environ
vocabularies HIDDEN, NUMBERS, RLVECT_1, RLSUB_1, SUBSET_1, RLVECT_2, RLVECT_3, TARSKI, CARD_3, XBOOLE_0, ARYTM_1, ARYTM_3, SUPINF_2, FUNCT_1, CARD_1, FINSEQ_1, STRUCT_0, FUNCT_2, RELAT_1, VALUED_1, NAT_1, PARTFUN1, ORDINAL4, CLASSES1, FINSET_1, XXREAL_0, FINSEQ_4, RLSUB_2, RLVECT_5;
notations HIDDEN, TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, CARD_1, ORDINAL1, NUMBERS, XCMPLX_0, REAL_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, NAT_1, FINSET_1, FINSEQ_1, FINSEQ_3, FINSEQ_4, DOMAIN_1, STRUCT_0, CLASSES1, RLVECT_1, RLVECT_2, RLSUB_1, RLSUB_2, RLVECT_3, XXREAL_0;
definitions TARSKI;
theorems TARSKI, ENUMSET1, SUBSET_1, NAT_1, CARD_1, CARD_2, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FINSET_1, RFINSEQ, RLVECT_1, RLVECT_2, RLVECT_3, RLSUB_1, RLSUB_2, XBOOLE_0, XBOOLE_1, ZFMISC_1, XCMPLX_0, XCMPLX_1, XREAL_1, XXREAL_0, STRUCT_0, PARTFUN1, RELAT_1, XREAL_0;
schemes NAT_1, FINSEQ_1, FUNCT_2;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, FUNCT_2, FINSET_1, NUMBERS, NAT_1, FINSEQ_1, STRUCT_0, RLVECT_3, CARD_1, RELSET_1, RLVECT_2, XREAL_0;
constructors HIDDEN, PARTFUN1, XXREAL_0, REAL_1, NAT_1, FINSEQ_3, FINSEQ_4, REALSET1, RFINSEQ, RLSUB_2, RLVECT_2, CLASSES1, RLVECT_3, RELSET_1;
requirements HIDDEN, REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
equalities ;
expansions TARSKI;
Lm1:
for X, x being set st x in X holds
(X \ {x}) \/ {x} = X
Lm2:
for n being Element of NAT
for V being finite-dimensional RealLinearSpace st n <= dim V holds
ex W being strict Subspace of V st dim W = n