 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 Th10:
  A is closed & p in Fr A implies
    for h be Function of (TOP-REAL n) |A,(TOP-REAL n) |B st
        h is being_homeomorphism
    holds h.p in Fr B
proof
  set TRn=TOP-REAL n;
  assume that
A1:   A is closed
    and
A2:   p in Fr A;
A3:Fr A c= A by A1, TOPS_1:35;
  let h be Function of TRn|A,TRn|B such that
A4:  h is being_homeomorphism;
A5: [#](TRn|A) = A by PRE_TOPC:def 5;
  then
A6: dom h = A by A4,TOPS_2:def 5;
  then
A7: h.p in rng h by A3,A2,FUNCT_1:def 3;
A8: [#](TRn|B) = B by PRE_TOPC:def 5;
  then
A9: rng h = B by A4,TOPS_2:def 5;
  then reconsider hp=h.p as Point of TRn by A7;
  per cases;
    suppose n=0;
      hence thesis by Lm3,A4, A2;
    end;
    suppose
A10:    n>0;
      then reconsider n1=n-1 as Element of NAT by NAT_1:20;
A11:  the TopStruct of TRn=TopSpaceMetr Euclid n by EUCLID:def 8;
      for r st r>0 ex U be open Subset of TRn|B st
        hp in U & U c= Ball(hp,r) &
        for f be Function of TRn | (B\U),Tunit_circle(n) st
          f is continuous ex h be Function of TRn |B,Tunit_circle(n) st
            h is continuous & h| (B\U) = f
      proof
        reconsider P =p as Point of Euclid n by A11, TOPMETR:12;
        let r such that
A13:      r>0;
        reconsider BB=B/\Ball(hp,r) as Subset of TRn|B by A8,XBOOLE_1:17;
        Ball(hp,r) in the topology of TRn by PRE_TOPC:def 2;
        then BB in the topology of (TRn|B) by A8,PRE_TOPC:def 4;
        then reconsider BB as open Subset of TRn|B by PRE_TOPC:def 2;
        h"BB is open by A7,A4,A8, TOPS_2:43;
        then h"BB in the topology of TRn|A by PRE_TOPC:def 2;
        then consider U be Subset of TRn such that
A14:        U in the topology of TRn
          and
A15:        h"BB = U/\[#](TRn|A) by PRE_TOPC:def 4;
        reconsider U as open Subset of TRn by A14,PRE_TOPC:def 2;
A16:    Int U = U by TOPS_1:23;
        hp is Element of REAL n by EUCLID:22;
        then |. hp-hp .|=0;
        then hp in Ball(hp,r) by A13;
        then hp in BB by A7,A8,XBOOLE_0:def 4;
        then p in h"BB by A3,A2, A6,FUNCT_1:def 7;
        then p in U by A15,XBOOLE_0:def 4;
        then consider s be Real such that
A17:        s>0
          and
A18:        Ball(P,s) c= U by A16,GOBOARD6:5;
        consider W be open Subset of TRn|A such that
A19:        p in W
          and
A20:        W c= Ball(p,s)
          and
A21:        for f be Function of TRn | (A\W),Tunit_circle(n) st
              f is continuous ex h be Function of
                (TOP-REAL n) |A,Tunit_circle(n)
              st h is continuous & h| (A\W) = f by A1,A17,Th9, A2;
        Ball(p,s)=Ball(P,s) by TOPREAL9:13;
        then
A22:      Ball(p,s)/\A c= U/\A by A18, XBOOLE_1: 27;
        W/\A=W by A5,XBOOLE_1:28;
        then W c= Ball(p,s)/\A by A20, XBOOLE_1: 27;
        then W c= U/\A by A22;
        then h.:W c= h.:(U/\A) by RELAT_1:123;
        then
A23:      h.:W c= BB by FUNCT_1:77,A8,A9,A15,A5;
        TRn|B is non empty by A3,A2, A6;
        then
