:: GROUP_9 semantic presentation

REAL is non empty non trivial non finite complex-membered ext-real-membered real-membered V207() V210() V211() V213() set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V208() V210() Element of bool REAL
bool REAL is non empty non trivial non finite set
COMPLEX is non empty non trivial non finite complex-membered V207() set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V208() V210() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
RAT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V207() set
INT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V207() set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial non finite V191() set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial non finite V191() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:REAL,REAL:] is Relation-like non empty non trivial non finite V191() V192() V193() set
bool [:REAL,REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial non finite V191() V192() V193() set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial non finite V191() V192() V193() set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial non finite V191() V192() V193() set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V191() V192() V193() set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V191() V192() V193() set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V191() V192() V193() V194() set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V191() V192() V193() V194() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
K284() is set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
[:2,2:] is Relation-like RAT -valued INT -valued non empty finite V191() V192() V193() V194() set
[:[:2,2:],2:] is Relation-like RAT -valued INT -valued non empty finite V191() V192() V193() V194() set
bool [:[:2,2:],2:] is non empty finite V39() set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
3 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg 1 is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of NAT
idseq 2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
<*1,2*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*1*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,1] is set
{1,1} is non empty finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{1,1},{1}} is non empty finite V39() set
{[1,1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*2*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,2] is set
{{1,2},{1}} is non empty finite V39() set
{[1,2]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*1*> ^ <*2*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Sgm {} is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
len {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
Permutations 2 is non empty permutational set
<*2,1*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*2*> ^ <*1*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
{<*1,2*>,<*2,1*>} is functional non empty finite V39() set
<*> INT is Relation-like non-empty empty-yielding NAT -defined INT -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of INT
[:NAT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V191() V192() V193() set
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
[#] G is non proper Element of bool G
bool G is non empty set
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
s2 is Element of O
s19 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s1 . s2 is Relation-like Function-like set
s19 .: ([#] G) is Element of bool G
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
bool G is non empty set
s2 is Element of bool G
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
bool G is non empty Element of bool (bool G)
bool (bool G) is non empty set
s19 is set
meet s19 is set
[#] G is non proper Element of bool G
p is set
p is Element of bool G
f1 is set
i is set
s199 is Element of bool G
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
f1 is Element of O
i is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s1 . f1 is Relation-like Function-like set
s199 is set
i .: p is Element of bool G
s199 is set
i .: s199 is Element of bool G
f2 is Element of bool G
f1 is Element of bool G
f1 is Element of bool G
s19 is Element of bool G
s29 is Element of bool G
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
s2 . 1 is set
s1 . (s2 . 1) is Relation-like Function-like set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 . 1 is set
s1 . (s29 . 1) is Relation-like Function-like set
Seg s19 is finite s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s19 ) } is set
s29 | (Seg s19) is Relation-like NAT -defined Seg s19 -defined NAT -defined O -valued Function-like finite FinSubsequence-like Element of bool [:NAT,O:]
[:NAT,O:] is Relation-like set
bool [:NAT,O:] is non empty set
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s29) is finite len s29 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s29 ) } is set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng s29 is finite Element of bool O
bool O is non empty set
dom s1 is Element of bool O
rng s1 is functional Element of bool (Funcs (G,G))
bool (Funcs (G,G)) is non empty set
the Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:] is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like Function-like Element of Funcs (G,G)
<*s199*> is Relation-like NAT -defined Funcs (G,G) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on (Funcs (G,G))
1 -tuples_on (Funcs (G,G)) is FinSequenceSet of Funcs (G,G)
[1,s199] is set
{1,s199} is non empty finite set
{{1,s199},{1}} is non empty finite V39() set
{[1,s199]} is Relation-like Function-like constant non empty trivial finite 1 -element set
p is Relation-like Function-like set
dom p is set
rng p is set
len <*s199*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
<*s199*> . (len <*s199*>) is Relation-like Function-like set
<*s199*> . 1 is Relation-like Function-like set
f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
<*s199*> . p is Relation-like Function-like set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 . (p + 1) is set
s1 . (s29 . (p + 1)) is Relation-like Function-like set
<*s199*> . (p + 1) is Relation-like Function-like set
the Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:] * the Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:] is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s29 . (s19 + 1) is set
s1 . (s29 . (s19 + 1)) is Relation-like Function-like set
0 + s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (s19 + 1) is non empty finite s19 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s19 + 1 ) } is set
(Seg (s19 + 1)) /\ (Seg s19) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom s29) /\ (Seg s19) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len s29) is finite len s29 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s29 ) } is set
rng s29 is finite Element of bool O
bool O is non empty set
dom s1 is Element of bool O
rng s1 is functional Element of bool (Funcs (G,G))
bool (Funcs (G,G)) is non empty set
s199 is Relation-like Function-like set
dom s199 is set
rng s199 is set
p . 1 is set
s1 . (p . 1) is Relation-like Function-like set
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 . (len s199) is Relation-like Function-like set
s199 . 1 is Relation-like Function-like set
s199 . s19 is Relation-like Function-like set
Seg (len s199) is finite len s199 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s199 ) } is set
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is Relation-like Function-like set
dom H1 is set
rng H1 is set
i is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p * i is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
<*(p * i)*> is Relation-like NAT -defined bool [:G,G:] -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on (bool [:G,G:])
1 -tuples_on (bool [:G,G:]) is FinSequenceSet of bool [:G,G:]
[1,(p * i)] is set
{1,(p * i)} is non empty finite set
{{1,(p * i)},{1}} is non empty finite V39() set
{[1,(p * i)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s199 ^ <*(p * i)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
j is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j . (len j) is Relation-like Function-like set
j . 1 is Relation-like Function-like set
len <*(p * i)*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len p) + (len <*(p * i)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j . H2 is Relation-like Function-like set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 . (H2 + 1) is set
s1 . (s29 . (H2 + 1)) is Relation-like Function-like set
j . (H2 + 1) is Relation-like Function-like set
H1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s299 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
H1 * s299 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 . H2 is Relation-like Function-like set
p . (H2 + 1) is set
s1 . (p . (H2 + 1)) is Relation-like Function-like set
s199 . (H2 + 1) is Relation-like Function-like set
s299 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
H1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s299 * H1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
1 + s19 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 . 1 is set
s1 . (s19 . 1) is Relation-like Function-like set
s29 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s19 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 . (len s19) is Relation-like Function-like set
s19 . 1 is Relation-like Function-like set
s19 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
p . (len s2) is Relation-like Function-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p . 1 is Relation-like Function-like set
s29 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f1 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
f1 . (len s2) is Relation-like Function-like set
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 . 1 is Relation-like Function-like set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p . i is Relation-like Function-like set
f1 . i is Relation-like Function-like set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p . (i + 1) is Relation-like Function-like set
f1 . (i + 1) is Relation-like Function-like set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (i + 1) is set
s1 . (s2 . (i + 1)) is Relation-like Function-like set
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 * s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f2 * p is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p . 0 is Relation-like Function-like set
f1 . 0 is Relation-like Function-like set
s19 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s29 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
i is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
i . (len s2) is Relation-like Function-like set
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i . 1 is Relation-like Function-like set
f1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
s199 . (len s2) is Relation-like Function-like set
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 . 1 is Relation-like Function-like set
O is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
O is set
the non empty set is non empty set
[: the non empty set , the non empty set :] is Relation-like non empty set
[:[: the non empty set , the non empty set :], the non empty set :] is Relation-like non empty set
bool [:[: the non empty set , the non empty set :], the non empty set :] is non empty set
the Relation-like [: the non empty set , the non empty set :] -defined the non empty set -valued Function-like non empty total quasi_total Element of bool [:[: the non empty set , the non empty set :], the non empty set :] is Relation-like [: the non empty set , the non empty set :] -defined the non empty set -valued Function-like non empty total quasi_total Element of bool [:[: the non empty set , the non empty set :], the non empty set :]
Funcs ( the non empty set , the non empty set ) is functional non empty set
[:O,(Funcs ( the non empty set , the non empty set )):] is Relation-like set
bool [:O,(Funcs ( the non empty set , the non empty set )):] is non empty set
the Relation-like O -defined Funcs ( the non empty set , the non empty set ) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the non empty set , the non empty set )):] is Relation-like O -defined Funcs ( the non empty set , the non empty set ) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the non empty set , the non empty set )):]
(O, the non empty set , the Relation-like [: the non empty set , the non empty set :] -defined the non empty set -valued Function-like non empty total quasi_total Element of bool [:[: the non empty set , the non empty set :], the non empty set :], the Relation-like O -defined Funcs ( the non empty set , the non empty set ) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the non empty set , the non empty set )):]) is (O) (O)
O is set
O is set
G is set
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
{(id G)} is functional non empty trivial finite 1 -element set
[:O,{(id G)}:] is Relation-like set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
s2 is set
s19 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s2 is Relation-like O -defined Funcs (G,G) -valued Element of bool [:O,(Funcs (G,G)):]
dom s2 is Element of bool O
bool O is non empty set
s19 is set
s29 is set
p is set
[s19,p] is set
{s19,p} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,p},{s19}} is non empty finite V39() set
s19 is set
s29 is set
[s19,s29] is set
{s19,s29} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,s29},{s19}} is non empty finite V39() set
p is set
[s19,p] is set
{s19,p} is non empty finite set
{{s19,p},{s19}} is non empty finite V39() set
f1 is set
i is set
[f1,i] is set
{f1,i} is non empty finite set
{f1} is non empty trivial finite 1 -element set
{{f1,i},{f1}} is non empty finite V39() set
s199 is set
s199 is set
[s199,s199] is set
{s199,s199} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,s199},{s199}} is non empty finite V39() set
