let T be non empty RelStr ; :: thesis: for A being Subset of T st T is filled holds
for n being Nat holds Fdfl (A,n) c= A
let A be Subset of T; :: thesis: ( T is filled implies for n being Nat holds Fdfl (A,n) c= A )
defpred S1[ Nat] means (Fdfl A) . $1 c= A;
assume A1: T is filled ; :: thesis: for n being Nat holds Fdfl (A,n) c= A
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be 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 Nat; :: thesis: Fdfl (A,n) c= A
A4: S1[ 0 ] by Def8;
for n being Nat holds S1[n] from NAT_1:sch 2(A4, A2);
 hence Fdfl (A,n) c= A ; :: thesis: verum