        reconsider hW=h.:W as open Subset of TRn|B by A3,A2,A4,TOPGRP_1:25;
        take hW;
        BB c= Ball(hp,r) by XBOOLE_1:17;
        hence hp in hW & hW c= Ball(hp,r)
          by A3,A2,A6,FUNCT_1:def 6,A19,A23;
        set AW=A\W,haw=h| AW,T=Tunit_circle(n);
A24:    [#](TRn | (B\hW))=B\hW by PRE_TOPC:def 5;
        reconsider aw=AW as Subset of TRn|A by XBOOLE_1:36,A5;
        let f be Function of TRn | (B\hW),T such that
A25:      f is continuous;
        per cases;
          suppose
A26:          B\hW is empty;
            set h = the continuous Function of TRn |B,Tunit_circle(n1+1);
            reconsider H=h as Function of (TOP-REAL n) |B,T;
            take H;
            f={} by A26;
            hence thesis by A26;
          end;
          suppose
A27:          B\hW is non empty;
            set AW=A\W,haw=h| AW,T=Tunit_circle(n);
            reconsider haw as Function of (TRn |A) | aw,(TRn|B) | (h.:AW)
              by A3,A2,A7,JORDAN24:12;
A28:        h.:AW = (h.:A) \ (h.:W) by A4, FUNCT_1:64
                 .= B\hW by RELAT_1:113,A6,A9;
            then
A29:          (TRn|B) | (h.:AW) = TRn | (B\hW) by XBOOLE_1:36,PRE_TOPC:7;
A30:        (TRn |A) | aw = TRn |AW by PRE_TOPC:7, XBOOLE_1:36;
            then reconsider HAW =haw as Function of TRn |AW,TRn | (B\hW)
              by A29;
            reconsider fhW=f*HAW as Function of TRn | AW,Tunit_circle(n)
              by A27;
            fhW is continuous
              by A27,JORDAN24:13,A4,A3,A2,A7,A29,A30,A25,TOPS_2:46;
            then consider HW be Function of TRn |A,T such that
A31:            HW is continuous
              and
A32:            HW| AW = fhW by A21;
            reconsider HWh=HW*(h") as Function of TRn|B,T by A3,A2;
            take HWh;
            h" is continuous by A4,TOPS_2:def 5;
            hence HWh is continuous by TOPS_2:46, A3,A2,A31;
A33:        dom f = B\hW by A10,A24, FUNCT_2:def 1;
A34:        rng ((h") | (B \ hW)) = (h").:(B\hW) by RELAT_1:115
                                 .= h"(h.:AW ) by A28,A8,A9,A4,TOPS_2:55
                                 .= AW by FUNCT_1:94,A6,A4,XBOOLE_1:36;
            thus HWh | (B \ hW) = HW* ((h") | (B \ hW)) by RELAT_1:83
               .= HW * ((id (AW))*((h") | (B \ hW))) by A34, RELAT_1:53
               .= (HW * (id (AW)))*((h") | (B \ hW)) by RELAT_1:36
               .= (HW |AW)*((h") | (B \ hW)) by RELAT_1:65
               .=( (f*h) |AW ) * ((h") | (B \ hW)) by A32,RELAT_1:83
               .=( (f*h) *(id AW )) * ((h") | (B \ hW)) by RELAT_1:65
               .= (f*h) *((id AW ) * ((h") | (B \ hW))) by RELAT_1:36
               .= (f*h) *( (h") | (B \ hW)) by A34,RELAT_1:53
               .= ((f*h) * (h")) | (B \ hW) by RELAT_1:83
               .= (f*(h * h")) | (B \ hW) by RELAT_1:36
               .= (f*id B) | (B \ hW) by TOPS_2:52,A9, A4, A8
               .= f | (dom f) by A33, XBOOLE_1:36,RELAT_1:51
               .=f;
        end;
      end;
      hence thesis by Th8, A7,A8, A10;
    end;
end;