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like Element of bool [:O,(Funcs (G,G)):]
O is set
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
{(id the carrier of G)} is functional non empty trivial finite 1 -element set
[:O,{(id the carrier of G)}:] is Relation-like set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
s1 is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
(O, the carrier of G, the multF of G,s1) is (O) (O)
p is non empty (O)
1_ G is non being_of_order_0 Element of the carrier of G
f1 is non empty multMagma
the carrier of f1 is non empty set
s199 is Element of the carrier of f1
f2 is Element of the carrier of f1
p is Element of the carrier of G
p " is Element of the carrier of G
s199 is Element of the carrier of f1
f2 * s199 is Element of the carrier of f1
the multF of f1 is Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
[: the carrier of f1, the carrier of f1:] is Relation-like non empty set
[:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is Relation-like non empty set
bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
the multF of f1 . (f2,s199) is Element of the carrier of f1
p * (1_ G) is Element of the carrier of G
the multF of G . (p,(1_ G)) is Element of the carrier of G
s199 * f2 is Element of the carrier of f1
the multF of f1 . (s199,f2) is Element of the carrier of f1
(1_ G) * p is Element of the carrier of G
the multF of G . ((1_ G),p) is Element of the carrier of G
j is Element of the carrier of f1
j is Element of the carrier of f1
f2 * j is Element of the carrier of f1
the multF of f1 . (f2,j) is Element of the carrier of f1
p * (p ") is Element of the carrier of G
the multF of G . (p,(p ")) is Element of the carrier of G
j * f2 is Element of the carrier of f1
the multF of f1 . (j,f2) is Element of the carrier of f1
(p ") * p is Element of the carrier of G
the multF of G . ((p "),p) is Element of the carrier of G
the carrier of p is non empty set
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
multMagma(# the carrier of p, the multF of p #) is non empty strict multMagma
i is non empty unital Group-like associative multMagma
the carrier of i is non empty set
Funcs ( the carrier of i, the carrier of i) is functional non empty set
[:O,(Funcs ( the carrier of i, the carrier of i)):] is Relation-like set
bool [:O,(Funcs ( the carrier of i, the carrier of i)):] is non empty set
Funcs ( the carrier of p, the carrier of p) is functional non empty set
s199 is Relation-like O -defined Funcs ( the carrier of i, the carrier of i) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of i, the carrier of i)):]
the of p is Relation-like O -defined Funcs ( the carrier of p, the carrier of p) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of p, the carrier of p)):]
[:O,(Funcs ( the carrier of p, the carrier of p)):] is Relation-like set
bool [:O,(Funcs ( the carrier of p, the carrier of p)):] is non empty set
the multF of i is Relation-like [: the carrier of i, the carrier of i:] -defined the carrier of i -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of i, the carrier of i:], the carrier of i:]
[: the carrier of i, the carrier of i:] is Relation-like non empty set
[:[: the carrier of i, the carrier of i:], the carrier of i:] is Relation-like non empty set
bool [:[: the carrier of i, the carrier of i:], the carrier of i:] is non empty set
multMagma(# the carrier of i, the multF of i #) is non empty strict multMagma
s199 is Element of O
dom s1 is Element of bool O
bool O is non empty set
s1 . s199 is Relation-like Function-like set
[s199,(s1 . s199)] is set
{s199,(s1 . s199)} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,(s1 . s199)},{s199}} is non empty finite V39() set
id the carrier of i is Relation-like the carrier of i -defined the carrier of i -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of i, the carrier of i:]
bool [: the carrier of i, the carrier of i:] is non empty set
{(id the carrier of i)} is functional non empty trivial finite 1 -element set
[:O,{(id the carrier of i)}:] is Relation-like set
f2 is set
p is set
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
s199 . s199 is Relation-like Function-like set
i is Element of the carrier of f1
s199 is Element of the carrier of f1
s199 is Element of the carrier of f1
i * s199 is Element of the carrier of f1
the multF of f1 . (i,s199) is Element of the carrier of f1
(i * s199) * s199 is Element of the carrier of f1
the multF of f1 . ((i * s199),s199) is Element of the carrier of f1
f2 is Element of the carrier of G
p is Element of the carrier of G
f2 * p is Element of the carrier of G
the multF of G . (f2,p) is Element of the carrier of G
H1 is Element of the carrier of G
(f2 * p) * H1 is Element of the carrier of G
the multF of G . ((f2 * p),H1) is Element of the carrier of G
p * H1 is Element of the carrier of G
the multF of G . (p,H1) is Element of the carrier of G
f2 * (p * H1) is Element of the carrier of G
the multF of G . (f2,(p * H1)) is Element of the carrier of G
s199 * s199 is Element of the carrier of f1
the multF of f1 . (s199,s199) is Element of the carrier of f1
i * (s199 * s199) is Element of the carrier of f1
the multF of f1 . (i,(s199 * s199)) is Element of the carrier of f1
i is Element of the carrier of f1
O is set
the non empty strict unital Group-like associative multMagma is non empty strict unital Group-like associative multMagma
s1 is non empty (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
O is set
s1 is Element of O
G is non empty unital Group-like associative (O) (O)
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
the carrier of G is non empty set
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
the of G . s1 is Relation-like Function-like set
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
s2 is non empty unital Group-like associative multMagma
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
s19 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
bool [: the carrier of s2, the carrier of s2:] is non empty set
s19 . s1 is Relation-like Function-like set
s29 is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
f1 is Element of the carrier of G
i is Element of the carrier of G
p is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
f1 * i is Element of the carrier of G
the multF of G . (f1,i) is Element of the carrier of G
p . (f1 * i) is Element of the carrier of G
s199 is Element of the carrier of s2
s199 is Element of the carrier of s2
s199 * s199 is Element of the carrier of s2
the multF of s2 . (s199,s199) is Element of the carrier of s2
s29 . (s199 * s199) is Element of the carrier of s2
s29 . s199 is Element of the carrier of s2
s29 . s199 is Element of the carrier of s2
(s29 . s199) * (s29 . s199) is Element of the carrier of s2
the multF of s2 . ((s29 . s199),(s29 . s199)) is Element of the carrier of s2
p . f1 is Element of the carrier of G
p . i is Element of the carrier of G
the multF of G . ((p . f1),(p . i)) is Element of the carrier of G
(p . f1) * (p . i) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s2 is Element of O
(O,G,s2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s2) | the carrier of G is Relation-like the carrier of G -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
1_ G is non being_of_order_0 Element of the carrier of G
s1 is non empty multMagma
the carrier of s1 is non empty set
s19 is Element of the carrier of s1
p is Element of the carrier of s1
f1 is Element of the carrier of G
f1 " is Element of the carrier of G
s29 is Element of the carrier of s1
p * s29 is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (p,s29) is Element of the carrier of s1
f1 * (1_ G) is Element of the carrier of G
the multF of G . (f1,(1_ G)) is Element of the carrier of G
s29 * p is Element of the carrier of s1
the multF of s1 . (s29,p) is Element of the carrier of s1
(1_ G) * f1 is Element of the carrier of G
the multF of G . ((1_ G),f1) is Element of the carrier of G
s199 is Element of the carrier of s1
s199 is Element of the carrier of s1
p * s199 is Element of the carrier of s1
the multF of s1 . (p,s199) is Element of the carrier of s1
f1 * (f1 ") is Element of the carrier of G
the multF of G . (f1,(f1 ")) is Element of the carrier of G
s199 * p is Element of the carrier of s1
the multF of s1 . (s199,p) is Element of the carrier of s1
(f1 ") * f1 is Element of the carrier of G
the multF of G . ((f1 "),f1) is Element of the carrier of G
s2 is Element of the carrier of s1
s19 is Element of the carrier of s1
s29 is Element of the carrier of s1
s2 * s19 is Element of the carrier of s1
the multF of s1 . (s2,s19) is Element of the carrier of s1
(s2 * s19) * s29 is Element of the carrier of s1
the multF of s1 . ((s2 * s19),s29) is Element of the carrier of s1
p is Element of the carrier of G
f1 is Element of the carrier of G
p * f1 is Element of the carrier of G
the multF of G . (p,f1) is Element of the carrier of G
i is Element of the carrier of G
(p * f1) * i is Element of the carrier of G
the multF of G . ((p * f1),i) is Element of the carrier of G
f1 * i is Element of the carrier of G
the multF of G . (f1,i) is Element of the carrier of G
p * (f1 * i) is Element of the carrier of G
the multF of G . (p,(f1 * i)) is Element of the carrier of G
s19 * s29 is Element of the carrier of s1
the multF of s1 . (s19,s29) is Element of the carrier of s1
s2 * (s19 * s29) is Element of the carrier of s1
the multF of s1 . (s2,(s19 * s29)) is Element of the carrier of s1
s19 is Element of the carrier of s1
s2 is non empty unital Group-like associative (O) (O)
the carrier of s2 is non empty set
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s19 is non empty unital Group-like associative multMagma
the carrier of s19 is non empty set
Funcs ( the carrier of s19, the carrier of s19) is functional non empty set
[:O,(Funcs ( the carrier of s19, the carrier of s19)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s19, the carrier of s19)):] is non empty set
s29 is Relation-like O -defined Funcs ( the carrier of s19, the carrier of s19) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s19, the carrier of s19)):]
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is Element of O
s19 is non empty unital Group-like associative (O) (O)
(O,s19,s29) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
the carrier of s19 is non empty set
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s19:] is non empty set
the of s19 is Relation-like O -defined Funcs ( the carrier of s19, the carrier of s19) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s19, the carrier of s19)):]
Funcs ( the carrier of s19, the carrier of s19) is functional non empty set
[:O,(Funcs ( the carrier of s19, the carrier of s19)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s19, the carrier of s19)):] is non empty set
the of s19 . s29 is Relation-like Function-like set
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
s29 is Element of O
s19 is non empty unital Group-like associative (O) (O)
the carrier of s19 is non empty set
(O,s19,s29) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s19:] is non empty set
id the carrier of s19 is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s19, the carrier of s19:]
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
s29 is Element of O
s19 is non empty unital Group-like associative (O) (O)
the carrier of s19 is non empty set
(O,s19,s29) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s19:] is non empty set
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
s19 is non empty unital Group-like associative (O) (O)
the carrier of s19 is non empty set
s29 is Element of O
(O,s19,s29) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s19:] is non empty set
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
dom (O,s19,s29) is non empty Element of bool the carrier of s19
bool the carrier of s19 is non empty set
(O,G,s29) | the carrier of s19 is Relation-like the carrier of G -defined the carrier of s19 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued non empty Element of bool [: the carrier of G, the carrier of G:]
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of G || the carrier of s19 is set
the multF of G | [: the carrier of s19, the carrier of s19:] is Relation-like [: the carrier of s19, the carrier of s19:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
dom the of s2 is Element of bool O
bool O is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
dom the of s1 is Element of bool O
p is set
f1 is Element of O
(O,s1,f1) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
the of s1 . f1 is Relation-like Function-like set
the carrier of G is non empty set
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,f1) | the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
(O,s2,f1) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
the of s1 . p is Relation-like Function-like set
the of s2 . p is Relation-like Function-like set
s19 is non empty unital Group-like associative Subgroup of G
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty unital Group-like associative Subgroup of G
the carrier of s29 is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
O is set
G is non empty unital Group-like associative (O) (O)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
{(1_ G)} is non empty trivial finite 1 -element set
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
s2 is non empty (O)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s19 is non empty unital Group-like associative (O) (O) (O)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of s19 is set
the multF of G | [: the carrier of s19, the carrier of s19:] is Relation-like [: the carrier of s19, the carrier of s19:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
s29 is Element of O
(O,s19,s29) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
bool [: the carrier of s19, the carrier of s19:] is non empty set
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s29) | the carrier of s19 is Relation-like the carrier of G -defined the carrier of s19 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
