environ
vocabularies HIDDEN, PRE_TOPC, RLVECT_1, FUNCT_1, ARYTM_3, ARYTM_1, FINSEQ_1, FINSEQ_2, XXREAL_2, FINSET_1, MEMBERED, RVSUM_1, RELAT_1, COMPLEX1, SQUARE_1, RCOMP_1, XBOOLE_0, PARTFUN1, NORMSP_1, LOPBAN_1, FDIFF_1, PDIFF_1, REAL_NS1, BORSUK_1, EUCLID, NUMBERS, STRUCT_0, CFCONT_1, FUNCT_4, REAL_1, SUBSET_1, CARD_1, TARSKI, XXREAL_0, CARD_3, NAT_1, SEQ_4, UNIALG_1, FCONT_1, XCMPLX_0;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FINSET_1, ORDINAL1, NUMBERS, XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, SQUARE_1, MEMBERED, COMPLEX1, XXREAL_2, FINSEQ_1, FINSEQ_2, RVSUM_1, SEQ_4, FUNCT_7, STRUCT_0, PRE_TOPC, RLVECT_1, NORMSP_0, EUCLID, LOPBAN_1, NFCONT_1, NDIFF_1, REAL_NS1, PDIFF_1;
definitions LOPBAN_1;
theorems ABSVALUE, EUCLID, XREAL_1, COMPLEX1, FINSEQ_1, FINSEQ_2, FINSEQ_3, ORDINAL1, NAT_1, FINSET_1, XXREAL_0, FUNCT_7, RVSUM_1, RELAT_1, FUNCT_1, FUNCT_2, SQUARE_1, NORMSP_1, LOPBAN_1, PARTFUN1, NFCONT_1, NDIFF_1, REAL_NS1, VALUED_1, PDIFF_1, XXREAL_2, XCMPLX_1, PDIFF_6, SEQ_4, VECTSP_1, PDIFF_7, LOPBAN_2, JORDAN2B, SERIES_3, CARD_1, XREAL_0;
schemes NAT_1, FINSEQ_1;
registrations FUNCT_1, ORDINAL1, RELSET_1, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, NAT_1, MEMBERED, VALUED_0, XXREAL_2, FINSEQ_1, FINSEQ_2, STRUCT_0, MONOID_0, EUCLID, LOPBAN_1, JORDAN2B, REAL_NS1, PDIFF_6, CARD_1, SQUARE_1;
constructors HIDDEN, RELSET_1, REAL_1, SQUARE_1, NAT_D, BINOP_2, FINSEQOP, SEQ_4, FINSEQ_7, MONOID_0, RSSPACE3, NFCONT_1, NDIFF_1, PDIFF_1;
requirements HIDDEN, BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
equalities XBOOLE_0, SQUARE_1, FINSEQ_1, NORMSP_0, EUCLID, LOPBAN_1;
expansions ;
Lm1:
for m, n being non zero Element of NAT
for s, t being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS n)))
for i being Nat
for si, ti being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m),(REAL-NS 1))) st si = (Proj (i,n)) * s & ti = (Proj (i,n)) * t & 1 <= i & i <= n holds
si - ti = (Proj (i,n)) * (s - t)
:: of PartFunc of REAL-NS m,REAL-NS n