environ
vocabularies HIDDEN, NUMBERS, SUBSET_1, REAL_1, NORMSP_1, PARTFUN1, NAT_1, PRE_TOPC, TARSKI, RELAT_1, CARD_1, ARYTM_3, ARYTM_1, STRUCT_0, COMPLEX1, XBOOLE_0, XXREAL_0, VALUED_1, SUPINF_2, FUNCT_1, CFCONT_1, NFCONT_1, RCOMP_1, VALUED_0, SEQ_2, ORDINAL2, FCONT_1, FUNCT_2, SEQ_4, LOPBAN_1, ALI2, FUNCOP_1, POWER, SEQ_1, RSSPACE3, FUNCT_7, NFCONT_2;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FUNCOP_1, XCMPLX_0, XXREAL_0, XREAL_0, ORDINAL1, NUMBERS, COMPLEX1, REAL_1, NAT_1, RLVECT_1, STRUCT_0, PRE_TOPC, VALUED_1, SEQ_1, SEQ_2, VALUED_0, SEQ_4, ABIAN, POWER, VFUNCT_1, NORMSP_0, NORMSP_1, LOPBAN_1, RSSPACE3, NFCONT_1, RECDEF_1;
definitions ;
theorems TARSKI, ABSVALUE, XBOOLE_1, RLVECT_1, XCMPLX_1, FUNCOP_1, SEQ_1, SEQ_2, SERIES_1, NAT_1, POWER, RELAT_1, VFUNCT_1, LOPBAN_1, NORMSP_1, SEQ_4, FUNCT_2, RSSPACE3, FUNCT_7, LOPBAN_3, PARTFUN2, NFCONT_1, XREAL_1, COMPLEX1, XXREAL_0, ORDINAL1, PARTFUN1, VALUED_0, NORMSP_0;
schemes SEQ_1, NAT_1, FUNCT_2;
registrations XBOOLE_0, FUNCT_1, ORDINAL1, RELSET_1, XXREAL_0, XREAL_0, MEMBERED, STRUCT_0, VALUED_1, FUNCT_2, NUMBERS, VALUED_0, NORMSP_1, NAT_1;
constructors HIDDEN, REAL_1, SQUARE_1, COMPLEX1, SEQ_2, SEQ_4, POWER, ABIAN, VFUNCT_1, RSSPACE3, LOPBAN_3, NFCONT_1, RECDEF_1, RELSET_1, COMSEQ_2;
requirements HIDDEN, SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
equalities RLVECT_1;
expansions ;
Lm1:
( ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| = 0 iff x = y ) ) & ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| <> 0 iff x <> y ) ) & ( for X being RealNormSpace
for x, y being Point of X holds
( ||.(x - y).|| > 0 iff x <> y ) ) & ( for X being RealNormSpace
for x, y being Point of X holds ||.(x - y).|| = ||.(y - x).|| ) & ( for X being RealNormSpace
for x, y, z being Point of X
for e being Real st e > 0 & ||.(x - z).|| < e / 2 & ||.(z - y).|| < e / 2 holds
||.(x - y).|| < e ) & ( for X being RealNormSpace
for x, y, z being Point of X
for e being Real st e > 0 & ||.(x - z).|| < e / 2 & ||.(y - z).|| < e / 2 holds
||.(x - y).|| < e ) & ( for X being RealNormSpace
for x being Point of X st ( for e being Real st e > 0 holds
||.x.|| < e ) holds
x = 0. X ) & ( for X being RealNormSpace
for x, y being Point of X st ( for e being Real st e > 0 holds
||.(x - y).|| < e ) holds
x = y ) )
Lm2:
for K, L, e being Real st 0 < K & K < 1 & 0 < e holds
ex n being Nat st |.(L * (K to_power n)).| < e