f1 is Relation-like Function-like set
dom f1 is set
id the carrier of s19 is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s19, the carrier of s19:]
(O,G,s29) * (id the carrier of s19) is Relation-like the carrier of s19 -defined the carrier of G -valued Function-like Element of bool [: the carrier of s19, the carrier of G:]
[: the carrier of s19, the carrier of G:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of G:] is non empty set
dom ((O,G,s29) * (id the carrier of s19)) is Element of bool the carrier of s19
bool the carrier of s19 is non empty set
dom (O,G,s29) is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
(dom (O,G,s29)) /\ the carrier of s19 is Element of bool the carrier of G
the carrier of G /\ the carrier of s19 is set
the carrier of ((1). G) is non empty trivial finite 1 -element set
i is set
dom (id the carrier of s19) is non empty Element of bool the carrier of s19
1_ s19 is non being_of_order_0 Element of the carrier of s19
p is Relation-like Function-like set
p . i is set
f1 . i is set
((O,G,s29) * (id the carrier of s19)) . i is set
(id the carrier of s19) . i is set
(O,G,s29) . ((id the carrier of s19) . i) is set
(O,G,s29) . i is set
dom p is set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
s1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G)
p is non empty strict unital Group-like associative Subgroup of G
s19 is non empty unital Group-like associative (O) (O,G)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty unital Group-like associative multMagma
(1). s29 is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of s29
the multF of ((1). s29) is Relation-like [: the carrier of ((1). s29), the carrier of ((1). s29):] -defined the carrier of ((1). s29) -valued Function-like non empty total quasi_total associative finite Element of bool [:[: the carrier of ((1). s29), the carrier of ((1). s29):], the carrier of ((1). s29):]
the carrier of ((1). s29) is non empty trivial finite 1 -element set
[: the carrier of ((1). s29), the carrier of ((1). s29):] is Relation-like non empty finite set
[:[: the carrier of ((1). s29), the carrier of ((1). s29):], the carrier of ((1). s29):] is Relation-like non empty finite set
bool [:[: the carrier of ((1). s29), the carrier of ((1). s29):], the carrier of ((1). s29):] is non empty finite V39() set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
the carrier of s29 is non empty set
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
the multF of s29 || the carrier of ((1). s29) is set
the multF of s29 | [: the carrier of ((1). s29), the carrier of ((1). s29):] is Relation-like [: the carrier of ((1). s29), the carrier of ((1). s29):] -defined [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like finite set
1_ s29 is non being_of_order_0 Element of the carrier of s29
{(1_ s29)} is non empty trivial finite 1 -element set
the carrier of (O,G) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1)
s2 is non empty unital Group-like associative (O) (O)
s19 is non empty unital Group-like associative (O) (O,s1)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative Subgroup of s1
the carrier of s29 is non empty set
p is non empty unital Group-like associative multMagma
1_ p is non being_of_order_0 Element of the carrier of p
the carrier of p is non empty set
{(1_ p)} is non empty trivial finite 1 -element set
(1). p is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of p
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G)
s1 is non empty unital Group-like associative multMagma
s2 is non empty strict unital Group-like associative Subgroup of G
the carrier of (O,G) is non empty set
the multF of (O,G) is Relation-like [: the carrier of (O,G), the carrier of (O,G):] -defined the carrier of (O,G) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):]
[: the carrier of (O,G), the carrier of (O,G):] is Relation-like non empty set
[:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is Relation-like non empty set
bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is non empty set
multMagma(# the carrier of (O,G), the multF of (O,G) #) is non empty strict multMagma
the carrier of s2 is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
{(1_ G)} is non empty trivial finite 1 -element set
s19 is non empty strict unital Group-like associative Subgroup of s1
(1). s1 is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of s1
(O,G) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s1 is non empty unital Group-like associative multMagma
s2 is non empty strict unital Group-like associative Subgroup of G
the carrier of (O,G) is non empty set
the multF of (O,G) is Relation-like [: the carrier of (O,G), the carrier of (O,G):] -defined the carrier of (O,G) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):]
[: the carrier of (O,G), the carrier of (O,G):] is Relation-like non empty set
[:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is Relation-like non empty set
bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is non empty set
multMagma(# the carrier of (O,G), the multF of (O,G) #) is non empty strict multMagma
s19 is non empty strict unital Group-like associative Subgroup of s1
(Omega). s1 is non empty strict unital Group-like associative normal Subgroup of s1
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
s1 is set
s19 is set
p is non empty unital Group-like associative (O) (O) (O,G)
s2 is set
the carrier of p is non empty set
s29 is set
f1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of f1 is non empty set
s2 is Relation-like Function-like set
s19 is set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
p is set
[s19,p] is set
{s19,p} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,p},{s19}} is non empty finite V39() set
s29 is set
[s19,s29] is set
{s19,s29} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,s29},{s19}} is non empty finite V39() set
dom s2 is set
rng s2 is set
s19 is set
s29 is set
s2 . s29 is set
[s29,s19] is set
{s29,s19} is non empty finite set
{s29} is non empty trivial finite 1 -element set
{{s29,s19},{s29}} is non empty finite V39() set
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
p is set
[p,s19] is set
{p,s19} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,s19},{p}} is non empty finite V39() set
s2 . p is set
s1 is set
s2 is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
Group_of_Perm 2 is non empty strict unital Group-like associative multMagma
G is non empty strict unital Group-like associative normal Subgroup of Group_of_Perm 2
(Omega). (Group_of_Perm 2) is non empty strict unital Group-like associative normal Subgroup of Group_of_Perm 2
the carrier of (Group_of_Perm 2) is non empty set
the multF of (Group_of_Perm 2) is Relation-like [: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):] -defined the carrier of (Group_of_Perm 2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):]
[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):] is non empty set
multMagma(# the carrier of (Group_of_Perm 2), the multF of (Group_of_Perm 2) #) is non empty strict multMagma
(1). (Group_of_Perm 2) is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of Group_of_Perm 2
1_ (Group_of_Perm 2) is non being_of_order_0 Element of the carrier of (Group_of_Perm 2)
the carrier of G is non empty set
{(1_ (Group_of_Perm 2))} is non empty trivial finite 1 -element set
{<*1,2*>} is functional non empty trivial finite V39() 1 -element set
{<*2,1*>} is functional non empty trivial finite V39() 1 -element set
<*2,1*> . 1 is set
<*1,2*> . 1 is set
{<*2,1*>} is functional non empty trivial finite V39() 1 -element set
G is set
{G} is non empty trivial finite 1 -element set
<*1,2*> . 1 is set
O is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
1_ s1 is non being_of_order_0 Element of the carrier of s1
s19 is non empty multMagma
the carrier of s19 is non empty set
p is Element of the carrier of s19
i is Element of the carrier of s19
s199 is Element of the carrier of s1
s199 " is Element of the carrier of s1
f1 is Element of the carrier of s19
i * f1 is Element of the carrier of s19
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of s19 . (i,f1) is Element of the carrier of s19
s199 * (1_ s1) is Element of the carrier of s1
the multF of s1 . (s199,(1_ s1)) is Element of the carrier of s1
f1 * i is Element of the carrier of s19
the multF of s19 . (f1,i) is Element of the carrier of s19
(1_ s1) * s199 is Element of the carrier of s1
the multF of s1 . ((1_ s1),s199) is Element of the carrier of s1
f2 is Element of the carrier of s19
p is Element of the carrier of s19
i * p is Element of the carrier of s19
the multF of s19 . (i,p) is Element of the carrier of s19
s199 * (s199 ") is Element of the carrier of s1
the multF of s1 . (s199,(s199 ")) is Element of the carrier of s1
p * i is Element of the carrier of s19
the multF of s19 . (p,i) is Element of the carrier of s19
(s199 ") * s199 is Element of the carrier of s1
the multF of s1 . ((s199 "),s199) is Element of the carrier of s1
p is Element of the carrier of s19
s29 is non empty unital Group-like multMagma
the carrier of s29 is non empty set
the carrier of G is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of s29 is set
the multF of G | [: the carrier of s29, the carrier of s29:] is Relation-like [: the carrier of s29, the carrier of s29:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
p is non empty unital Group-like associative Subgroup of G
O is non empty unital Group-like associative multMagma
the carrier of O is non empty set
bool the carrier of O is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
the multF of O is Relation-like [: the carrier of O, the carrier of O:] -defined the carrier of O -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of O, the carrier of O:], the carrier of O:]
[: the carrier of O, the carrier of O:] is Relation-like non empty set
[:[: the carrier of O, the carrier of O:], the carrier of O:] is Relation-like non empty set
bool [:[: the carrier of O, the carrier of O:], the carrier of O:] is non empty set
multMagma(# the carrier of O, the multF of O #) is non empty strict multMagma
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
s1 is Element of bool the carrier of O
s2 is Element of bool the carrier of G
s19 is non empty strict unital Group-like associative Subgroup of O
s1 * s19 is Element of bool the carrier of O
carr s19 is Element of bool the carrier of O
the carrier of s19 is non empty set
s1 * (carr s19) is Element of bool the carrier of O
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of O : ( b₁ in s1 & b₂ in carr s19 ) } is set
s19 * s1 is Element of bool the carrier of O
(carr s19) * s1 is Element of bool the carrier of O
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of O : ( b₁ in carr s19 & b₂ in s1 ) } is set
s29 is non empty strict unital Group-like associative Subgroup of G
s2 * s29 is Element of bool the carrier of G
carr s29 is Element of bool the carrier of G
the carrier of s29 is non empty set
s2 * (carr s29) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in s2 & b₂ in carr s29 ) } is set
s29 * s2 is Element of bool the carrier of G
(carr s29) * s2 is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s29 & b₂ in s2 ) } is set
p is Element of bool the carrier of O
f1 is Element of bool the carrier of O
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of O : ( b₁ in p & b₂ in f1 ) } is set
i is Element of bool the carrier of G
s199 is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in i & b₂ in s199 ) } is set
p is set
H1 is Element of the carrier of O
j is Element of the carrier of O
H1 * j is Element of the carrier of O
the multF of O . (H1,j) is Element of the carrier of O
s299 is Element of the carrier of G
H1 is Element of the carrier of G
s299 * H1 is Element of the carrier of G
the multF of G . (s299,H1) is Element of the carrier of G
p is set
H1 is Element of the carrier of G
j is Element of the carrier of G
H1 * j is Element of the carrier of G
the multF of G . (H1,j) is Element of the carrier of G
j is Element of the carrier of O
H2 is Element of the carrier of O
j * H2 is Element of the carrier of O
the multF of O . (j,H2) is Element of the carrier of O
O is set
s1 is non empty (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty unital Group-like associative (O) (O) (O)
(O,s2) is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
(O, the carrier of s2, the multF of s2, the of s2) is (O) (O)
(O,s2) is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
s29 is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
s29 is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
the carrier of s29 is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
s19 is non empty unital Group-like associative multMagma
p is non empty strict unital Group-like associative normal Subgroup of s2
f1 is non empty strict unital Group-like associative normal Subgroup of s19
the carrier of f1 is non empty set
the carrier of s19 is non empty set
the multF of f1 is Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
[: the carrier of f1, the carrier of f1:] is Relation-like non empty set
[:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is Relation-like non empty set
bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of s19 || the carrier of f1 is set
the multF of s19 | [: the carrier of f1, the carrier of f1:] is Relation-like [: the carrier of f1, the carrier of f1:] -defined [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like set
the carrier of (Group_of_Perm 2) is non empty set
bool the carrier of (Group_of_Perm 2) is non empty set
bool the carrier of s19 is non empty set
s199 is Element of bool the carrier of (Group_of_Perm 2)
s199 is non empty strict unital Group-like associative Subgroup of Group_of_Perm 2
s199 * s199 is Element of bool the carrier of (Group_of_Perm 2)
carr s199 is Element of bool the carrier of (Group_of_Perm 2)
the carrier of s199 is non empty set
s199 * (carr s199) is Element of bool the carrier of (Group_of_Perm 2)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (Group_of_Perm 2) : ( b₁ in s199 & b₂ in carr s199 ) } is set
