let n be Element of NAT ; :: thesis: the_Complex_Space n is regular

let p be Point of (the_Complex_Space n); :: according to COMPTS_1:def 2 :: thesis: for b_{1} being Element of K10( the carrier of (the_Complex_Space n)) holds

( b_{1} = {} or not b_{1} is closed or not p in b_{1} ` or ex b_{2}, b_{3} being Element of K10( the carrier of (the_Complex_Space n)) st

( b_{2} is open & b_{3} is open & p in b_{2} & b_{1} c= b_{3} & b_{2} misses b_{3} ) )

let P be Subset of (the_Complex_Space n); :: thesis: ( P = {} or not P is closed or not p in P ` or ex b_{1}, b_{2} being Element of K10( the carrier of (the_Complex_Space n)) st

( b_{1} is open & b_{2} is open & p in b_{1} & P c= b_{2} & b_{1} misses b_{2} ) )

assume that

A1: P <> {} and

A2: ( P is closed & p in P ` ) ; :: thesis: ex b_{1}, b_{2} being Element of K10( the carrier of (the_Complex_Space n)) st

( b_{1} is open & b_{2} is open & p in b_{1} & P c= b_{2} & b_{1} misses b_{2} )

reconsider A = P as Subset of (COMPLEX n) ;

reconsider z1 = p as Element of COMPLEX n ;

set d = (dist (z1,A)) / 2;

reconsider K1 = Ball (z1,((dist (z1,A)) / 2)), K2 = Ball (A,((dist (z1,A)) / 2)) as Subset of (the_Complex_Space n) ;

take K1 ; :: thesis: ex b_{1} being Element of K10( the carrier of (the_Complex_Space n)) st

( K1 is open & b_{1} is open & p in K1 & P c= b_{1} & K1 misses b_{1} )

take K2 ; :: thesis: ( K1 is open & K2 is open & p in K1 & P c= K2 & K1 misses K2 )

A3: Ball (z1,((dist (z1,A)) / 2)) is open by SEQ_4:112;

Ball (A,((dist (z1,A)) / 2)) is open by A1, SEQ_4:122;

hence ( K1 is open & K2 is open ) by A3; :: thesis: ( p in K1 & P c= K2 & K1 misses K2 )

( A is closed & not p in P ) by A2, Th5, XBOOLE_0:def 5;

then 0 < (dist (z1,A)) / 2 by A1, SEQ_4:117, XREAL_1:215;

hence ( p in K1 & P c= K2 ) by SEQ_4:111, SEQ_4:121; :: thesis: K1 misses K2

assume K1 /\ K2 <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction

then consider x being Element of COMPLEX n such that

A4: x in (Ball (z1,((dist (z1,A)) / 2))) /\ (Ball (A,((dist (z1,A)) / 2))) by SUBSET_1:4;

x in K2 by A4, XBOOLE_0:def 4;

then A5: dist (x,A) < (dist (z1,A)) / 2 by SEQ_4:119;

x in K1 by A4, XBOOLE_0:def 4;

then |.(z1 - x).| < (dist (z1,A)) / 2 by SEQ_4:110;

then |.(z1 - x).| + (dist (x,A)) < ((dist (z1,A)) / 2) + ((dist (z1,A)) / 2) by A5, XREAL_1:8;

hence contradiction by A1, SEQ_4:118; :: thesis: verum

let p be Point of (the_Complex_Space n); :: according to COMPTS_1:def 2 :: thesis: for b

( b

( b

let P be Subset of (the_Complex_Space n); :: thesis: ( P = {} or not P is closed or not p in P ` or ex b

( b

assume that

A1: P <> {} and

A2: ( P is closed & p in P ` ) ; :: thesis: ex b

( b

reconsider A = P as Subset of (COMPLEX n) ;

reconsider z1 = p as Element of COMPLEX n ;

set d = (dist (z1,A)) / 2;

reconsider K1 = Ball (z1,((dist (z1,A)) / 2)), K2 = Ball (A,((dist (z1,A)) / 2)) as Subset of (the_Complex_Space n) ;

take K1 ; :: thesis: ex b

( K1 is open & b

take K2 ; :: thesis: ( K1 is open & K2 is open & p in K1 & P c= K2 & K1 misses K2 )

A3: Ball (z1,((dist (z1,A)) / 2)) is open by SEQ_4:112;

Ball (A,((dist (z1,A)) / 2)) is open by A1, SEQ_4:122;

hence ( K1 is open & K2 is open ) by A3; :: thesis: ( p in K1 & P c= K2 & K1 misses K2 )

( A is closed & not p in P ) by A2, Th5, XBOOLE_0:def 5;

then 0 < (dist (z1,A)) / 2 by A1, SEQ_4:117, XREAL_1:215;

hence ( p in K1 & P c= K2 ) by SEQ_4:111, SEQ_4:121; :: thesis: K1 misses K2

assume K1 /\ K2 <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction

then consider x being Element of COMPLEX n such that

A4: x in (Ball (z1,((dist (z1,A)) / 2))) /\ (Ball (A,((dist (z1,A)) / 2))) by SUBSET_1:4;

x in K2 by A4, XBOOLE_0:def 4;

then A5: dist (x,A) < (dist (z1,A)) / 2 by SEQ_4:119;

x in K1 by A4, XBOOLE_0:def 4;

then |.(z1 - x).| < (dist (z1,A)) / 2 by SEQ_4:110;

then |.(z1 - x).| + (dist (x,A)) < ((dist (z1,A)) / 2) + ((dist (z1,A)) / 2) by A5, XREAL_1:8;

hence contradiction by A1, SEQ_4:118; :: thesis: verum