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 );
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: ( dom (iter (f,k)) = dom f & rng (iter (f,k)) c= dom f ) ; :: thesis: S₁[k + 1]
iter (f,(k + 1)) = (iter (f,k)) * f by Th71;
then A3: rng (iter (f,(k + 1))) c= rng (iter (f,k)) by RELAT_1:45;
iter (f,(k + 1)) = f * (iter (f,k)) by Th73;
hence S₁[k + 1] by A2, A3, RELAT_1:46, XBOOLE_1:1; :: thesis: verum

end;

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 Lm4;
then A4: S₁[ 0 ] by RELAT_1:71;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A1);
hence ( dom (iter (f,n)) = dom f & rng (iter (f,n)) c= dom f ) ; :: thesis: verum