f2 is Element of bool the carrier of s19
f2 * f1 is Element of bool the carrier of s19
carr f1 is Element of bool the carrier of s19
f2 * (carr f1) is Element of bool the carrier of s19
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s19 : ( b₁ in f2 & b₂ in carr f1 ) } is set
f1 * f2 is Element of bool the carrier of s19
(carr f1) * f2 is Element of bool the carrier of s19
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s19 : ( b₁ in carr f1 & b₂ in f2 ) } is set
s199 * s199 is Element of bool the carrier of (Group_of_Perm 2)
(carr s199) * s199 is Element of bool the carrier of (Group_of_Perm 2)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (Group_of_Perm 2) : ( b₁ in carr s199 & b₂ in s199 ) } is set
1_ (Group_of_Perm 2) is non being_of_order_0 Element of the carrier of (Group_of_Perm 2)
p is Element of the carrier of s2
f2 is Element of the carrier of s2
p * f2 is Element of the carrier of s2
the multF of s2 . (p,f2) is Element of the carrier of s2
H1 is Element of the carrier of (Group_of_Perm 2)
H1 * (1_ (Group_of_Perm 2)) is Element of the carrier of (Group_of_Perm 2)
the multF of (Group_of_Perm 2) is Relation-like [: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):] -defined the carrier of (Group_of_Perm 2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):]
[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm 2), the carrier of (Group_of_Perm 2):], the carrier of (Group_of_Perm 2):] is non empty set
the multF of (Group_of_Perm 2) . (H1,(1_ (Group_of_Perm 2))) is Element of the carrier of (Group_of_Perm 2)
f2 * p is Element of the carrier of s2
the multF of s2 . (f2,p) is Element of the carrier of s2
(1_ (Group_of_Perm 2)) * H1 is Element of the carrier of (Group_of_Perm 2)
the multF of (Group_of_Perm 2) . ((1_ (Group_of_Perm 2)),H1) is Element of the carrier of (Group_of_Perm 2)
s199 is non empty strict unital Group-like associative normal Subgroup of Group_of_Perm 2
(1). (Group_of_Perm 2) is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of Group_of_Perm 2
the carrier of s199 is non empty set
{(1_ (Group_of_Perm 2))} is non empty trivial finite 1 -element set
1_ s2 is non being_of_order_0 Element of the carrier of s2
{(1_ s2)} is non empty trivial finite 1 -element set
(Omega). (Group_of_Perm 2) is non empty strict unital Group-like associative normal Subgroup of Group_of_Perm 2
multMagma(# the carrier of (Group_of_Perm 2), the multF of (Group_of_Perm 2) #) is non empty strict multMagma
(O,s2) is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
(O, the carrier of s2, the multF of s2, the of s2) is (O) (O)
(O,s2) is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
s29 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s29 is non empty Element of bool (bool the carrier of G)
s19 is set
s29 is set
s2 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is set
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
s19 is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
s29 is non empty strict unital Group-like associative normal Subgroup of G
CosOp s29 is Relation-like [:(Left_Cosets s29),(Left_Cosets s29):] -defined Left_Cosets s29 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s29),(Left_Cosets s29):],(Left_Cosets s29):]
Left_Cosets s29 is non empty Element of bool (bool the carrier of G)
[:(Left_Cosets s29),(Left_Cosets s29):] is Relation-like non empty set
[:[:(Left_Cosets s29),(Left_Cosets s29):],(Left_Cosets s29):] is Relation-like non empty set
bool [:[:(Left_Cosets s29),(Left_Cosets s29):],(Left_Cosets s29):] is non empty set
s19 is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
s29 is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
s2 is non empty strict unital Group-like associative normal Subgroup of G
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
O is non empty unital Group-like associative multMagma
the carrier of O is non empty set
bool the carrier of O is non empty set
G is non empty unital Group-like associative normal Subgroup of O
Left_Cosets G is non empty Element of bool (bool the carrier of O)
bool (bool the carrier of O) is non empty set
s1 is Element of Left_Cosets G
s2 is Element of the carrier of O
s2 * G is Element of bool the carrier of O
carr G is Element of bool the carrier of O
the carrier of G is non empty set
s2 * (carr G) is Element of bool the carrier of O
K350( the carrier of O,s2) is non empty trivial finite 1 -element Element of bool the carrier of O
K350( the carrier of O,s2) * (carr G) is Element of bool the carrier of O
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of O : ( b₁ in K350( the carrier of O,s2) & b₂ in carr G ) } is set
s19 is Element of the carrier of O
s19 * G is Element of bool the carrier of O
s19 * (carr G) is Element of bool the carrier of O
K350( the carrier of O,s19) is non empty trivial finite 1 -element Element of bool the carrier of O
K350( the carrier of O,s19) * (carr G) is Element of bool the carrier of O
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of O : ( b₁ in K350( the carrier of O,s19) & b₂ in carr G ) } is set
s29 is Element of the carrier of O
s19 * s29 is Element of the carrier of O
the multF of O is Relation-like [: the carrier of O, the carrier of O:] -defined the carrier of O -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of O, the carrier of O:], the carrier of O:]
[: the carrier of O, the carrier of O:] is Relation-like non empty set
[:[: the carrier of O, the carrier of O:], the carrier of O:] is Relation-like non empty set
bool [:[: the carrier of O, the carrier of O:], the carrier of O:] is non empty set
the multF of O . (s19,s29) is Element of the carrier of O
s2 " is Element of the carrier of O
(s2 ") * s19 is Element of the carrier of O
the multF of O . ((s2 "),s19) is Element of the carrier of O
s29 " is Element of the carrier of O
s19 " is Element of the carrier of O
(s29 ") * (s19 ") is Element of the carrier of O
the multF of O . ((s29 "),(s19 ")) is Element of the carrier of O
((s29 ") * (s19 ")) * s19 is Element of the carrier of O
the multF of O . (((s29 ") * (s19 ")),s19) is Element of the carrier of O
(s19 ") * s19 is Element of the carrier of O
the multF of O . ((s19 "),s19) is Element of the carrier of O
(s29 ") * ((s19 ") * s19) is Element of the carrier of O
the multF of O . ((s29 "),((s19 ") * s19)) is Element of the carrier of O
1_ O is non being_of_order_0 Element of the carrier of O
(s29 ") * (1_ O) is Element of the carrier of O
the multF of O . ((s29 "),(1_ O)) is Element of the carrier of O
s2 * (1_ O) is Element of the carrier of O
the multF of O . (s2,(1_ O)) is Element of the carrier of O
O is set
G is Element of O
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
(O,s1,G) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O,s1)
s19 is Element of the carrier of s1
(O,s1,G) . s19 is Element of the carrier of s1
the carrier of s2 is non empty set
(O,s1,G) | the carrier of s2 is Relation-like the carrier of s1 -defined the carrier of s2 -defined the carrier of s1 -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of s1, the carrier of s1:]
((O,s1,G) | the carrier of s2) . s19 is set
(O,s2,G) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
(O,s2,G) . s19 is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is set
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
the carrier of G is non empty set
id (O,G,s1) is Relation-like (O,G,s1) -defined (O,G,s1) -valued Function-like one-to-one total quasi_total Element of bool [:(O,G,s1),(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
bool [:(O,G,s1),(O,G,s1):] is non empty set
{(id (O,G,s1))} is functional non empty trivial finite 1 -element set
[:{},{(id (O,G,s1))}:] is Relation-like finite set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : for b₃ being Element of O holds
( not a₁ = b₃ or ex b₄, b₅ being Element of the carrier of G st
( b₄ in b₁ & b₅ in b₂ & b₅ = (O,G,b₃) . b₄ ) ) } is set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
s19 is set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : for b₃ being Element of O holds
( not s19 = b₃ or ex b₄, b₅ being Element of the carrier of G st
( b₄ in b₁ & b₅ in b₂ & b₅ = (O,G,b₃) . b₄ ) ) } is set
p is set
f1 is Element of (O,G,s1)
i is Element of (O,G,s1)
[f1,i] is set
{f1,i} is non empty finite set
{f1} is non empty trivial finite 1 -element set
{{f1,i},{f1}} is non empty finite V39() set
s199 is set
s199 is set
f2 is set
p is set
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
f1 is set
i is set
[f1,i] is set
{f1,i} is non empty finite set
{f1} is non empty trivial finite 1 -element set
{{f1,i},{f1}} is non empty finite V39() set
s199 is Element of (O,G,s1)
f2 is Element of (O,G,s1)
[s199,f2] is set
{s199,f2} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,f2},{s199}} is non empty finite V39() set
s199 is set
[f1,s199] is set
{f1,s199} is non empty finite set
{{f1,s199},{f1}} is non empty finite V39() set
p is Element of (O,G,s1)
H1 is Element of (O,G,s1)
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
p is Element of O
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
j is Element of the carrier of G
H2 is Element of the carrier of G
(O,G,p) . j is Element of the carrier of G
s299 is Element of the carrier of G
H1 is Element of the carrier of G
(O,G,p) . s299 is Element of the carrier of G
s299 is Element of Left_Cosets s2
s299 * s2 is Element of bool the carrier of G
carr s2 is Element of bool the carrier of G
the carrier of s2 is non empty set
s299 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,s299) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,s299) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,s299) & b₂ in carr s2 ) } is set
H2 is Element of Left_Cosets s2
j * s2 is Element of bool the carrier of G
j * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,j) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,j) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,j) & b₂ in carr s2 ) } is set
s299 " is Element of the carrier of G
(s299 ") * j is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((s299 "),j) is Element of the carrier of G
(O,G,p) . ((s299 ") * j) is Element of the carrier of G
(O,G,p) . (s299 ") is Element of the carrier of G
((O,G,p) . (s299 ")) * ((O,G,p) . j) is Element of the carrier of G
the multF of G . (((O,G,p) . (s299 ")),((O,G,p) . j)) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * H2 is Element of the carrier of G
the multF of G . ((H1 "),H2) is Element of the carrier of G
j is Element of Left_Cosets s2
H1 * s2 is Element of bool the carrier of G
H1 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,H1) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H1) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H1) & b₂ in carr s2 ) } is set
i is Element of Left_Cosets s2
H2 * s2 is Element of bool the carrier of G
H2 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,H2) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H2) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H2) & b₂ in carr s2 ) } is set
f1 is set
s199 is Element of the carrier of G
s199 * s2 is Element of bool the carrier of G
s199 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,s199) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,s199) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,s199) & b₂ in carr s2 ) } is set
s2 * s199 is Element of bool the carrier of G
(carr s2) * s199 is Element of bool the carrier of G
(carr s2) * K350( the carrier of G,s199) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s2 & b₂ in K350( the carrier of G,s199) ) } is set
i is Element of O
(O,G,i) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,i) . s199 is Element of the carrier of G
((O,G,i) . s199) * s2 is Element of bool the carrier of G
((O,G,i) . s199) * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . s199)) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . s199)) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,((O,G,i) . s199)) & b₂ in carr s2 ) } is set
p is Element of (O,G,s1)
H1 is set
j is Element of O
H2 is Element of the carrier of G
s199 is Element of (O,G,s1)
s299 is Element of the carrier of G
(O,G,j) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,j) . H2 is Element of the carrier of G
j is set
[f1,j] is set
{f1,j} is non empty finite set
{f1} is non empty trivial finite 1 -element set
{{f1,j},{f1}} is non empty finite V39() set
p is Relation-like Function-like set
i is set
[f1,i] is set
{f1,i} is non empty finite set
{{f1,i},{f1}} is non empty finite V39() set
s199 is Element of (O,G,s1)
s199 is Element of (O,G,s1)
[s199,s199] is set
{s199,s199} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,s199},{s199}} is non empty finite V39() set
dom p is set
f1 is set
rng p is set
i is set
[i,f1] is set
{i,f1} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,f1},{i}} is non empty finite V39() set
s199 is Element of (O,G,s1)
s199 is Element of (O,G,s1)
[s199,s199] is set
{s199,s199} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,s199},{s199}} is non empty finite V39() set
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s29 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
f1 is Element of O
i is set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : for b₃ being Element of O holds
( not i = b₃ or ex b₄, b₅ being Element of the carrier of G st
( b₄ in b₁ & b₅ in b₂ & b₅ = (O,G,b₃) . b₄ ) ) } is set
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : ex b₃, b₄ being Element of the carrier of G st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,G,f1) . b₃ ) } is set
f2 is set
p is Element of (O,G,s1)
H1 is Element of (O,G,s1)
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
j is Element of the carrier of G
j is Element of the carrier of G
(O,G,f1) . j is Element of the carrier of G
p is Element of (O,G,s1)
H1 is Element of (O,G,s1)
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
j is Element of O
(O,G,j) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
j is Element of the carrier of G
