 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;
reserve M,M1,M2 for non empty TopSpace;

theorem Th8:
  for N,M be locally_euclidean non empty TopSpace holds
    Int [:N,M:] = [:Int N,Int M:]
proof
  let N,M be locally_euclidean non empty TopSpace;
  set NM=[:N,M:],IN=Int N,IM=Int M;
  thus Int NM c= [:IN,IM:]
  proof
    let z be object;
    assume
A1: z in Int NM;
    then reconsider p=z as Point of NM;
    z in the carrier of NM by A1;
    then z in [:the carrier of N,the carrier of M:] by BORSUK_1:def 2;
    then consider x,y be object such that
A2: x in the carrier of N and
A3: y in the carrier of M and
A4: z=[x,y] by ZFMISC_1: def 2;
    reconsider y as Point of M by A3;
    reconsider x as Point of N by A2;
    assume
A5: not z in [:IN,IM:];
    per cases by A5,A4,ZFMISC_1:87;
      suppose
A6:       not x in IN;
        consider W be a_neighborhood of y,m be Nat such that
A7:     M|W,Tdisk(0.TOP-REAL m,1) are_homeomorphic by Def2;
        x in [#]N\IN by A6,XBOOLE_0:def 5;
        then x in Fr N by SUBSET_1:def 4;
        then consider U be a_neighborhood of x,n be Nat, h be Function of
          N|U,Tdisk(0.TOP-REAL n,1) such that
A8:       h is being_homeomorphism
        and
A9:       h.x in Sphere(0.TOP-REAL n,1) by Th5;
A10:    y in Int W by CONNSP_2:def 1;
        reconsider mn=m+n as Nat;
        set TRm=TOP-REAL m,TRn=TOP-REAL n,TRnm=TOP-REAL mn;
        consider f be Function of M|W,Tdisk(0.TRm,1) such that
A11:      f is being_homeomorphism by A7,T_0TOPSP:def 1;
A12:    not h.x in Ball(0.TRn,1) by TOPREAL9:19,A9,XBOOLE_0:3;
        consider hf be Function of [: Tdisk(0.TRn,1),Tdisk( 0.TRm,1):],
          Tdisk(0.TRnm,1) such that
A13:    hf is being_homeomorphism
        and
A14:    hf.:[:Ball(0.TRn,1), Ball(0.TRm,1):] = Ball(0.TRnm,1) by TIETZE_2:19;
        set H=hf*[:h,f:];
        [:h,f:] is being_homeomorphism by A8,A11,TIETZE_2:15;
        then
A15:    H is being_homeomorphism by A13,TOPS_2:57;
        then
A16:    rng H = [#]Tdisk(0.TRnm,1) by TOPS_2:def 5;
A17:    Int W c= W by TOPS_1:16;
A18:    Int U c= U by TOPS_1:16;
        x in Int U by CONNSP_2:def 1;
        then
A19:    [x,y] in [:U,W:] by A18,A17,A10,ZFMISC_1:87;
        n+m in NAT by ORDINAL1:def 12;
        then
A20:    [#]Tdisk(0.TRnm,1) = cl_Ball(0.TRnm,1) by BROUWER:3;
A21:    [:N|U,M|W:] = NM| [:U,W:] by BORSUK_3:22;
        then dom H = [#](NM| [:U,W:]) by A15,TOPS_2:def 5;
        then
A22:    p in dom H by A19,A4,PRE_TOPC:def 5;
        then
A23:    [:h,f:].p in dom hf by FUNCT_1:11;
        dom [:h,f:] = [:dom h,dom f:] by FUNCT_3:def 8;
        then
A24:     [:h,f:].(x,y) = [h.x,f.y] by A22,FUNCT_1:11,A4,FUNCT_3:65;
A25:    H.p = hf.([:h,f:].p) by A22,FUNCT_1:12;
        H.p in Sphere(0.TRnm,1)
        proof
          H.p in cl_Ball(0.TRnm,1) by A16,A20,A22,FUNCT_1:def 3;
          then
A26:      H.p in Sphere(0.TRnm,1)\/Ball(0.TRnm,1) by TOPREAL9:18;
          assume not H.p in Sphere(0.TRnm,1);
          then H.p in hf.:[:Ball(0.TRn,1), Ball(0.TRm,1):]
            by A26,XBOOLE_0:def 3,A14;
          then consider w be object such that
A27:        w in dom hf
          and
A28:        w in [:Ball(0.TRn,1), Ball(0.TRm,1):]
          and
A29:        hf.w =H.p by FUNCT_1:def 6;
          w = [:h,f:].p by A25,A23,A13,A27,A29,FUNCT_1:def 4;
          hence thesis by A28,A4,A24,A12,ZFMISC_1:87;
        end;
        then p in Fr NM by A21,A4,Th5,A15;
        then p in [#]NM\Int NM by SUBSET_1:def 4;
        hence thesis by XBOOLE_0:def 5,A1;
      end;
