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 Nat ex G being Function of (),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( 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 ) ) )

let A, B be closed Subset of T; :: thesis: ( A <> {} & A misses B implies for n being Nat ex G being Function of (),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( 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 ) ) ) )

assume that
A1: A <> {} and
A2: A misses B ; :: thesis: for n being Nat ex G being Function of (),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( 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 ) ) )

defpred S1[ Nat] means ex G being Function of (dyadic \$1),(bool the carrier of T) st
( A c= G . 0 & B = ([#] T) \ (G . 1) & ( for r1, r2 being Element of dyadic \$1 st r1 < r2 holds
( G . r1 is open & G . r2 is open & Cl (G . r1) c= G . r2 ) ) );
A3: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
given G being Function of (),(bool the carrier of T) such that A4: ( A c= G . 0 & B = ([#] T) \ (G . 1) & ( 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 ) ) ) ; :: thesis: S1[n + 1]
consider F being Function of (dyadic (n + 1)),(bool the carrier of T) such that
A5: ( 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 ;
take F ; :: thesis: ( A c= F . 0 & B = ([#] T) \ (F . 1) & ( 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 ) ) )

thus ( A c= F . 0 & B = ([#] T) \ (F . 1) & ( 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 A5; :: thesis: verum
end;
A6: S1[ 0 ] by A1, A2, Lm1;
thus for n being Nat holds S1[n] from NAT_1:sch 2(A6, A3); :: thesis: verum