H2 is Element of the carrier of G
(O,G,f1) . j is Element of the carrier of G
p is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
p . f1 is Relation-like Function-like set
s2 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s2 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s2 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s29 is set
dom s2 is Element of bool O
bool O is non empty set
p is Element of O
s2 . p is Relation-like Function-like set
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : ex b₃, b₄ being Element of the carrier of G st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,G,p) . b₃ ) } is set
s2 . s29 is Relation-like Function-like set
s19 . s29 is Relation-like Function-like set
dom s19 is Element of bool O
s2 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s19 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
s29 is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
p is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
the carrier of (O,G,s1) is non empty set
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
1_ (G ./. s2) is non being_of_order_0 Element of the carrier of (G ./. s2)
the carrier of (G ./. s2) is non empty set
p is Element of the carrier of (O,G,s1)
i is Element of the carrier of (O,G,s1)
s199 is Element of the carrier of (G ./. s2)
s199 " is Element of the carrier of (G ./. s2)
f1 is Element of the carrier of (O,G,s1)
i * f1 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (i,f1) is Element of the carrier of (O,G,s1)
s199 * (1_ (G ./. s2)) is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) is Relation-like [: the carrier of (G ./. s2), the carrier of (G ./. s2):] -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):]
[: the carrier of (G ./. s2), the carrier of (G ./. s2):] is Relation-like non empty set
[:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is non empty set
the multF of (G ./. s2) . (s199,(1_ (G ./. s2))) is Element of the carrier of (G ./. s2)
f1 * i is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (f1,i) is Element of the carrier of (O,G,s1)
(1_ (G ./. s2)) * s199 is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . ((1_ (G ./. s2)),s199) is Element of the carrier of (G ./. s2)
p is Element of the carrier of (O,G,s1)
H1 is Element of the carrier of (O,G,s1)
i * H1 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (i,H1) is Element of the carrier of (O,G,s1)
s199 * (s199 ") is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . (s199,(s199 ")) is Element of the carrier of (G ./. s2)
H1 * i is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (H1,i) is Element of the carrier of (O,G,s1)
(s199 ") * s199 is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . ((s199 "),s199) is Element of the carrier of (G ./. s2)
s29 is non empty unital Group-like associative multMagma
the carrier of s29 is non empty set
Funcs ( the carrier of s29, the carrier of s29) is functional non empty set
[:O,(Funcs ( the carrier of s29, the carrier of s29)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s29, the carrier of s29)):] is non empty set
Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) is functional non empty set
p is Relation-like O -defined Funcs ( the carrier of s29, the carrier of s29) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s29, the carrier of s29)):]
the of (O,G,s1) is Relation-like O -defined Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):]
[:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
f1 is Element of O
p . f1 is Relation-like Function-like set
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : ex b₃, b₄ being Element of the carrier of G st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,G,f1) . b₃ ) } is set
i is Relation-like Function-like set
dom i is set
rng i is set
bool [: the carrier of s29, the carrier of s29:] is non empty set
s199 is Element of the carrier of s29
f2 is Element of the carrier of s29
s199 * f2 is Element of the carrier of s29
the multF of s29 . (s199,f2) is Element of the carrier of s29
s199 is Relation-like the carrier of s29 -defined the carrier of s29 -valued Function-like non empty total quasi_total Element of bool [: the carrier of s29, the carrier of s29:]
s199 . s199 is Element of the carrier of s29
s199 . f2 is Element of the carrier of s29
s199 . (s199 * f2) is Element of the carrier of s29
[s199,(s199 . s199)] is set
{s199,(s199 . s199)} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,(s199 . s199)},{s199}} is non empty finite V39() set
H2 is Element of (O,G,s1)
s299 is Element of (O,G,s1)
[H2,s299] is set
{H2,s299} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,s299},{H2}} is non empty finite V39() set
[f2,(s199 . f2)] is set
{f2,(s199 . f2)} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,(s199 . f2)},{f2}} is non empty finite V39() set
H1 is Element of (O,G,s1)
H2 is Element of (O,G,s1)
[H1,H2] is set
{H1,H2} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,H2},{H1}} is non empty finite V39() set
[(s199 * f2),(s199 . (s199 * f2))] is set
{(s199 * f2),(s199 . (s199 * f2))} is non empty finite set
{(s199 * f2)} is non empty trivial finite 1 -element set
{{(s199 * f2),(s199 . (s199 * f2))},{(s199 * f2)}} is non empty finite V39() set
s299 is Element of (O,G,s1)
i is Element of (O,G,s1)
[s299,i] is set
{s299,i} is non empty finite set
{s299} is non empty trivial finite 1 -element set
{{s299,i},{s299}} is non empty finite V39() set
j is Element of the carrier of G
H is Element of the carrier of G
(O,G,f1) . j is Element of the carrier of G
j is Element of the carrier of G
H is Element of the carrier of G
(O,G,f1) . j is Element of the carrier of G
K is Element of the carrier of G
H9 is Element of the carrier of G
(O,G,f1) . K is Element of the carrier of G
K is Element of the carrier of G
H9 is Element of the carrier of G
(O,G,f1) . K is Element of the carrier of G
K9 is Element of the carrier of G
i9 is Element of the carrier of G
(O,G,f1) . K9 is Element of the carrier of G
K9 is Element of the carrier of G
i9 is Element of the carrier of G
(O,G,f1) . K9 is Element of the carrier of G
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
(nat_hom s2) . K9 is Element of the carrier of (G ./. s2)
@ ((nat_hom s2) . K9) is Element of bool the carrier of G
(nat_hom s2) . K is Element of the carrier of (G ./. s2)
@ ((nat_hom s2) . K) is Element of bool the carrier of G
K9 * s2 is Element of bool the carrier of G
carr s2 is Element of bool the carrier of G
the carrier of s2 is non empty set
K9 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,K9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,K9) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,K9) & b₂ in carr s2 ) } is set
K * s2 is Element of bool the carrier of G
K * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,K) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,K) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,K) & b₂ in carr s2 ) } is set
JK is Element of Left_Cosets s2
JK is Element of Left_Cosets s2
j * s2 is Element of bool the carrier of G
j * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,j) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,j) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,j) & b₂ in carr s2 ) } is set
JH is Element of Left_Cosets s2
H9 * s2 is Element of bool the carrier of G
H9 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,H9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H9) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H9) & b₂ in carr s2 ) } is set
j9 is Element of Left_Cosets s2
JH is Element of Left_Cosets s2
i9 is Element of the carrier of (G ./. s2)
H2 is Element of the carrier of (G ./. s2)
the multF of s29 . (i9,H2) is set
i9 * H2 is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . (i9,H2) is Element of the carrier of (G ./. s2)
@ (i9 * H2) is Element of bool the carrier of G
@ i9 is Element of bool the carrier of G
@ H2 is Element of bool the carrier of G
(@ i9) * (@ H2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in @ i9 & b₂ in @ H2 ) } is set
(K9 * s2) * (K * s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K9 * s2 & b₂ in K * s2 ) } is set
((nat_hom s2) . K9) * ((nat_hom s2) . K) is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . (((nat_hom s2) . K9),((nat_hom s2) . K)) is Element of the carrier of (G ./. s2)
K9 * K is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (K9,K) is Element of the carrier of G
(nat_hom s2) . (K9 * K) is Element of the carrier of (G ./. s2)
(K9 * K) * s2 is Element of bool the carrier of G
(K9 * K) * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,(K9 * K)) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,(K9 * K)) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,(K9 * K)) & b₂ in carr s2 ) } is set
j " is Element of the carrier of G
(j ") * (K9 * K) is Element of the carrier of G
the multF of G . ((j "),(K9 * K)) is Element of the carrier of G
(O,G,f1) . ((j ") * (K9 * K)) is Element of the carrier of G
(O,G,f1) . (j ") is Element of the carrier of G
(O,G,f1) . (K9 * K) is Element of the carrier of G
((O,G,f1) . (j ")) * ((O,G,f1) . (K9 * K)) is Element of the carrier of G
the multF of G . (((O,G,f1) . (j ")),((O,G,f1) . (K9 * K))) is Element of the carrier of G
((O,G,f1) . K9) * ((O,G,f1) . K) is Element of the carrier of G
the multF of G . (((O,G,f1) . K9),((O,G,f1) . K)) is Element of the carrier of G
((O,G,f1) . (j ")) * (((O,G,f1) . K9) * ((O,G,f1) . K)) is Element of the carrier of G
the multF of G . (((O,G,f1) . (j ")),(((O,G,f1) . K9) * ((O,G,f1) . K))) is Element of the carrier of G
H " is Element of the carrier of G
i9 * H9 is Element of the carrier of G
the multF of G . (i9,H9) is Element of the carrier of G
(H ") * (i9 * H9) is Element of the carrier of G
the multF of G . ((H "),(i9 * H9)) is Element of the carrier of G
(nat_hom s2) . i9 is Element of the carrier of (G ./. s2)
i9 * s2 is Element of bool the carrier of G
i9 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,i9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,i9) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,i9) & b₂ in carr s2 ) } is set
(nat_hom s2) . H9 is Element of the carrier of (G ./. s2)
k19 is Element of Left_Cosets s2
H * s2 is Element of bool the carrier of G
H * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,H) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H) & b₂ in carr s2 ) } is set
@ ((nat_hom s2) . i9) is Element of bool the carrier of G
@ ((nat_hom s2) . H9) is Element of bool the carrier of G
(s199 . s199) * (s199 . f2) is Element of the carrier of s29
the multF of s29 . ((s199 . s199),(s199 . f2)) is Element of the carrier of s29
H1 is Element of the carrier of (G ./. s2)
H1 is Element of the carrier of (G ./. s2)
the multF of s29 . (H1,H1) is set
H1 * H1 is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . (H1,H1) is Element of the carrier of (G ./. s2)
@ (H1 * H1) is Element of bool the carrier of G
@ H1 is Element of bool the carrier of G
@ H1 is Element of bool the carrier of G
(@ H1) * (@ H1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in @ H1 & b₂ in @ H1 ) } is set
(i9 * s2) * (H9 * s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in i9 * s2 & b₂ in H9 * s2 ) } is set
((nat_hom s2) . i9) * ((nat_hom s2) . H9) is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . (((nat_hom s2) . i9),((nat_hom s2) . H9)) is Element of the carrier of (G ./. s2)
(nat_hom s2) . (i9 * H9) is Element of the carrier of (G ./. s2)
(i9 * H9) * s2 is Element of bool the carrier of G
(i9 * H9) * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,(i9 * H9)) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,(i9 * H9)) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,(i9 * H9)) & b₂ in carr s2 ) } is set
the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is non empty set
s29 is Element of the carrier of (O,G,s1)
p is Element of the carrier of (O,G,s1)
f1 is Element of the carrier of (O,G,s1)
i is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
s199 is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
i * s199 is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
the multF of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is Relation-like [: the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #), the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):] -defined the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #), the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):], the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):]
[: the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #), the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):] is Relation-like non empty set
[:[: the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #), the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):], the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):] is Relation-like non empty set
bool [:[: the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #), the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):], the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #):] is non empty set
the multF of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) . (i,s199) is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
s199 is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
(i * s199) * s199 is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
the multF of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) . ((i * s199),s199) is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
s29 * p is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (s29,p) is Element of the carrier of (O,G,s1)
(s29 * p) * f1 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . ((s29 * p),f1) is Element of the carrier of (O,G,s1)
s199 * s199 is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
the multF of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) . (s199,s199) is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
i * (s199 * s199) is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
