begin
theorem
canceled;
theorem Th2:
:: deftheorem Def1 defines non-zero TOPRNS_1:def 1 :
theorem Th3:
theorem Th4:
:: deftheorem Def2 defines + TOPRNS_1:def 2 :
:: deftheorem Def3 defines * TOPRNS_1:def 3 :
:: deftheorem Def4 defines - TOPRNS_1:def 4 :
:: deftheorem defines - TOPRNS_1:def 5 :
:: deftheorem TOPRNS_1:def 6 :
canceled;
:: deftheorem Def7 defines |. TOPRNS_1:def 7 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th8:
theorem
theorem
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem
theorem Th17:
theorem Th18:
theorem
theorem
theorem
theorem Th22:
theorem
theorem Th24:
theorem Th25:
theorem
theorem
theorem Th28:
Lm1:
for N being Element of NAT
for w1, w2 being Point of st |.(w1 - w2).| = 0 holds
w1 = w2
Lm2:
for N being Element of NAT
for w1, w2 being Point of st w1 = w2 holds
|.(w1 - w2).| = 0
theorem
theorem Th30:
theorem
theorem
theorem Th33:
theorem
theorem
theorem
canceled;
theorem
canceled;
theorem
canceled;
:: deftheorem Def8 defines bounded TOPRNS_1:def 8 :
theorem Th39:
:: deftheorem Def9 defines convergent TOPRNS_1:def 9 :
:: deftheorem Def10 defines lim TOPRNS_1:def 10 :
theorem
canceled;
theorem Th41:
theorem Th42:
theorem Th43:
theorem Th44:
theorem Th45:
theorem Th46:
theorem
theorem
theorem
canceled;
theorem
theorem