let g be Function; :: thesis: for m being Nat
for f being Function st f c= g holds
iter (f,m) c= iter (g,m)
let m be Nat; :: thesis: for f being Function st f c= g holds
iter (f,m) c= iter (g,m)
let f be Function; :: thesis: ( f c= g implies iter (f,m) c= iter (g,m) )
assume A1: f c= g ; :: thesis: iter (f,m) c= iter (g,m)
defpred S₁[ Nat] means iter (f,$1) c= iter (g,$1);
A2: S₁[ 0 ]

proof

A3: ( iter (f,0) = id (field f) & iter (g,0) = id (field g) ) by FUNCT_7:68;
( dom f c= dom g & rng f c= rng g ) by RELAT_1:11, A1;
then (dom f) \/ (rng f) c= (dom g) \/ (rng g) by XBOOLE_1:13;
hence S₁[ 0 ] by A3, FUNCT_4:3; :: thesis: verum

end;

A4: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then f * (iter (f,n)) c= g * (iter (g,n)) by A1, RELAT_1:31;
then iter (f,(n + 1)) c= g * (iter (g,n)) by FUNCT_7:71;
hence S₁[n + 1] by FUNCT_7:71; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A4);
hence iter (f,m) c= iter (g,m) ; :: thesis: verum