the multF of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) . (i,(s199 * s199)) is Element of the carrier of multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #)
p * f1 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (p,f1) is Element of the carrier of (O,G,s1)
s29 * (p * f1) is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (s29,(p * f1)) is Element of the carrier of (O,G,s1)
s29 is Element of the carrier of (O,G,s1)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
1: (G,s1) is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of s1:]
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of s1:]
s29 is Element of O
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,s1,s29) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
p is Element of the carrier of G
(O,G,s29) . p is Element of the carrier of G
s19 . ((O,G,s29) . p) is Element of the carrier of s1
s19 . p is Element of the carrier of s1
(O,s1,s29) . (s19 . p) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
(O,s1,s29) . (1_ s1) is Element of the carrier of s1
O is set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O)
the carrier of s2 is non empty set
[: the carrier of s1, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s2:] is non empty set
[: the carrier of G, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s2:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s29 is Relation-like the carrier of s1 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,s2) Element of bool [: the carrier of s1, the carrier of s2:]
s29 * s19 is Relation-like the carrier of G -defined the carrier of G -defined the carrier of s2 -valued the carrier of s2 -valued Function-like non empty total total quasi_total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of s2:]
p is Relation-like the carrier of G -defined the carrier of s2 -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of s2:]
f1 is Element of O
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
i is Element of the carrier of G
(O,G,f1) . i is Element of the carrier of G
p . ((O,G,f1) . i) is Element of the carrier of s2
s19 . ((O,G,f1) . i) is Element of the carrier of s1
s29 . (s19 . ((O,G,f1) . i)) is Element of the carrier of s2
(O,s1,f1) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
s19 . i is Element of the carrier of s1
(O,s1,f1) . (s19 . i) is Element of the carrier of s1
s29 . ((O,s1,f1) . (s19 . i)) is Element of the carrier of s2
(O,s2,f1) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
s29 . (s19 . i) is Element of the carrier of s2
(O,s2,f1) . (s29 . (s19 . i)) is Element of the carrier of s2
p . i is Element of the carrier of s2
(O,s2,f1) . (p . i) is Element of the carrier of s2
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O)
the carrier of s2 is non empty set
[: the carrier of s1, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s2:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s29 is Relation-like the carrier of s1 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,s2) Element of bool [: the carrier of s1, the carrier of s2:]
s19 * s29 is Relation-like the carrier of G -defined the carrier of s2 -valued Function-like set
[: the carrier of G, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s2:] is non empty set
O is set
s2 is non empty unital Group-like associative (O) (O)
the carrier of s2 is non empty set
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
s19 is non empty unital Group-like associative multMagma
the carrier of s19 is non empty set
id the carrier of s19 is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s19, the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s19:] is non empty set
p is Element of O
(O,s2,p) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
f1 is Element of the carrier of s2
(O,s2,p) . f1 is Element of the carrier of s2
(id the carrier of s19) . ((O,s2,p) . f1) is set
(id the carrier of s19) . f1 is set
(O,s2,p) . ((id the carrier of s19) . f1) is set
p is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s2,s2) Element of bool [: the carrier of s2, the carrier of s2:]
rng p is non empty Element of bool the carrier of s2
bool the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s2 " is Relation-like Function-like set
rng s2 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
dom (s2 ") is set
dom s2 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
rng (s2 ") is set
[: the carrier of s1, the carrier of G:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of G:] is non empty set
s29 is Relation-like the carrier of s1 -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of s1, the carrier of G:]
f1 is Element of the carrier of s1
s29 . f1 is Element of the carrier of G
p is Element of O
(O,s1,p) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,p) . f1 is Element of the carrier of s1
s29 . ((O,s1,p) . f1) is Element of the carrier of G
s2 . (s29 . f1) is Element of the carrier of s1
(O,s1,p) . (s2 . (s29 . f1)) is Element of the carrier of s1
s29 . ((O,s1,p) . (s2 . (s29 . f1))) is Element of the carrier of G
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,p) . (s29 . f1) is Element of the carrier of G
s2 . ((O,G,p) . (s29 . f1)) is Element of the carrier of s1
s29 . (s2 . ((O,G,p) . (s29 . f1))) is Element of the carrier of G
p is Element of the carrier of s1
s29 . p is Element of the carrier of G
f1 is Element of the carrier of s1
s29 . f1 is Element of the carrier of G
s2 . (s29 . p) is Element of the carrier of s1
s2 . (s29 . f1) is Element of the carrier of s1
p * f1 is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (p,f1) is Element of the carrier of s1
s29 . (p * f1) is Element of the carrier of G
(s29 . p) * (s29 . f1) is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((s29 . p),(s29 . f1)) is Element of the carrier of G
s2 . ((s29 . p) * (s29 . f1)) is Element of the carrier of s1
s29 . (s2 . ((s29 . p) * (s29 . f1))) is Element of the carrier of G
p is Relation-like the carrier of s1 -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,G) Element of bool [: the carrier of s1, the carrier of G:]
O is set
s2 is non empty unital Group-like associative (O) (O)
s19 is non empty unital Group-like associative (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of (O,G,s1) is non empty set
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s19 is non empty strict unital Group-like associative normal Subgroup of G
nat_hom s19 is Relation-like the carrier of G -defined the carrier of (G ./. s19) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s19):]
G ./. s19 is non empty strict unital Group-like associative multMagma
Left_Cosets s19 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s19 is Relation-like [:(Left_Cosets s19),(Left_Cosets s19):] -defined Left_Cosets s19 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):]
[:(Left_Cosets s19),(Left_Cosets s19):] is Relation-like non empty set
[:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):] is Relation-like non empty set
bool [:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):] is non empty set
multMagma(# (Left_Cosets s19),(CosOp s19) #) is non empty strict multMagma
the carrier of (G ./. s19) is non empty set
[: the carrier of G, the carrier of (G ./. s19):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s19):] is non empty set
p is non empty unital Group-like associative (O) (O)
the carrier of p is non empty set
[: the carrier of G, the carrier of p:] is Relation-like non empty set
bool [: the carrier of G, the carrier of p:] is non empty set
f1 is Relation-like the carrier of G -defined the carrier of p -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of p:]
i is Element of the carrier of G
s199 is Element of the carrier of G
i * s199 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (i,s199) is Element of the carrier of G
f1 . (i * s199) is Element of the carrier of p
(nat_hom s19) . i is Element of the carrier of (G ./. s19)
(nat_hom s19) . s199 is Element of the carrier of (G ./. s19)
((nat_hom s19) . i) * ((nat_hom s19) . s199) is Element of the carrier of (G ./. s19)
the multF of (G ./. s19) is Relation-like [: the carrier of (G ./. s19), the carrier of (G ./. s19):] -defined the carrier of (G ./. s19) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):]
[: the carrier of (G ./. s19), the carrier of (G ./. s19):] is Relation-like non empty set
[:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):] is non empty set
the multF of (G ./. s19) . (((nat_hom s19) . i),((nat_hom s19) . s199)) is Element of the carrier of (G ./. s19)
f1 . i is Element of the carrier of p
f1 . s199 is Element of the carrier of p
(f1 . i) * (f1 . s199) is Element of the carrier of p
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the multF of p . ((f1 . i),(f1 . s199)) is Element of the carrier of p
the of p is Relation-like O -defined Funcs ( the carrier of p, the carrier of p) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of p, the carrier of p)):]
Funcs ( the carrier of p, the carrier of p) is functional non empty set
[:O,(Funcs ( the carrier of p, the carrier of p)):] is Relation-like set
bool [:O,(Funcs ( the carrier of p, the carrier of p)):] is non empty set
i is Element of O
the of p . i is Relation-like Function-like set
Funcs ( the carrier of p, the carrier of p) is functional non empty FUNCTION_DOMAIN of the carrier of p, the carrier of p
s199 is Relation-like Function-like set
dom s199 is set
rng s199 is set
(O,G,i) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : ex b₃, b₄ being Element of the carrier of G st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,G,i) . b₃ ) } is set
s199 is Element of the carrier of G
f1 . s199 is Element of the carrier of p
s199 . (f1 . s199) is set
[(f1 . s199),(s199 . (f1 . s199))] is set
{(f1 . s199),(s199 . (f1 . s199))} is non empty finite set
{(f1 . s199)} is non empty trivial finite 1 -element set
{{(f1 . s199),(s199 . (f1 . s199))},{(f1 . s199)}} is non empty finite V39() set
f2 is Element of (O,G,s1)
p is Element of (O,G,s1)
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
H1 is Element of the carrier of G
j is Element of the carrier of G
(O,G,i) . H1 is Element of the carrier of G
H1 is Element of the carrier of G
j is Element of the carrier of G
(O,G,i) . H1 is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * s199 is Element of the carrier of G
the multF of G . ((H1 "),s199) is Element of the carrier of G
(O,G,i) . ((H1 ") * s199) is Element of the carrier of G
(O,G,i) . (H1 ") is Element of the carrier of G
(O,G,i) . s199 is Element of the carrier of G
((O,G,i) . (H1 ")) * ((O,G,i) . s199) is Element of the carrier of G
the multF of G . (((O,G,i) . (H1 ")),((O,G,i) . s199)) is Element of the carrier of G
((O,G,i) . H1) " is Element of the carrier of G
(((O,G,i) . H1) ") * ((O,G,i) . s199) is Element of the carrier of G
the multF of G . ((((O,G,i) . H1) "),((O,G,i) . s199)) is Element of the carrier of G
j is Element of Left_Cosets s19
H1 * s19 is Element of bool the carrier of G
carr s19 is Element of bool the carrier of G
the carrier of s19 is non empty set
H1 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,H1) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H1) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H1) & b₂ in carr s19 ) } is set
s199 * s19 is Element of bool the carrier of G
s199 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,s199) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,s199) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,s199) & b₂ in carr s19 ) } is set
(O,p,i) is Relation-like the carrier of p -defined the carrier of p -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of p, the carrier of p:]
bool [: the carrier of p, the carrier of p:] is non empty set
(O,p,i) . (f1 . s199) is Element of the carrier of p
f1 . ((O,G,i) . s199) is Element of the carrier of p
((O,G,i) . s199) * s19 is Element of bool the carrier of G
((O,G,i) . s199) * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . s199)) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . s199)) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,((O,G,i) . s199)) & b₂ in carr s19 ) } is set
((O,G,i) . H1) * s19 is Element of bool the carrier of G
((O,G,i) . H1) * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . H1)) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,((O,G,i) . H1)) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,((O,G,i) . H1)) & b₂ in carr s19 ) } is set
H2 is Element of Left_Cosets s19
i is Element of O
(O,G,i) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
s199 is Element of the carrier of G
(O,G,i) . s199 is Element of the carrier of G
(O,p,i) is Relation-like the carrier of p -defined the carrier of p -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of p, the carrier of p:]
bool [: the carrier of p, the carrier of p:] is non empty set
id the carrier of p is Relation-like the carrier of p -defined the carrier of p -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of p, the carrier of p:]
f1 . ((O,G,i) . s199) is Element of the carrier of p
f1 . s199 is Element of the carrier of p
(O,p,i) . (f1 . s199) is Element of the carrier of p
i is Relation-like the carrier of G -defined the carrier of p -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,p) Element of bool [: the carrier of G, the carrier of p:]
s199 is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
s199 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s199 is non empty strict unital Group-like associative multMagma
Left_Cosets s199 is non empty Element of bool (bool the carrier of G)
CosOp s199 is Relation-like [:(Left_Cosets s199),(Left_Cosets s199):] -defined Left_Cosets s199 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s199),(Left_Cosets s199):],(Left_Cosets s199):]
[:(Left_Cosets s199),(Left_Cosets s199):] is Relation-like non empty set
[:[:(Left_Cosets s199),(Left_Cosets s199):],(Left_Cosets s199):] is Relation-like non empty set
bool [:[:(Left_Cosets s199),(Left_Cosets s199):],(Left_Cosets s199):] is non empty set
multMagma(# (Left_Cosets s199),(CosOp s199) #) is non empty strict multMagma
