environ
vocabularies HIDDEN, NUMBERS, XBOOLE_0, FINSEQ_1, ARYTM_1, GROUP_1, STRUCT_0, GROUP_2, SETFAM_1, SUBSET_1, RELAT_1, INT_1, TARSKI, GROUP_3, QC_LANG1, NEWTON, ARYTM_3, CARD_1, XXREAL_0, COMPLEX1, ALGSTR_0, CARD_3, FINSOP_1, BINOP_1, ORDINAL4, SETWISEO, FINSEQ_2, NAT_1, FUNCT_1, PARTFUN1, FUNCT_2, ZFMISC_1, RLSUB_1, BVFUNC_2, EQREL_1, PRE_TOPC, RLSUB_2, LATTICES, GROUP_4;
notations HIDDEN, TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XCMPLX_0, FINSOP_1, ORDINAL1, NUMBERS, INT_1, SETWISEO, SETFAM_1, FUNCT_1, PARTFUN1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, BINOP_1, STRUCT_0, ALGSTR_0, GROUP_2, GROUP_3, LATTICES, GROUP_1, DOMAIN_1, XXREAL_0, NAT_1, INT_2;
definitions TARSKI, XBOOLE_0;
theorems BINOP_1, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FINSOP_1, FUNCT_1, GROUP_1, GROUP_2, GROUP_3, INT_1, LATTICES, NAT_1, RLSUB_2, SETFAM_1, SUBSET_1, TARSKI, ZFMISC_1, RELAT_1, XBOOLE_0, XBOOLE_1, ORDERS_1, XREAL_1, XXREAL_0, NAT_D, STRUCT_0, PARTFUN1, ABSVALUE, CARD_1;
schemes BINOP_1, DOMAIN_1, FINSEQ_1, FINSEQ_2, NAT_1, ORDERS_1, XBOOLE_0, XFAMILY;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, NUMBERS, XREAL_0, INT_1, FINSEQ_1, STRUCT_0, LATTICES, GROUP_1, GROUP_2, GROUP_3, ORDINAL1, CARD_1, FINSEQ_2;
constructors HIDDEN, PARTFUN1, SETFAM_1, BINOP_1, SETWISEO, XXREAL_0, NAT_1, NAT_D, FINSEQ_3, FINSEQ_4, FINSOP_1, REALSET1, REAL_1, LATTICES, GROUP_3, RELSET_1, FINSEQ_2, BINOP_2;
requirements HIDDEN, NUMERALS, REAL, BOOLE, SUBSET, ARITHM;
equalities GROUP_2, SUBSET_1, ALGSTR_0;
expansions TARSKI, XBOOLE_0;
Lm3:
for F1 being FinSequence
for y being Element of NAT st y in dom F1 holds
( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 )
theorem
for
i1,
i2,
i3 being
Integer for
G being
Group for
a,
b,
c being
Element of
G holds
<*a,b,c*> |^ <*(@ i1),(@ i2),(@ i3)*> = <*(a |^ i1),(b |^ i2),(c |^ i3)*>
theorem
for
G being
Group for
H1,
H2,
H3 being
Subgroup of
G holds
(H1 * H2) * H3 = H1 * (H2 * H3)
Lm4:
for G being Group
for H1, H2 being Subgroup of G holds H1 is Subgroup of H1 "\/" H2
Lm5:
for G being Group
for H1, H2, H3 being Subgroup of G holds (H1 "\/" H2) "\/" H3 is Subgroup of H1 "\/" (H2 "\/" H3)
Lm6:
for G being Group holds
( LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is Lattice & LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is 0_Lattice & LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is 1_Lattice )
:: Frattini subgroup.
::