begin
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th5:
:: deftheorem Def1 defines dominated_by_0 CATALAN2:def 1 :
theorem Th6:
theorem Th7:
Lm1:
for n, m, k being Nat st n <= m holds
(m --> k) | n = n --> k
Lm3:
for k being Nat holds k --> 0 is dominated_by_0
theorem
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem
begin
:: deftheorem Def2 defines Domin_0 CATALAN2:def 2 :
theorem Th24:
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
theorem Th32:
theorem Th33:
theorem Th34:
theorem
theorem
theorem Th37:
theorem Th38:
Lm5:
for D being set holds D ^omega is functional
:: deftheorem Def3 defines OMEGA CATALAN2:def 3 :
theorem
theorem Th40:
theorem Th41:
theorem Th42:
theorem Th43:
theorem Th44:
theorem
theorem
begin
definition
canceled;
end;
:: deftheorem CATALAN2:def 4 :
canceled;
Lm7:
for Fr being XFinSequence of REAL st ( dom Fr = 1 or len Fr = 1 ) holds
Sum Fr = Fr . 0
Lm9:
for Fr1, Fr2 being XFinSequence of REAL st dom Fr1 = dom Fr2 & ( for n being Nat st n in len Fr1 holds
Fr1 . n = Fr2 . ((len Fr1) -' (1 + n)) ) holds
Sum Fr1 = Sum Fr2
:: deftheorem Def5 defines (##) CATALAN2:def 5 :
theorem
canceled;
theorem
theorem Th49:
theorem
theorem
theorem Th52:
theorem Th53:
theorem Th54:
theorem Th55:
theorem Th56:
theorem Th57:
theorem Th58:
theorem
canceled;
begin
theorem