begin
:: deftheorem GROUP_4:def 1 :
canceled;
:: deftheorem defines @ GROUP_4:def 2 :
theorem
canceled;
theorem
canceled;
theorem Th3:
theorem
theorem Th5:
theorem Th6:
:: deftheorem defines Product GROUP_4:def 3 :
theorem
canceled;
theorem Th8:
theorem
theorem Th10:
theorem Th11:
theorem
theorem
theorem
theorem
theorem
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 Th17:
theorem
theorem
theorem
theorem Th21:
:: deftheorem Def4 defines |^ GROUP_4:def 4 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
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
theorem
theorem
:: deftheorem Def5 defines gr GROUP_4:def 5 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th37:
theorem Th38:
theorem
theorem Th40:
theorem Th41:
theorem
theorem Th43:
theorem Th44:
:: deftheorem Def6 defines generating GROUP_4:def 6 :
theorem
canceled;
theorem
:: deftheorem Def7 defines maximal GROUP_4:def 7 :
theorem
canceled;
theorem Th48:
:: deftheorem Def8 defines Phi GROUP_4:def 8 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th52:
theorem
theorem Th54:
theorem Th55:
theorem
:: deftheorem defines * GROUP_4:def 9 :
theorem
theorem
canceled;
theorem
for
G being
Group for
H1,
H2,
H3 being
Subgroup of
G holds
(H1 * H2) * H3 = H1 * (H2 * H3)
theorem
theorem
theorem
theorem
:: deftheorem defines "\/" GROUP_4:def 10 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
theorem
theorem Th69:
theorem
theorem Th71:
theorem
theorem
theorem
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)
theorem Th75:
theorem
theorem Th77:
theorem Th78:
theorem Th79:
theorem
theorem
theorem
theorem
theorem Th84:
theorem Th85:
theorem
:: deftheorem Def11 defines SubJoin GROUP_4:def 11 :
:: deftheorem Def12 defines SubMeet GROUP_4:def 12 :
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 )
:: deftheorem defines lattice GROUP_4:def 13 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
theorem
theorem
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
theorem