let T be non empty normal TopSpace; :: thesis:
let A, B be closed Subset of T; :: thesis:
assume that
A1: A <> {} and
A2: A misses B ; :: thesis:
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]

let n be Nat; :: thesis:
given G being Function of (dyadic n),(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:
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 A1, A4, URYSOHN1:24;
take F ; :: thesis:
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:

end;