let T be non empty normal TopSpace; :: thesis: for A, B being closed Subset of T st A <> {} & A misses B holds
for n being Element of NAT
for G being Drizzle of A,B,n ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r
let A, B be closed Subset of T; :: thesis: ( A <> {} & A misses B implies for n being Element of NAT
for G being Drizzle of A,B,n ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r )
assume A1: ( A <> {} & A misses B ) ; :: thesis: for n being Element of NAT
for G being Drizzle of A,B,n ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r
let n be Element of NAT ; :: thesis: for G being Drizzle of A,B,n ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r
let G be Drizzle of A,B,n; :: thesis: ex F being Drizzle of A,B,n + 1 st
for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r
A2: ( A c= G . 0 & B = ([#] T) \ (G . 1) ) by A1, Def1;
for r1, r2 being Element of dyadic n st r1 < r2 holds
( G . r1 is open & G . r2 is open & Cl (G . r1) c= G . r2 ) by A1, Def1;
then consider F being Function of (dyadic (n + 1)),(bool the carrier of T) such that
A3: ( A c= F . 0 & B = ([#] T) \ (F . 1) & ( for r1, r2, r being Element of dyadic (n + 1) st r1 < r2 holds
( F . r1 is open & F . r2 is open & Cl (F . r1) c= F . r2 & ( r in dyadic n implies F . r = G . r ) ) ) ) by A1, A2, URYSOHN1:32;
for r1, r2 being Element of dyadic (n + 1) st r1 < r2 holds
( F . r1 is open & F . r2 is open & Cl (F . r1) c= F . r2 ) by A3;
then reconsider F = F as Drizzle of A,B,n + 1 by A1, A3, Def1;
take F ; :: thesis: for r being Element of dyadic (n + 1) st r in dyadic n holds
F . r = G . r
let r be Element of dyadic (n + 1); :: thesis: ( r in dyadic n implies F . r = G . r )
assume A4: r in dyadic n ; :: thesis: F . r = G . r
( 0 in dyadic (n + 1) & 1 in dyadic (n + 1) ) by URYSOHN1:12;
then consider r1, r2 being Element of dyadic (n + 1) such that
A5: ( r1 = 0 & r2 = 1 ) ;
thus F . r = G . r by A3, A4, A5; :: thesis: verum