let T be non empty RelStr ; :: thesis: for A, B being Subset of T
for n being Nat holds Finf ((A \/ B),n) = (Finf (A,n)) \/ (Finf (B,n))
let A, B be Subset of T; :: thesis: for n being Nat holds Finf ((A \/ B),n) = (Finf (A,n)) \/ (Finf (B,n))
defpred S₁[ Nat] means (Finf (A \/ B)) . $1 = ((Finf A) . $1) \/ ((Finf B) . $1);
let n be Nat; :: thesis: Finf ((A \/ B),n) = (Finf (A,n)) \/ (Finf (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]
(Finf (A \/ B)) . (k + 1) = (Finf ((A \/ B),k)) ^f by Def6
.= ((Finf (A,k)) ^f) \/ ((Finf (B,k)) ^f) by A2, Th11
.= (Finf (A,(k + 1))) \/ ((Finf (B,k)) ^f) by Def6
.= ((Finf A) . (k + 1)) \/ ((Finf B) . (k + 1)) by Def6 ;
hence S₁[k + 1] ; :: thesis: verum

end;

(Finf (A \/ B)) . 0 = A \/ B by Def6
.= ((Finf A) . 0) \/ B by Def6
.= ((Finf A) . 0) \/ ((Finf B) . 0) by Def6 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence Finf ((A \/ B),n) = (Finf (A,n)) \/ (Finf (B,n)) ; :: thesis: verum