let T be non empty RelStr ; :: thesis: for A being Subset of T
for n being Nat holds Fdfl (A,n) = (Finf ((A `),n)) `
let A be Subset of T; :: thesis: for n being Nat holds Fdfl (A,n) = (Finf ((A `),n)) `
defpred S1[ Nat] means (Fdfl A) . $1 = ((Finf (A `)) . (In ($1,NAT))) ` ;
let n be Nat; :: thesis: Fdfl (A,n) = (Finf ((A `),n)) `
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
reconsider kk = k as Element of NAT by ORDINAL1:def 12;
assume A2: S1[k] ; :: thesis: S1[k + 1]
(Fdfl A) . (k + 1) = (Fdfl (A,k)) ^d by Def8;
then (Fdfl A) . (k + 1) = ((((Fdfl A) . kk) `) ^f) ` by Th4
.= ((Finf (A `)) . (In ((k + 1),NAT))) ` by A2, Def6 ;
hence S1[k + 1] ; :: thesis: verum
end;
((Finf (A `)) . 0) ` = (A `) ` by Def6
.= A ;
then A3: S1[ 0 ] by Def8;
reconsider n = n as Element of NAT by ORDINAL1:def 12;
for n being Nat holds S1[n] from NAT_1:sch 2(A3, A1);
 then S1[n] ;
hence Fdfl (A,n) = (Finf ((A `),n)) ` ; :: thesis: verum