environ
vocabularies HIDDEN, NUMBERS, BHSP_1, PRE_TOPC, REAL_1, COMPLEX1, ARYTM_1, ARYTM_3, XXREAL_0, CARD_1, SUBSET_1, RELAT_1, HAHNBAN, BHSP_6, FINSET_1, XBOOLE_0, TARSKI, BHSP_5, STRUCT_0, BINOP_2, FUNCT_1, ZFMISC_1, NAT_1, SEQ_2, ALGSTR_0, BINOP_1, SETWISEO, PROB_2, FINSEQ_1, FINSOP_1, SQUARE_1, NORMSP_1, SEMI_AF1, SUPINF_2, RSSPACE2;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ZFMISC_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, BINOP_2, STRUCT_0, ALGSTR_0, NORMSP_0, COMPLEX1, REAL_1, NAT_1, FUNCT_1, FUNCT_2, RELAT_1, PRE_TOPC, BHSP_1, SQUARE_1, SEQ_2, BINOP_1, FINSET_1, SETWISEO, HAHNBAN, FINSEQ_1, FINSOP_1, BHSP_5, BHSP_6, RSSPACE2, PARTFUN1, XXREAL_0;
definitions TARSKI;
theorems XBOOLE_1, SQUARE_1, ZFMISC_1, ABSVALUE, FUNCT_1, NAT_1, FUNCT_2, RLVECT_1, SEQ_2, SEQ_4, BHSP_1, BHSP_5, BHSP_6, UNIFORM1, CHAIN_1, XCMPLX_1, BINOP_2, XREAL_1, COMPLEX1, XXREAL_0, ORDINAL1;
schemes NAT_1, FUNCT_2;
registrations XBOOLE_0, FUNCT_1, ORDINAL1, RELSET_1, FUNCT_2, FINSET_1, NUMBERS, XREAL_0, BINOP_2, STRUCT_0, BHSP_5, SQUARE_1, NAT_1, VALUED_0;
constructors HIDDEN, BINOP_1, DOMAIN_1, SETWISEO, REAL_1, SQUARE_1, BINOP_2, COMPLEX1, FINSOP_1, BHSP_3, BHSP_5, BHSP_6, RELSET_1, COMSEQ_2;
requirements HIDDEN, SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
equalities SQUARE_1, RLVECT_1;
expansions TARSKI;
Lm1:
for x, y, z, e being Real st |.(x - y).| < e / 2 & |.(y - z).| < e / 2 holds
|.(x - z).| < e
Lm2:
for p being Real st p > 0 holds
ex k being Nat st 1 / (k + 1) < p
Lm3:
for p being Real
for m being Nat st p > 0 holds
ex i being Nat st
( 1 / (i + 1) < p & i >= m )
theorem Th2:
for
X being
RealUnitarySpace st the
addF of
X is
commutative & the
addF of
X is
associative & the
addF of
X is
having_a_unity holds
for
S being
finite OrthogonalFamily of
X st not
S is
empty holds
for
I being
Function of the
carrier of
X, the
carrier of
X st
S c= dom I & ( for
y being
Point of
X st
y in S holds
I . y = y ) holds
for
H being
Function of the
carrier of
X,
REAL st
S c= dom H & ( for
y being
Point of
X st
y in S holds
H . y = y .|. y ) holds
(setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) = setopfunc (
S, the
carrier of
X,
REAL,
H,
addreal)
Lm4:
for p1, p2 being Real st ( for e being Real st 0 < e holds
|.(p1 - p2).| < e ) holds
p1 = p2