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 )
defpred S₁[ Element of NAT ] means (Fdfl A) . $1 c= A;
assume A1: T is filled ; :: thesis: for n being Element of NAT holds Fdfl (A,n) c= A
A2: 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 A3: Fdfl (A,(k + 1)) c= A ^d by Def8;
A ^d c= A by A1, Th3;
hence S₁[k + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

let n be Element of NAT ; :: thesis: Fdfl (A,n) c= A
A4: S₁[ 0 ] by Def8;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A2);
hence Fdfl (A,n) c= A ; :: thesis: verum