let n be Nat; :: thesis: iter ({},n) = {}
defpred S₁[ Nat] means iter ({},$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 iter ({},k) = {} ; :: thesis: S₁[k + 1]
thus iter ({},(k + 1)) = (iter ({},k)) * {} by Th68
.= {} ; :: thesis: verum

end;

iter ({},0) = id (field {}) by Th67
.= {} ;
then A2: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A1);
hence iter ({},n) = {} ; :: thesis: verum