begin
Lm1:
{} in omega
by ORDINAL1:def 12;
Lm2:
omega is limit_ordinal
by ORDINAL1:def 12;
Lm3:
1 = succ {}
;
theorem
:: deftheorem Def1 defines ^ ORDINAL4:def 1 :
for fi, psi, b3 being T-Sequence holds
( b3 = fi ^ psi iff ( dom b3 = (dom fi) +^ (dom psi) & ( for A being Ordinal st A in dom fi holds
b3 . A = fi . A ) & ( for A being Ordinal st A in dom psi holds
b3 . ((dom fi) +^ A) = psi . A ) ) );
theorem Th2:
theorem Th3:
theorem
Lm4:
for fi being Ordinal-Sequence
for A being Ordinal st A is_limes_of fi holds
dom fi <> {}
theorem Th5:
theorem Th6:
theorem
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem Th16:
Lm5:
for f, g being Function
for X being set st rng f c= X holds
(g | X) * f = g * f
theorem
theorem
theorem Th19:
theorem Th20:
Lm6:
for A being Ordinal st A <> {} & A is limit_ordinal holds
for fi being Ordinal-Sequence st dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp ({},B) ) holds
{} is_limes_of fi
Lm7:
for A being Ordinal st A <> {} holds
for fi being Ordinal-Sequence st dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp (1,B) ) holds
1 is_limes_of fi
Lm8:
for C, A being Ordinal st A <> {} & A is limit_ordinal holds
ex fi being Ordinal-Sequence st
( dom fi = A & ( for B being Ordinal st B in A holds
fi . B = exp (C,B) ) & ex D being Ordinal st D is_limes_of fi )
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem
theorem
theorem
theorem
theorem Th30:
theorem
theorem
:: deftheorem ORDINAL4:def 2 :
canceled;
:: deftheorem ORDINAL4:def 3 :
canceled;
:: deftheorem defines 0-element_of ORDINAL4:def 4 :
for W being Universe holds 0-element_of W = {} ;
:: deftheorem defines 1-element_of ORDINAL4:def 5 :
for W being Universe holds 1-element_of W = 1;
theorem
canceled;
theorem
canceled;
theorem
theorem Th36:
theorem Th37:
theorem
begin
theorem
theorem