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

proof

A3: S₁[ 0 ] by Def8;
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 (Fdfl A,k) ^d c= A ^d by Th6;
then A5: Fdfl A,(k + 1) c= A ^d by Def8;
A ^d c= A by A1, Th3;
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: Fdfl A,n c= A
thus Fdfl A,n c= A by A2; :: thesis: verum