let T be non empty RelStr ; :: thesis: for A, B being Subset of T
for n being Nat holds Fint ((A /\ B),n) = (Fint (A,n)) /\ (Fint (B,n))
let A, B be Subset of T; :: thesis: for n being Nat holds Fint ((A /\ B),n) = (Fint (A,n)) /\ (Fint (B,n))
defpred S₁[ Nat] means (Fint (A /\ B)) . $1 = ((Fint A) . $1) /\ ((Fint B) . $1);
let n be Nat; :: thesis: Fint ((A /\ B),n) = (Fint (A,n)) /\ (Fint (B,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] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
(Fint (A /\ B)) . (k + 1) = (Fint ((A /\ B),k)) ^i by Def4
.= ((Fint (A,k)) ^i) /\ ((Fint (B,k)) ^i) by A2, Th10
.= (Fint (A,(k + 1))) /\ ((Fint (B,k)) ^i) by Def4
.= ((Fint A) . (k + 1)) /\ ((Fint B) . (k + 1)) by Def4 ;
hence S₁[k + 1] ; :: thesis: verum

end;

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