environ
vocabularies HIDDEN, ORDINAL2, ORDINAL1, FUNCT_1, XBOOLE_0, RELAT_1, TARSKI, ORDINAL3, SUBSET_1, CLASSES2, ZFMISC_1, CARD_1, ORDINAL4;
notations HIDDEN, TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, ORDINAL1, ORDINAL2, ORDINAL3, CARD_1, CLASSES2;
definitions ORDINAL1, TARSKI, ORDINAL2, XBOOLE_0, RELAT_1;
theorems ZFMISC_1, FUNCT_1, GRFUNC_1, ORDINAL1, ORDINAL2, ORDINAL3, CARD_2, CLASSES2, RELAT_1, CLASSES1, XBOOLE_0, XBOOLE_1, FUNCT_2, RELSET_1, CARD_1;
schemes ORDINAL1, ORDINAL2, FINSET_1;
registrations XBOOLE_0, FUNCT_1, ORDINAL1, ORDINAL2, ORDINAL3, CARD_1, CLASSES2, RELSET_1;
constructors HIDDEN, WELLORD2, FUNCOP_1, ORDINAL3, CARD_1, CLASSES1, CLASSES2, RELSET_1;
requirements HIDDEN, SUBSET, BOOLE, NUMERALS;
equalities ORDINAL1, ORDINAL2;
expansions TARSKI, ORDINAL2, XBOOLE_0;
Lm1:
{} in omega
by ORDINAL1:def 11;
Lm2:
omega is limit_ordinal
by ORDINAL1:def 11;
Lm3:
1 = succ {}
;
Lm4:
for fi being Ordinal-Sequence
for A being Ordinal st A is_limes_of fi holds
dom fi <> {}
Lm5:
for f, g being Function
for X being set st rng f c= X holds
(g | X) * f = g * f
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
0 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 )
Lm9:
0 = {}
;