begin
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem Th5:
:: deftheorem Def1 defines dominated_by_0 CATALAN2:def 1 :
theorem Th6:
theorem Th7:
Lm1:
for n, m, k being Element of NAT st n <= m holds
(m --> k) | n = n --> k
Lm2:
for k, m being Element of NAT holds Sum (k --> m) = k * m
Lm3:
for k being Element of NAT holds k --> 0 is dominated_by_0
theorem
Lm4:
for p, q being XFinSequence of NAT holds Sum (p ^ q) = (Sum p) + (Sum q)
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
:: deftheorem defines Sum CATALAN2:def 4 :
Lm6:
for Fr being XFinSequence of REAL st dom Fr = 0 holds
Sum Fr = 0
Lm7:
for Fr being XFinSequence of REAL st ( dom Fr = 1 or len Fr = 1 ) holds
Sum Fr = Fr . 0
Lm8:
for Fr being XFinSequence of REAL
for n being Element of NAT st n in dom Fr holds
(Sum (Fr | n)) + (Fr . n) = Sum (Fr | (n + 1))
Lm9:
for Fr1, Fr2 being XFinSequence of REAL st dom Fr1 = dom Fr2 & ( for n being Element of 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
theorem
theorem Th49:
theorem
theorem
theorem Th52:
theorem Th53:
theorem Th54:
theorem Th55:
theorem Th56:
theorem Th57:
theorem Th58:
theorem Th59:
begin
theorem