let T be non empty RelStr ; :: thesis: for A being Subset of T st T is filled holds
for n being Element of NAT holds A c= Fcl A,n
let A be Subset of T; :: thesis: ( T is filled implies for n being Element of NAT holds A c= Fcl A,n )
assume A1: T is filled ; :: thesis: for n being Element of NAT holds A c= Fcl A,n
defpred S₁[ Element of NAT ] means A c= (Fcl A) . $1;
A2: for n being Element of NAT holds S₁[n]

proof

A3: S₁[ 0 ] by Def2;
A4: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then A ^b c= (Fcl A,k) ^b by FIN_TOPO:19;
then A5: A ^b c= Fcl A,(k + 1) by Def2;
A c= A ^b by A1, FIN_TOPO:18;
hence S₁[k + 1] by A5, XBOOLE_1:1; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A4);
hence for n being Element of NAT holds S₁[n] ; :: thesis: verum

end;

let n be Element of NAT ; :: thesis: A c= Fcl A,n
thus A c= Fcl A,n by A2; :: thesis: verum