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= Finf (A,n)
let A be Subset of T; :: thesis: ( T is filled implies for n being Element of NAT holds A c= Finf (A,n) )
defpred S₁[ Element of NAT ] means A c= (Finf A) . $1;
assume A1: T is filled ; :: thesis: for n being Element of NAT holds A c= Finf (A,n)
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 A ^f c= (Finf (A,k)) ^f by Th5;
then A3: A ^f c= Finf (A,(k + 1)) by Def6;
A c= A ^f by A1, Th1;
hence S₁[k + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

let n be Element of NAT ; :: thesis: A c= Finf (A,n)
A4: S₁[ 0 ] by Def6;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A2);
hence A c= Finf (A,n) ; :: thesis: verum