the carrier of (G ./. s199) is non empty set
nat_hom s199 is Relation-like the carrier of G -defined the carrier of (G ./. s199) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s199):]
[: the carrier of G, the carrier of (G ./. s199):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s199):] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
s29 is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
the carrier of (G ./. s2) is non empty set
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
1_ s1 is non being_of_order_0 Element of the carrier of s1
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s2 . (1_ G) is Element of the carrier of s1
(1_ G) * (1_ G) is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((1_ G),(1_ G)) is Element of the carrier of G
s2 . ((1_ G) * (1_ G)) is Element of the carrier of s1
(s2 . (1_ G)) * (s2 . (1_ G)) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((s2 . (1_ G)),(s2 . (1_ G))) is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Element of the carrier of G
s2 " is Element of the carrier of G
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s19 . (s2 ") is Element of the carrier of s1
s19 . s2 is Element of the carrier of s1
(s19 . s2) " is Element of the carrier of s1
(s19 . (s2 ")) * (s19 . s2) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((s19 . (s2 ")),(s19 . s2)) is Element of the carrier of s1
(s2 ") * s2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((s2 "),s2) is Element of the carrier of G
s19 . ((s2 ") * s2) is Element of the carrier of s1
1_ G is non being_of_order_0 Element of the carrier of G
s19 . (1_ G) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
s2 is non empty strict unital Group-like associative Subgroup of G
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
p is non empty (O)
f1 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
(O, the carrier of s2, the multF of s2,f1) is (O) (O)
the carrier of p is non empty set
i is Element of the carrier of p
s199 is Element of the carrier of p
s199 is Element of the carrier of p
i * s199 is Element of the carrier of p
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the multF of p . (i,s199) is Element of the carrier of p
(i * s199) * s199 is Element of the carrier of p
the multF of p . ((i * s199),s199) is Element of the carrier of p
f2 is Element of the carrier of s2
p is Element of the carrier of s2
f2 * p is Element of the carrier of s2
the multF of s2 . (f2,p) is Element of the carrier of s2
H1 is Element of the carrier of s2
(f2 * p) * H1 is Element of the carrier of s2
the multF of s2 . ((f2 * p),H1) is Element of the carrier of s2
p * H1 is Element of the carrier of s2
the multF of s2 . (p,H1) is Element of the carrier of s2
f2 * (p * H1) is Element of the carrier of s2
the multF of s2 . (f2,(p * H1)) is Element of the carrier of s2
s199 * s199 is Element of the carrier of p
the multF of p . (s199,s199) is Element of the carrier of p
i * (s199 * s199) is Element of the carrier of p
the multF of p . (i,(s199 * s199)) is Element of the carrier of p
1_ s2 is non being_of_order_0 Element of the carrier of s2
s199 is Element of the carrier of p
f2 is Element of the carrier of p
p is Element of the carrier of s2
p " is Element of the carrier of s2
s199 is Element of the carrier of p
f2 * s199 is Element of the carrier of p
the multF of p . (f2,s199) is Element of the carrier of p
p * (1_ s2) is Element of the carrier of s2
the multF of s2 . (p,(1_ s2)) is Element of the carrier of s2
s199 * f2 is Element of the carrier of p
the multF of p . (s199,f2) is Element of the carrier of p
(1_ s2) * p is Element of the carrier of s2
the multF of s2 . ((1_ s2),p) is Element of the carrier of s2
j is Element of the carrier of p
j is Element of the carrier of p
f2 * j is Element of the carrier of p
the multF of p . (f2,j) is Element of the carrier of p
p * (p ") is Element of the carrier of s2
the multF of s2 . (p,(p ")) is Element of the carrier of s2
j * f2 is Element of the carrier of p
the multF of p . (j,f2) is Element of the carrier of p
(p ") * p is Element of the carrier of s2
the multF of s2 . ((p "),p) is Element of the carrier of s2
i is Element of the carrier of p
id the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s2, the carrier of s2:]
bool [: the carrier of s2, the carrier of s2:] is non empty set
{(id the carrier of s2)} is functional non empty trivial finite 1 -element set
[:{},{(id the carrier of s2)}:] is Relation-like finite set
f1 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
(O, the carrier of s2, the multF of s2,f1) is (O) (O)
s199 is non empty unital Group-like associative multMagma
the carrier of s199 is non empty set
Funcs ( the carrier of s199, the carrier of s199) is functional non empty set
[:O,(Funcs ( the carrier of s199, the carrier of s199)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):] is non empty set
s199 is non empty (O)
the carrier of s199 is non empty set
Funcs ( the carrier of s199, the carrier of s199) is functional non empty set
f2 is Relation-like O -defined Funcs ( the carrier of s199, the carrier of s199) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):]
the of s199 is Relation-like O -defined Funcs ( the carrier of s199, the carrier of s199) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):]
[:O,(Funcs ( the carrier of s199, the carrier of s199)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):] is non empty set
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
multMagma(# the carrier of s199, the multF of s199 #) is non empty strict multMagma
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
multMagma(# the carrier of s199, the multF of s199 #) is non empty strict multMagma
bool [: the carrier of s199, the carrier of s199:] is non empty set
p is Element of O
f2 . p is Relation-like Function-like set
s199 is non empty unital Group-like associative (O) (O)
the carrier of s199 is non empty set
p is set
H1 is set
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(id the carrier of G) | the carrier of s199 is Relation-like the carrier of G -defined the carrier of s199 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
id the carrier of s199 is Relation-like the carrier of s199 -defined the carrier of s199 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s199, the carrier of s199:]
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
bool [: the carrier of s199, the carrier of s199:] is non empty set
p is set
H1 is set
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
f2 is Element of O
(O,s199,f2) is Relation-like the carrier of s199 -defined the carrier of s199 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s199, the carrier of s199:]
(O,G,f2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,f2) | the carrier of s199 is Relation-like the carrier of G -defined the carrier of s199 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of s199 is set
the multF of G | [: the carrier of s199, the carrier of s199:] is Relation-like [: the carrier of s199, the carrier of s199:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
f2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of f2 is non empty set
{ [b₁,((O,G,b₁) | the carrier of s2)] where b₁ is Element of O : verum } is set
f1 is set
i is Element of O
(O,G,i) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,i) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[i,((O,G,i) | the carrier of s2)] is set
{i,((O,G,i) | the carrier of s2)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,((O,G,i) | the carrier of s2)},{i}} is non empty finite V39() set
i is set
s199 is Element of O
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s199) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
s199 is set
f2 is set
[i,f2] is set
{i,f2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,f2},{i}} is non empty finite V39() set
f1 is Relation-like set
i is set
s199 is set
[i,s199] is set
{i,s199} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s199},{i}} is non empty finite V39() set
s199 is Element of O
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,s199) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[s199,((O,G,s199) | the carrier of s2)] is set
{s199,((O,G,s199) | the carrier of s2)} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,((O,G,s199) | the carrier of s2)},{s199}} is non empty finite V39() set
dom f1 is set
i is set
s199 is set
[i,s199] is set
{i,s199} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s199},{i}} is non empty finite V39() set
f2 is Element of O
(O,G,f2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,f2) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[f2,((O,G,f2) | the carrier of s2)] is set
{f2,((O,G,f2) | the carrier of s2)} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,((O,G,f2) | the carrier of s2)},{f2}} is non empty finite V39() set
s199 is set
[i,s199] is set
{i,s199} is non empty finite set
{{i,s199},{i}} is non empty finite V39() set
p is Element of O
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,p) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[p,((O,G,p) | the carrier of s2)] is set
{p,((O,G,p) | the carrier of s2)} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,((O,G,p) | the carrier of s2)},{p}} is non empty finite V39() set
s199 is set
i is Relation-like Function-like set
rng i is set
dom i is set
s199 is set
i . s199 is set
[s199,s199] is set
{s199,s199} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,s199},{s199}} is non empty finite V39() set
f2 is Element of O
(O,G,f2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,f2) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[f2,((O,G,f2) | the carrier of s2)] is set
{f2,((O,G,f2) | the carrier of s2)} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,((O,G,f2) | the carrier of s2)},{f2}} is non empty finite V39() set
p is Relation-like Function-like set
dom ((O,G,f2) | the carrier of s2) is Element of bool the carrier of G
id the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of s2, the carrier of s2:]
bool [: the carrier of s2, the carrier of s2:] is non empty set
(O,G,f2) * (id the carrier of s2) is Relation-like the carrier of s2 -defined the carrier of G -valued Function-like Element of bool [: the carrier of s2, the carrier of G:]
[: the carrier of s2, the carrier of G:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of G:] is non empty set
dom ((O,G,f2) * (id the carrier of s2)) is Element of bool the carrier of s2
bool the carrier of s2 is non empty set
dom (O,G,f2) is non empty Element of bool the carrier of G
(dom (O,G,f2)) /\ the carrier of s2 is Element of bool the carrier of G
the carrier of G /\ the carrier of s2 is set
H1 is Relation-like Function-like set
dom H1 is set
j is set
rng H1 is set
j is set
H1 . j is set
((O,G,f2) * (id the carrier of s2)) . j is set
(id the carrier of s2) . j is set
(O,G,f2) . ((id the carrier of s2) . j) is set
(O,G,f2) . j is set
Funcs ( the carrier of s2, the carrier of s2) is functional non empty FUNCTION_DOMAIN of the carrier of s2, the carrier of s2
p is Relation-like Function-like set
dom p is set
rng p is set
s199 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
(O, the carrier of s2, the multF of s2,s199) is (O) (O)
s199 is non empty (O)
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
the carrier of s199 is non empty set
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of s199 is set
the multF of G | [: the carrier of s199, the carrier of s199:] is Relation-like [: the carrier of s199, the carrier of s199:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
f2 is non empty unital Group-like associative multMagma
the carrier of f2 is non empty set
Funcs ( the carrier of f2, the carrier of f2) is functional non empty set
[:O,(Funcs ( the carrier of f2, the carrier of f2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of f2, the carrier of f2)):] is non empty set
Funcs ( the carrier of s199, the carrier of s199) is functional non empty set
p is Relation-like O -defined Funcs ( the carrier of f2, the carrier of f2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of f2, the carrier of f2)):]
the of s199 is Relation-like O -defined Funcs ( the carrier of s199, the carrier of s199) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):]
[:O,(Funcs ( the carrier of s199, the carrier of s199)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s199, the carrier of s199)):] is non empty set
the multF of f2 is Relation-like [: the carrier of f2, the carrier of f2:] -defined the carrier of f2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of f2, the carrier of f2:], the carrier of f2:]
[: the carrier of f2, the carrier of f2:] is Relation-like non empty set
[:[: the carrier of f2, the carrier of f2:], the carrier of f2:] is Relation-like non empty set
bool [:[: the carrier of f2, the carrier of f2:], the carrier of f2:] is non empty set
multMagma(# the carrier of f2, the multF of f2 #) is non empty strict multMagma
multMagma(# the carrier of s199, the multF of s199 #) is non empty strict multMagma
H1 is Element of O
dom p is Element of bool O
bool O is non empty set
p . H1 is Relation-like Function-like set
rng p is functional Element of bool (Funcs ( the carrier of f2, the carrier of f2))
bool (Funcs ( the carrier of f2, the carrier of f2)) is non empty set
j is Relation-like Function-like set
dom j is set
rng j is set
bool [: the carrier of f2, the carrier of f2:] is non empty set
[H1,(p . H1)] is set
{H1,(p . H1)} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,(p . H1)},{H1}} is non empty finite V39() set
H2 is Element of O
(O,G,H2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,H2) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[H2,((O,G,H2) | the carrier of s2)] is set
{H2,((O,G,H2) | the carrier of s2)} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,((O,G,H2) | the carrier of s2)},{H2}} is non empty finite V39() set
H1 is Element of the carrier of f2
dom (id the carrier of s2) is non empty Element of bool the carrier of s2
