begin
theorem Th1:
theorem Th2:
theorem Th3:
:: deftheorem defines max_diff_index EUCLID_9:def 1 :
for n being Nat
for f1, f2 being real-valued b1 -element FinSequence holds max_diff_index (f1,f2) = the Element of (abs (f1 - f2)) " {(sup (rng (abs (f1 - f2))))};
theorem
theorem Th5:
theorem Th6:
theorem
:: deftheorem defines @ EUCLID_9:def 2 :
for n being Nat
for e being Point of (Euclid n) holds @ e = e;
theorem Th8:
theorem Th9:
theorem Th10:
:: deftheorem Def3 defines Intervals EUCLID_9:def 3 :
for f being real-valued Function
for r being real number
for b3 being Function holds
( b3 = Intervals (f,r) iff ( dom b3 = dom f & ( for x being set st x in dom f holds
b3 . x = ].((f . x) - r),((f . x) + r).[ ) ) );
:: deftheorem defines OpenHypercube EUCLID_9:def 4 :
for n being Nat
for e being Point of (Euclid n)
for r being real number holds OpenHypercube (e,r) = product (Intervals (e,r));
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem
theorem
deffunc H1( Nat, Point of (Euclid $1)) -> set = { (OpenHypercube ($2,(1 / m))) where m is non zero Element of NAT : verum } ;
:: deftheorem defines OpenHypercubes EUCLID_9:def 5 :
for n being Nat
for e being Point of (Euclid n) holds OpenHypercubes e = { (OpenHypercube (e,(1 / m))) where m is non zero Element of NAT : verum } ;
theorem Th25:
theorem Th26:
theorem Th27:
theorem