      suppose
        not y in IM;
        then y in [#]M\IM by XBOOLE_0:def 5;
        then y in Fr M by SUBSET_1:def 4;
        then consider W be a_neighborhood of y,m be Nat, f be Function of
          M|W,Tdisk(0.TOP-REAL m,1) such that
A30:      f is being_homeomorphism
        and
A31:      f.y in Sphere(0.TOP-REAL m,1) by Th5;
A32:    y in Int W by CONNSP_2:def 1;
        consider U be a_neighborhood of x,n be Nat such that
A33:    N|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by Def2;
        reconsider mn = n+m as Nat;
        set TRm=TOP-REAL m,TRn=TOP-REAL n,TRnm=TOP-REAL mn;
        consider h be Function of N|U,Tdisk(0.TRn,1) such that
A34:    h is being_homeomorphism by A33,T_0TOPSP:def 1;
A35:    not f.y in Ball(0.TRm,1) by TOPREAL9:19,A31,XBOOLE_0:3;
        consider hf be Function of [: Tdisk(0.TRn,1),Tdisk( 0.TRm,1):],
          Tdisk(0.TRnm,1) such that
A36:    hf is being_homeomorphism
        and
A37:    hf.:[:Ball(0.TRn,1), Ball(0.TRm,1):] = Ball(0.TRnm,1) by TIETZE_2:19;
        set H=hf*[:h,f:];
        [:h,f:] is being_homeomorphism by A30,A34,TIETZE_2:15;
        then
A38:    H is being_homeomorphism by A36,TOPS_2:57;
        then
A39:    rng H = [#]Tdisk(0.TRnm,1) by TOPS_2:def 5;
A40:    Int W c= W by TOPS_1:16;
A41:    Int U c= U by TOPS_1:16;
        x in Int U by CONNSP_2:def 1;
        then
A42:    [x,y] in [:U,W:] by A41,A40,A32,ZFMISC_1:87;
        n+m in NAT by ORDINAL1:def 12;
        then
A43:    [#]Tdisk(0.TRnm,1) = cl_Ball(0.TRnm,1) by BROUWER:3;
A44:    [:N|U,M|W:] = NM| [:U,W:] by BORSUK_3:22;
        then dom H = [#](NM| [:U,W:]) by A38,TOPS_2:def 5;
        then
A45:    p in dom H by A42,A4,PRE_TOPC:def 5;
        then
A46:    [:h,f:].p in dom hf by FUNCT_1:11;
        dom [:h,f:] = [:dom h,dom f:] by FUNCT_3:def 8;
        then
A47:     [:h,f:].(x,y) = [h.x,f.y] by A45,FUNCT_1:11,A4,FUNCT_3:65;
A48:    H.p = hf.([:h,f:].p) by A45,FUNCT_1:12;
        H.p in Sphere(0.TRnm,1)
        proof
          H.p in cl_Ball(0.TRnm,1) by A39,A43,A45,FUNCT_1:def 3;
          then
A49:        H.p in Sphere(0.TRnm,1)\/Ball(0.TRnm,1) by TOPREAL9:18;
          assume not H.p in Sphere(0.TRnm,1);
          then H.p in hf.:[:Ball(0.TRn,1), Ball(0.TRm,1):]
            by A49,XBOOLE_0:def 3,A37;
          then consider w be object such that
A50:        w in dom hf
          and
A51:        w in [:Ball(0.TRn,1), Ball(0.TRm,1):]
          and
A52:        hf.w =H.p by FUNCT_1:def 6;
          w = [:h,f:].p by A48,A46,A36,A50,A52,FUNCT_1:def 4;
         hence thesis by A51,A4,A47,A35,ZFMISC_1:87;
       end;
       then p in Fr NM by A44,A4,Th5,A38;
       then p in [#]NM\Int NM by SUBSET_1:def 4;
       hence thesis by XBOOLE_0:def 5,A1;
     end;
   end;
   let z be object;
   assume
A53: z in [:IN,IM:];
   then consider x,y be object such that
A54: x in IN
   and
A55: y in IM
   and
A56: z=[x,y] by ZFMISC_1:def 2;
   reconsider x as Point of N by A54;
   consider U be a_neighborhood of x, n such that
A57:N|U,Tball(0.TOP-REAL n,1) are_homeomorphic by A54,Def4;
   reconsider y as Point of M by A55;
   consider W be a_neighborhood of y, m such that
A58:M|W,Tball(0.TOP-REAL m,1) are_homeomorphic by A55,Def4;
   reconsider p=z as Point of NM by A53;
   set TRn=TOP-REAL n,TRm=TOP-REAL m;
    reconsider mn=m+n as Nat;
   [:N|U,M|W:], (TOP-REAL mn) | Ball(0.TOP-REAL mn,1)
     are_homeomorphic by A57,A58,TIETZE_2:20;
   then NM| [:U,W:],Tball(0.TOP-REAL mn,1) are_homeomorphic
     by BORSUK_3:22;
   hence thesis by Def4, A56;
end;
