let f be Function; :: thesis: for n being Nat st rng f c= dom f holds
( dom (iter (f,n)) = dom f & rng (iter (f,n)) c= dom f )
let n be Nat; :: thesis: ( rng f c= dom f implies ( dom (iter (f,n)) = dom f & rng (iter (f,n)) c= dom f ) )
defpred S₁[ Nat] means ( dom (iter (f,$1)) = dom f & rng (iter (f,$1)) c= dom f );
assume rng f c= dom f ; :: thesis: ( dom (iter (f,n)) = dom f & rng (iter (f,n)) c= dom f )
then iter (f,0) = id (dom f) by Lm1;
then A1: S₁[ 0 ] by RELAT_1:45;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: ( dom (iter (f,k)) = dom f & rng (iter (f,k)) c= dom f ) ; :: thesis: S₁[k + 1]
( iter (f,(k + 1)) = f * (iter (f,k)) & iter (f,(k + 1)) = (iter (f,k)) * f ) by FUNCT_7:69, FUNCT_7:71;
then ( dom (iter (f,(k + 1))) = dom (iter (f,k)) & rng (iter (f,(k + 1))) c= rng (iter (f,k)) ) by A3, RELAT_1:26, RELAT_1:27;
hence S₁[k + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence ( dom (iter (f,n)) = dom f & rng (iter (f,n)) c= dom f ) ; :: thesis: verum