environ
vocabularies HIDDEN, NUMBERS, NAT_1, INT_1, FINSEQ_1, ARYTM_3, XXREAL_0, CARD_1, SUBSET_1, ARYTM_1, INT_2, COMPLEX1, RELAT_1, TARSKI, FUNCT_1, CLASSES1, CARD_3, ORDINAL4, RFINSEQ, XBOOLE_0, NEWTON, EULER_1, FINSET_1;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, CARD_1, RELAT_1, FUNCT_1, RECDEF_1, INT_1, INT_2, FINSET_1, NAT_1, NAT_D, NEWTON, EULER_1, CLASSES1, RFINSEQ, RVSUM_1, FINSEQ_1, FINSEQ_3, TREES_4, WSIERP_1;
definitions TARSKI, FUNCT_1, XBOOLE_0;
theorems INT_2, FINSEQ_1, NAT_1, ABSVALUE, INT_1, FUNCT_1, EULER_1, PREPOWER, RFINSEQ, FINSEQ_2, RVSUM_1, FINSEQ_3, NEWTON, RELAT_1, XBOOLE_1, NUMBERS, XCMPLX_1, XREAL_1, FINSEQ_5, XXREAL_0, ORDINAL1, NAT_D, CLASSES1;
schemes NAT_1, FINSEQ_1;
registrations FUNCT_1, ORDINAL1, RELSET_1, XXREAL_0, XREAL_0, NAT_1, INT_1, FINSEQ_1, NEWTON, VALUED_0, FINSET_1, CARD_1;
constructors HIDDEN, REAL_1, SQUARE_1, NAT_1, NAT_D, NEWTON, RFINSEQ, WSIERP_1, EULER_1, RECDEF_1, CLASSES1, BINOP_2, RELSET_1, NUMBERS;
requirements HIDDEN, REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
equalities RELAT_1;
expansions FUNCT_1;
Lm1:
for t being Integer holds
( t < 1 iff t <= 0 )
Lm2:
for a being Nat st a <> 0 holds
a - 1 >= 0
Lm3:
for z being Integer holds 1 gcd z = 1
Lm4:
for m being Nat
for z being Integer st 1 - m <= z & z <= m - 1 & m divides z holds
z = 0