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 ()
for pB being Point of () 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 ()
for pB being Point of () 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 ()
for pB being Point of () 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 ()
for pB being Point of () 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 ()
for pB being Point of () 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 ;
let pA be Point of (); :: thesis: for pB being Point of () 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 (); :: 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 ;
A10: (T | B) | IB = T | ((Int A) /\ (Int B)) by ;
A11: [#] (Tdisk (pB,s)) = cl_Ball (pB,s) by PRE_TOPC:def 5;
then reconsider hBI = hB .: IB as Subset of () 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 ;
then A15: hB . p in hB .: IB by ;
p in Int B by ;
then not Tdisk (pB,s) is empty by ;
then reconsider f = hB | IB as Function of ((T | B) | IB),((Tdisk (pB,s)) | (hB .: IB)) by ;
A16: Int A c= A by TOPS_1:16;
IB /\ B = IB by ;
then IB in the topology of (T | B) by ;
then IB is open by PRE_TOPC:def 2;
then hB .: IB is open by ;
then not Int hBI is empty by ;
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) = () | hBI by ;
then reconsider F = f as Function of (T | ((Int A) /\ (Int B))),(() | hBI) by A10;
F is being_homeomorphism by ;
then A20: F " is being_homeomorphism by ;
consider hA being Function of (T | A),(Tdisk (pA,r)) such that
A21: hA is being_homeomorphism by ;
A22: [#] (T | A) = A by PRE_TOPC:def 5;
then reconsider IA = (Int A) /\ (Int B) as Subset of (T | A) by ;
A23: (T | A) | IA = T | ((Int A) /\ (Int B)) by ;
A24: dom hA = [#] (T | A) by ;
then A25: hA . p in hA .: IA by ;
p in Int A by ;
then not Tdisk (pA,r) is empty by ;
then reconsider g = hA | IA as Function of ((T | A) | IA),((Tdisk (pA,r)) | (hA .: IA)) by ;
A26: [#] (Tdisk (pA,r)) = cl_Ball (pA,r) by PRE_TOPC:def 5;
then reconsider hAI = hA .: IA as Subset of () by XBOOLE_1:1;
A27: (Tdisk (pA,r)) | (hA .: IA) = () | hAI by ;
then reconsider G = g as Function of (T | ((Int A) /\ (Int B))),(() | hAI) by A23;
reconsider GF = G * (F ") as Function of (() | hBI),(() | hAI) by A13;
G is being_homeomorphism by ;
then GF is being_homeomorphism by ;
then hBI,hAI are_homeomorphic by ;
then A28: ind hBI = ind hAI by TOPDIM_1:27;
IA /\ A = IA by ;
then IA in the topology of (T | A) by ;
then IA is open by PRE_TOPC:def 2;
then hA .: IA is open by ;
then not Int hAI is empty by ;
then not hAI is boundary ;
hence n = m by ; :: thesis: verum