let X be set ; :: thesis: for n being Element of NAT holds iter ((id X),n) = id X
let n be Element of NAT ; :: thesis: iter ((id X),n) = id X
defpred S₁[ Element of NAT ] means iter ((id X),$1) = id X;
A1: 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 A2: S₁[k] ; :: thesis: S₁[k + 1]
thus iter ((id X),(k + 1)) = (iter ((id X),k)) * (id X) by Th71
.= id X by A2, FUNCT_2:17 ; :: thesis: verum

end;

( dom (id X) = X & rng (id X) = X ) by RELAT_1:45;
then id (field (id X)) = id X ;
then A3: S₁[ 0 ] by Th70;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A3, A1);
hence iter ((id X),n) = id X ; :: thesis: verum