begin
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.
::