let T be non empty RelStr ; :: thesis: for A, B being Subset of T
for n being Element of NAT holds Finf (A \/ B),n = (Finf A,n) \/ (Finf B,n)
let A, B be Subset of T; :: thesis: for n being Element of NAT holds Finf (A \/ B),n = (Finf A,n) \/ (Finf B,n)
defpred S₁[ Element of NAT ] means (Finf (A \/ B)) . $1 = ((Finf A) . $1) \/ ((Finf B) . $1);
A1: for n being Element of NAT holds S₁[n]

proof

A2: S₁[ 0 ]

proof

(Finf (A \/ B)) . 0 = A \/ B by Def6
.= ((Finf A) . 0 ) \/ B by Def6
.= ((Finf A) . 0 ) \/ ((Finf B) . 0 ) by Def6 ;
hence S₁[ 0 ] ; :: thesis: verum

end;

A3: 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 A4: 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 A4, 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;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence for n being Element of NAT holds S₁[n] ; :: thesis: verum

end;

let n be Element of NAT ; :: thesis: Finf (A \/ B),n = (Finf A,n) \/ (Finf B,n)
thus Finf (A \/ B),n = (Finf A,n) \/ (Finf B,n) by A1; :: thesis: verum