let T be non empty RelStr ; :: thesis: for A being Subset of T
for n being Nat holds (Fint ((A `),n)) ` = Fcl (A,n)
let A be Subset of T; :: thesis: for n being Nat holds (Fint ((A `),n)) ` = Fcl (A,n)
defpred S₁[ Nat] means (Fint ((A `),$1)) ` = Fcl (A,$1);
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
Fcl (A,(k + 1)) = (Fcl (A,k)) ^b by Def2
.= ((((Fint ((A `),k)) `) `) ^i) ` by A2, FIN_TOPO:16
.= (Fint ((A `),(k + 1))) ` by Def4 ;
hence S₁[k + 1] ; :: thesis: verum

end;

(Fint ((A `),0)) ` = (A `) ` by Def4
.= Fcl (A,0) by Def2 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence for n being Nat holds (Fint ((A `),n)) ` = Fcl (A,n) ; :: thesis: verum