let n be Element of NAT ; :: thesis: for r1 being Real
for A being Subset of (COMPLEX n) st A <> {} holds
Ball A,r1 is open
let r1 be Real; :: thesis: for A being Subset of (COMPLEX n) st A <> {} holds
Ball A,r1 is open
let A be Subset of (COMPLEX n); :: thesis: ( A <> {} implies Ball A,r1 is open )
assume A1: A <> {} ; :: thesis: Ball A,r1 is open
let x be Element of COMPLEX n; :: according to COMPLSP1:def 15 :: thesis: ( x in Ball A,r1 implies ex r being Real st
( 0 < r & ( for z being Element of COMPLEX n st |.z.| < r holds
x + z in Ball A,r1 ) ) )
assume x in Ball A,r1 ; :: thesis: ex r being Real st
( 0 < r & ( for z being Element of COMPLEX n st |.z.| < r holds
x + z in Ball A,r1 ) )
then A2: dist x,A < r1 by Th90;
take r = r1 - (dist x,A); :: thesis: ( 0 < r & ( for z being Element of COMPLEX n st |.z.| < r holds
x + z in Ball A,r1 ) )
thus 0 < r by A2, XREAL_1:52; :: thesis: for z being Element of COMPLEX n st |.z.| < r holds
x + z in Ball A,r1
let z be Element of COMPLEX n; :: thesis: ( |.z.| < r implies x + z in Ball A,r1 )
assume |.z.| < r ; :: thesis: x + z in Ball A,r1
then A3: |.z.| + (dist x,A) < r + (dist x,A) by XREAL_1:8;
dist (x + z),A <= |.z.| + (dist x,A) by A1, Th86;
then dist (x + z),A < r + (dist x,A) by A3, XXREAL_0:2;
hence x + z in Ball A,r1 ; :: thesis: verum