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