let n be Nat; :: thesis: for R being Relation holds
( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R )
let R be Relation; :: thesis: ( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R )
defpred S₁[ Nat] means ( dom (iter (R,$1)) c= field R & rng (iter (R,$1)) c= field R );
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
iter (R,(k + 1)) = (iter (R,k)) * R by Th70;
then A2: dom (iter (R,(k + 1))) c= dom (iter (R,k)) by RELAT_1:25;
iter (R,(k + 1)) = R * (iter (R,k)) by Th68;
then A3: rng (iter (R,(k + 1))) c= rng (iter (R,k)) by RELAT_1:26;
assume ( dom (iter (R,k)) c= field R & rng (iter (R,k)) c= field R ) ; :: thesis: S₁[k + 1]
hence S₁[k + 1] by A2, A3, XBOOLE_1:1; :: thesis: verum

end;

iter (R,0) = id (field R) by Th67;
then A4: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A1);
hence ( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R ) ; :: thesis: verum