let n, m be Nat; :: thesis: for T being non empty TopSpace

for A, B being Subset of T

for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let T be non empty TopSpace; :: thesis: for A, B being Subset of T

for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let A, B be Subset of T; :: thesis: for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let r, s be Real; :: thesis: ( r > 0 & s > 0 implies for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m )

assume that

A1: r > 0 and

A2: s > 0 ; :: thesis: for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

A3: Int B c= B by TOPS_1:16;

A4: (Int A) /\ (Int B) c= Int B by XBOOLE_1:17;

A5: [#] (T | B) = B by PRE_TOPC:def 5;

then reconsider IB = (Int A) /\ (Int B) as Subset of (T | B) by A3, A4, XBOOLE_1:1;

let pA be Point of (TOP-REAL n); :: thesis: for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let pB be Point of (TOP-REAL m); :: thesis: ( T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B implies n = m )

assume that

A6: T | A, Tdisk (pA,r) are_homeomorphic and

A7: T | B, Tdisk (pB,s) are_homeomorphic and

A8: Int A meets Int B ; :: thesis: n = m

consider hB being Function of (T | B),(Tdisk (pB,s)) such that

A9: hB is being_homeomorphism by A7, T_0TOPSP:def 1;

A10: (T | B) | IB = T | ((Int A) /\ (Int B)) by A3, A4, XBOOLE_1:1, PRE_TOPC:7;

A11: [#] (Tdisk (pB,s)) = cl_Ball (pB,s) by PRE_TOPC:def 5;

then reconsider hBI = hB .: IB as Subset of (TOP-REAL m) by XBOOLE_1:1;

A12: (Int A) /\ (Int B) in the topology of T by PRE_TOPC:def 2;

not (Int A) /\ (Int B) is empty by A8;

then consider p being set such that

A13: p in (Int A) /\ (Int B) ;

reconsider p = p as Point of T by A13;

A14: dom hB = [#] (T | B) by A9, TOPS_2:def 5;

then A15: hB . p in hB .: IB by A13, FUNCT_1:def 6;

p in Int B by A13, XBOOLE_0:def 4;

then not Tdisk (pB,s) is empty by A14, A3;

then reconsider f = hB | IB as Function of ((T | B) | IB),((Tdisk (pB,s)) | (hB .: IB)) by A13, JORDAN24:12;

A16: Int A c= A by TOPS_1:16;

IB /\ B = IB by A3, A4, XBOOLE_1:1, XBOOLE_1:28;

then IB in the topology of (T | B) by A12, A5, PRE_TOPC:def 4;

then IB is open by PRE_TOPC:def 2;

then hB .: IB is open by A13, A9, TOPGRP_1:25, A2;

then not Int hBI is empty by A13, A2, Th13;

then not hBI is boundary ;

then A17: ind hBI = m by Th6;

A18: (Int A) /\ (Int B) c= Int A by XBOOLE_1:17;

A19: (Tdisk (pB,s)) | (hB .: IB) = (TOP-REAL m) | hBI by PRE_TOPC:7, A11;

then reconsider F = f as Function of (T | ((Int A) /\ (Int B))),((TOP-REAL m) | hBI) by A10;

F is being_homeomorphism by A9, METRIZTS:2, A19, A10;

then A20: F " is being_homeomorphism by TOPS_2:56, A15;

consider hA being Function of (T | A),(Tdisk (pA,r)) such that

A21: hA is being_homeomorphism by A6, T_0TOPSP:def 1;

A22: [#] (T | A) = A by PRE_TOPC:def 5;

then reconsider IA = (Int A) /\ (Int B) as Subset of (T | A) by A16, A18, XBOOLE_1:1;

A23: (T | A) | IA = T | ((Int A) /\ (Int B)) by A16, A18, XBOOLE_1:1, PRE_TOPC:7;

A24: dom hA = [#] (T | A) by A21, TOPS_2:def 5;

then A25: hA . p in hA .: IA by A13, FUNCT_1:def 6;

p in Int A by A13, XBOOLE_0:def 4;

then not Tdisk (pA,r) is empty by A24, A16;

then reconsider g = hA | IA as Function of ((T | A) | IA),((Tdisk (pA,r)) | (hA .: IA)) by A13, JORDAN24:12;

A26: [#] (Tdisk (pA,r)) = cl_Ball (pA,r) by PRE_TOPC:def 5;

then reconsider hAI = hA .: IA as Subset of (TOP-REAL n) by XBOOLE_1:1;

A27: (Tdisk (pA,r)) | (hA .: IA) = (TOP-REAL n) | hAI by PRE_TOPC:7, A26;

then reconsider G = g as Function of (T | ((Int A) /\ (Int B))),((TOP-REAL n) | hAI) by A23;

reconsider GF = G * (F ") as Function of ((TOP-REAL m) | hBI),((TOP-REAL n) | hAI) by A13;

G is being_homeomorphism by A21, METRIZTS:2, A27, A23;

then GF is being_homeomorphism by A20, TOPS_2:57, A25, A15, A13;

then hBI,hAI are_homeomorphic by T_0TOPSP:def 1, METRIZTS:def 1;

then A28: ind hBI = ind hAI by TOPDIM_1:27;

IA /\ A = IA by A16, A18, XBOOLE_1:1, XBOOLE_1:28;

then IA in the topology of (T | A) by A12, A22, PRE_TOPC:def 4;

then IA is open by PRE_TOPC:def 2;

then hA .: IA is open by A13, A21, TOPGRP_1:25, A1;

then not Int hAI is empty by A13, A1, Th13;

then not hAI is boundary ;

hence n = m by A17, Th6, A28; :: thesis: verum

for A, B being Subset of T

for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let T be non empty TopSpace; :: thesis: for A, B being Subset of T

for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let A, B be Subset of T; :: thesis: for r, s being Real st r > 0 & s > 0 holds

for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let r, s be Real; :: thesis: ( r > 0 & s > 0 implies for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m )

assume that

A1: r > 0 and

A2: s > 0 ; :: thesis: for pA being Point of (TOP-REAL n)

for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

A3: Int B c= B by TOPS_1:16;

