environ
vocabularies HIDDEN, FINSEQ_1, CARD_1, ORDINAL4, SUBSET_1, NUMBERS, ARYTM_3, TARSKI, RELAT_1, XBOOLE_0, FUNCT_1, QC_LANG1, XBOOLEAN, HILBERT1;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, NAT_1, FUNCT_1, FINSEQ_1;
definitions TARSKI;
theorems TARSKI, FINSEQ_1, XBOOLE_0, XBOOLE_1;
schemes XFAMILY;
registrations SUBSET_1, ORDINAL1, FUNCT_1, FINSEQ_1, XCMPLX_0;
constructors HIDDEN, NAT_1, FINSEQ_1, NUMBERS;
requirements HIDDEN, NUMERALS, BOOLE, SUBSET;
equalities ;
expansions TARSKI;
Lm1:
for D being set st D is HP-closed holds
not D is empty
Lm2:
for X being Subset of HP-WFF holds CnPos (CnPos X) c= CnPos X
Lm3:
for X being Subset of HP-WFF holds CnPos X is Hilbert_theory
by Th1, Th4, Th3, Th5, Th6, Th2, Th7;
Lm4:
for p, q, r, s being Element of HP-WFF holds (((q => r) => (p => r)) => s) => ((p => q) => s) in HP_TAUT
Lm5:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => q in HP_TAUT
Lm6:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' ((p '&' q) '&' s)) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lm7:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lm8:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' s) in HP_TAUT
Lm9:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' q) => ((p '&' s) '&' q) in HP_TAUT
Lm10:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((p '&' s) '&' q) in HP_TAUT
Lm11:
for p, q, s being Element of HP-WFF holds ((p '&' s) '&' q) => ((s '&' p) '&' q) in HP_TAUT
Lm12:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' p) '&' q) in HP_TAUT
Lm13:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' q) '&' p) in HP_TAUT
Lm14:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' (s '&' q)) in HP_TAUT
Lm15:
for p, q, s being Element of HP-WFF holds (p '&' (s '&' q)) => (p '&' (q '&' s)) in HP_TAUT
Lm16:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((s '&' q) '&' p) in HP_TAUT
Lm17:
for p, q, s being Element of HP-WFF holds ((s '&' q) '&' p) => ((q '&' s) '&' p) in HP_TAUT
Lm18:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((q '&' s) '&' p) in HP_TAUT
Lm19:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((p '&' s) '&' q) in HP_TAUT
Lm20:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => (p '&' (s '&' q)) in HP_TAUT