;
end;