A4: (Int A) /\ (Int B) c= Int B by XBOOLE_1:17;

A5: [#] (T | B) = B by PRE_TOPC:def 5;

then reconsider IB = (Int A) /\ (Int B) as Subset of (T | B) by A3, A4, XBOOLE_1:1;

let pA be Point of (TOP-REAL n); :: thesis: for pB being Point of (TOP-REAL m) st T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B holds

n = m

let pB be Point of (TOP-REAL m); :: thesis: ( T | A, Tdisk (pA,r) are_homeomorphic & T | B, Tdisk (pB,s) are_homeomorphic & Int A meets Int B implies n = m )

assume that

A6: T | A, Tdisk (pA,r) are_homeomorphic and

A7: T | B, Tdisk (pB,s) are_homeomorphic and

A8: Int A meets Int B ; :: thesis: n = m

consider hB being Function of (T | B),(Tdisk (pB,s)) such that

A9: hB is being_homeomorphism by A7, T_0TOPSP:def 1;

A10: (T | B) | IB = T | ((Int A) /\ (Int B)) by A3, A4, XBOOLE_1:1, PRE_TOPC:7;

A11: [#] (Tdisk (pB,s)) = cl_Ball (pB,s) by PRE_TOPC:def 5;

then reconsider hBI = hB .: IB as Subset of (TOP-REAL m) by XBOOLE_1:1;

A12: (Int A) /\ (Int B) in the topology of T by PRE_TOPC:def 2;

not (Int A) /\ (Int B) is empty by A8;

then consider p being set such that

A13: p in (Int A) /\ (Int B) ;

reconsider p = p as Point of T by A13;

A14: dom hB = [#] (T | B) by A9, TOPS_2:def 5;

then A15: hB . p in hB .: IB by A13, FUNCT_1:def 6;

p in Int B by A13, XBOOLE_0:def 4;

then not Tdisk (pB,s) is empty by A14, A3;

then reconsider f = hB | IB as Function of ((T | B) | IB),((Tdisk (pB,s)) | (hB .: IB)) by A13, JORDAN24:12;

A16: Int A c= A by TOPS_1:16;

IB /\ B = IB by A3, A4, XBOOLE_1:1, XBOOLE_1:28;

then IB in the topology of (T | B) by A12, A5, PRE_TOPC:def 4;

then IB is open by PRE_TOPC:def 2;

then hB .: IB is open by A13, A9, TOPGRP_1:25, A2;

then not Int hBI is empty by A13, A2, Th13;

then not hBI is boundary ;

then A17: ind hBI = m by Th6;

A18: (Int A) /\ (Int B) c= Int A by XBOOLE_1:17;

A19: (Tdisk (pB,s)) | (hB .: IB) = (TOP-REAL m) | hBI by PRE_TOPC:7, A11;

then reconsider F = f as Function of (T | ((Int A) /\ (Int B))),((TOP-REAL m) | hBI) by A10;

F is being_homeomorphism by A9, METRIZTS:2, A19, A10;

then A20: F " is being_homeomorphism by TOPS_2:56, A15;

consider hA being Function of (T | A),(Tdisk (pA,r)) such that

A21: hA is being_homeomorphism by A6, T_0TOPSP:def 1;

A22: [#] (T | A) = A by PRE_TOPC:def 5;

then reconsider IA = (Int A) /\ (Int B) as Subset of (T | A) by A16, A18, XBOOLE_1:1;

A23: (T | A) | IA = T | ((Int A) /\ (Int B)) by A16, A18, XBOOLE_1:1, PRE_TOPC:7;

A24: dom hA = [#] (T | A) by A21, TOPS_2:def 5;

then A25: hA . p in hA .: IA by A13, FUNCT_1:def 6;

p in Int A by A13, XBOOLE_0:def 4;

then not Tdisk (pA,r) is empty by A24, A16;

then reconsider g = hA | IA as Function of ((T | A) | IA),((Tdisk (pA,r)) | (hA .: IA)) by A13, JORDAN24:12;

A26: [#] (Tdisk (pA,r)) = cl_Ball (pA,r) by PRE_TOPC:def 5;

then reconsider hAI = hA .: IA as Subset of (TOP-REAL n) by XBOOLE_1:1;

A27: (Tdisk (pA,r)) | (hA .: IA) = (TOP-REAL n) | hAI by PRE_TOPC:7, A26;

then reconsider G = g as Function of (T | ((Int A) /\ (Int B))),((TOP-REAL n) | hAI) by A23;

reconsider GF = G * (F ") as Function of ((TOP-REAL m) | hBI),((TOP-REAL n) | hAI) by A13;

G is being_homeomorphism by A21, METRIZTS:2, A27, A23;

then GF is being_homeomorphism by A20, TOPS_2:57, A25, A15, A13;

then hBI,hAI are_homeomorphic by T_0TOPSP:def 1, METRIZTS:def 1;

then A28: ind hBI = ind hAI by TOPDIM_1:27;

IA /\ A = IA by A16, A18, XBOOLE_1:1, XBOOLE_1:28;

then IA in the topology of (T | A) by A12, A22, PRE_TOPC:def 4;

then IA is open by PRE_TOPC:def 2;

then hA .: IA is open by A13, A21, TOPGRP_1:25, A1;

then not Int hAI is empty by A13, A1, Th13;

then not hAI is boundary ;

hence n = m by A17, Th6, A28; :: thesis: verum