s299 is Element of the carrier of f2
H2 is Element of the carrier of s199
s299 is Element of the carrier of s199
(O,G,H1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
i is Element of the carrier of G
(O,G,H1) . i is Element of the carrier of G
j is Element of the carrier of G
(O,G,H1) . j is Element of the carrier of G
s299 * H1 is Element of the carrier of f2
the multF of f2 . (s299,H1) is Element of the carrier of f2
j is Relation-like the carrier of f2 -defined the carrier of f2 -valued Function-like non empty total quasi_total Element of bool [: the carrier of f2, the carrier of f2:]
j . H1 is Element of the carrier of f2
(O,G,H1) * (id the carrier of s2) is Relation-like the carrier of s2 -defined the carrier of G -valued Function-like Element of bool [: the carrier of s2, the carrier of G:]
((O,G,H1) * (id the carrier of s2)) . H1 is set
(id the carrier of s2) . H1 is set
(O,G,H1) . ((id the carrier of s2) . H1) is set
K is Element of the carrier of s199
j . s299 is Element of the carrier of f2
((O,G,H1) * (id the carrier of s2)) . s299 is set
(id the carrier of s2) . s299 is set
(O,G,H1) . ((id the carrier of s2) . s299) is set
H is Element of the carrier of s199
j . (s299 * H1) is Element of the carrier of f2
((O,G,H1) * (id the carrier of s2)) . (s299 * H1) is set
(id the carrier of s2) . (s299 * H1) is set
(O,G,H1) . ((id the carrier of s2) . (s299 * H1)) is set
H2 * s299 is Element of the carrier of s199
the multF of s199 . (H2,s299) is Element of the carrier of s199
(O,G,H1) . (H2 * s299) is set
i * j is Element of the carrier of G
the multF of G . (i,j) is Element of the carrier of G
(O,G,H1) . (i * j) is Element of the carrier of G
((O,G,H1) . i) * ((O,G,H1) . j) is Element of the carrier of G
the multF of G . (((O,G,H1) . i),((O,G,H1) . j)) is Element of the carrier of G
H * K is Element of the carrier of s199
the multF of s199 . (H,K) is Element of the carrier of s199
(j . s299) * (j . H1) is Element of the carrier of f2
the multF of f2 . ((j . s299),(j . H1)) is Element of the carrier of f2
p is Element of O
dom s199 is Element of bool O
s199 . p is Relation-like Function-like set
[p,(s199 . p)] is set
{p,(s199 . p)} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,(s199 . p)},{p}} is non empty finite V39() set
H1 is Element of O
(O,G,H1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,H1) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
[H1,((O,G,H1) | the carrier of s2)] is set
{H1,((O,G,H1) | the carrier of s2)} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,((O,G,H1) | the carrier of s2)},{H1}} is non empty finite V39() set
f2 is non empty unital Group-like associative (O) (O)
(O,f2,p) is Relation-like the carrier of f2 -defined the carrier of f2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of f2, the carrier of f2:]
the carrier of f2 is non empty set
[: the carrier of f2, the carrier of f2:] is Relation-like non empty set
bool [: the carrier of f2, the carrier of f2:] is non empty set
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,p) | the carrier of f2 is Relation-like the carrier of G -defined the carrier of f2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
1_ s1 is non being_of_order_0 Element of the carrier of s1
{ b₁ where b₁ is Element of the carrier of G : s2 . b₁ = 1_ s1 } is set
bool the carrier of G is non empty set
{ b₁ where b₁ is Element of the carrier of G : S₁[b₁] } is set
s29 is Element of the carrier of G
s19 is Element of bool the carrier of G
p is Element of the carrier of G
s29 * p is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s29,p) is Element of the carrier of G
s2 . (s29 * p) is Element of the carrier of s1
s2 . s29 is Element of the carrier of s1
s2 . p is Element of the carrier of s1
(s2 . s29) * (s2 . p) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((s2 . s29),(s2 . p)) is Element of the carrier of s1
f1 is Element of the carrier of G
s2 . f1 is Element of the carrier of s1
i is Element of the carrier of G
s2 . i is Element of the carrier of s1
s29 is Element of the carrier of G
s29 " is Element of the carrier of G
s2 . (s29 ") is Element of the carrier of s1
(1_ s1) " is Element of the carrier of s1
p is Element of the carrier of G
s2 . p is Element of the carrier of s1
p is Element of the carrier of G
s29 is Element of O
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s29) . p is Element of the carrier of G
s2 . ((O,G,s29) . p) is Element of the carrier of s1
(O,s1,s29) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,s29) . (1_ s1) is Element of the carrier of s1
f1 is Element of the carrier of G
s2 . f1 is Element of the carrier of s1
1_ G is non being_of_order_0 Element of the carrier of G
s2 . (1_ G) is Element of the carrier of s1
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative multMagma
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative multMagma
the carrier of s29 is non empty set
[: the carrier of s19, the carrier of s29:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s29:] is non empty set
f1 is Relation-like the carrier of s19 -defined the carrier of s29 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s29:]
Ker f1 is non empty strict unital Group-like associative normal Subgroup of s19
the carrier of (Ker f1) is non empty set
1_ s1 is non being_of_order_0 Element of the carrier of s1
{ b₁ where b₁ is Element of the carrier of G : s2 . b₁ = 1_ s1 } is set
p is non empty strict unital Group-like associative Subgroup of G
the carrier of (O,G,s1,s2) is non empty set
the multF of (O,G,s1,s2) is Relation-like [: the carrier of (O,G,s1,s2), the carrier of (O,G,s1,s2):] -defined the carrier of (O,G,s1,s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1,s2), the carrier of (O,G,s1,s2):], the carrier of (O,G,s1,s2):]
[: the carrier of (O,G,s1,s2), the carrier of (O,G,s1,s2):] is Relation-like non empty set
[:[: the carrier of (O,G,s1,s2), the carrier of (O,G,s1,s2):], the carrier of (O,G,s1,s2):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1,s2), the carrier of (O,G,s1,s2):], the carrier of (O,G,s1,s2):] is non empty set
multMagma(# the carrier of (O,G,s1,s2), the multF of (O,G,s1,s2) #) is non empty strict multMagma
the carrier of p is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
1_ s1 is non being_of_order_0 Element of the carrier of s1
s2 is non empty multMagma
the carrier of s2 is non empty set
s29 is Element of the carrier of s2
f1 is Element of the carrier of s2
i is Element of the carrier of s1
i " is Element of the carrier of s1
p is Element of the carrier of s2
f1 * p is Element of the carrier of s2
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the multF of s2 . (f1,p) is Element of the carrier of s2
i * (1_ s1) is Element of the carrier of s1
the multF of s1 . (i,(1_ s1)) is Element of the carrier of s1
p * f1 is Element of the carrier of s2
the multF of s2 . (p,f1) is Element of the carrier of s2
(1_ s1) * i is Element of the carrier of s1
the multF of s1 . ((1_ s1),i) is Element of the carrier of s1
s199 is Element of the carrier of s2
f2 is Element of the carrier of s2
f1 * f2 is Element of the carrier of s2
the multF of s2 . (f1,f2) is Element of the carrier of s2
i * (i ") is Element of the carrier of s1
the multF of s1 . (i,(i ")) is Element of the carrier of s1
f2 * f1 is Element of the carrier of s2
the multF of s2 . (f2,f1) is Element of the carrier of s2
(i ") * i is Element of the carrier of s1
the multF of s1 . ((i "),i) is Element of the carrier of s1
s29 is Element of the carrier of s2
s19 is non empty unital Group-like multMagma
the carrier of s19 is non empty set
the carrier of G is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of s19 is set
the multF of G | [: the carrier of s19, the carrier of s19:] is Relation-like [: the carrier of s19, the carrier of s19:] -defined [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s29 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s29 .: the carrier of s2 is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
s19 is non empty strict unital Group-like associative Subgroup of G
{ (s29 . b₁) where b₁ is Element of the carrier of G : b₁ in s19 } is set
1_ G is non being_of_order_0 Element of the carrier of G
s29 . (1_ G) is Element of the carrier of s1
i is set
f1 is non empty set
s199 is Element of the carrier of G
s29 . s199 is Element of the carrier of s1
s199 is Element of the carrier of s1
i is Element of bool the carrier of s1
s199 is Element of the carrier of s1
f2 is Element of the carrier of G
s29 . f2 is Element of the carrier of s1
p is Element of the carrier of G
s29 . p is Element of the carrier of s1
f2 * p is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (f2,p) is Element of the carrier of G
s29 . (f2 * p) is Element of the carrier of s1
s199 * s199 is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (s199,s199) is Element of the carrier of s1
s199 is Element of the carrier of s1
f2 is Element of the carrier of G
s29 . f2 is Element of the carrier of s1
the carrier of s19 is non empty set
s199 is Element of O
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s199) . f2 is Element of the carrier of G
(O,s1,s199) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,s199) . s199 is Element of the carrier of s1
s29 . ((O,G,s199) . f2) is Element of the carrier of s1
s199 is Element of the carrier of s1
s199 is Element of the carrier of G
s29 . s199 is Element of the carrier of s1
s199 " is Element of the carrier of G
s199 " is Element of the carrier of s1
s29 . (s199 ") is Element of the carrier of s1
s199 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s199 is non empty set
f2 is Element of the carrier of s1
H1 is Element of the carrier of G
s29 . H1 is Element of the carrier of s1
dom s29 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
[H1,f2] is set
{H1,f2} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,f2},{H1}} is non empty finite V39() set
p is Relation-like the carrier of G -defined the carrier of s1 -valued Element of bool [: the carrier of G, the carrier of s1:]
H1 is Element of the carrier of G
[H1,f2] is set
{H1,f2} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,f2},{H1}} is non empty finite V39() set
s29 . H1 is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s2 .: the carrier of G is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
s2 .: the carrier of s19 is Element of bool the carrier of s1
s29 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s29 is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s29 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty set
s2 is non empty unital Group-like associative Subgroup of G
carr s2 is Element of bool the carrier of G
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
O is set
G is non empty unital Group-like associative (O) (O)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is Element of the carrier of G
s19 is Element of the carrier of G
s2 * s19 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s2,s19) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is Element of the carrier of G
s2 " is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) /\ (O,G,s2) is Element of bool the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
s29 is Element of the carrier of G
p is Element of the carrier of G
s29 * p is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s29,p) is Element of the carrier of G
p is Element of the carrier of G
s29 is Element of O
(O,G,s29) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s29) . p is Element of the carrier of G
s29 is Element of the carrier of G
s29 " is Element of the carrier of G
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative (O) (O,G)
(O,G,s29) is Element of bool the carrier of G
the carrier of s29 is non empty set
p is non empty unital Group-like associative (O) (O,G)
(O,G,p) is Element of bool the carrier of G
the carrier of p is non empty set
(O,G,s29) /\ (O,G,p) is Element of bool the carrier of G
(O,G,p) /\ (O,G,s29) is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s2 is non empty set
p is Element of O
(O,s1,p) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
the carrier of G is non empty set
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,p) | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
(O,G,p) | the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
((O,G,p) | the carrier of s2) | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
(O,s2,p) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
(O,s2,p) | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of s2 -defined the carrier of s2 -valued Function-like Element of bool [: the carrier of s2, the carrier of s2:]
s19 is non empty unital Group-like associative Subgroup of G
s29 is non empty unital Group-like associative Subgroup of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
bool the carrier of G is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
s2 is set
meet s2 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
s29 is set
p is set
f1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,f1) is Element of bool the carrier of G
the carrier of f1 is non empty set
s29 is set
p is non empty unital Group-like associative (O) (O) (O,G)
(O,G,p) is Element of bool the carrier of G
the carrier of p is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
f1 is Element of the carrier of G
s29 is Element of bool the carrier of G
i is set
s199 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s199) is Element of bool the carrier of G
the carrier of s199 is non empty set
p is Element of O
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,p) . f1 is Element of the carrier of G
p is Element of the carrier of G
f1 is Element of the carrier of G
i is set
s199 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s199) is Element of bool the carrier of G
the carrier of s199 is non empty set
p * f1 is Element of the carrier of G
the multF of G . (p,f1) is Element of the carrier of G
p is Element of the carrier of G
f1 is set
i is non empty unital Group-like associative (O) (O) (O,G)
(O,G,i) is Element of bool the carrier of G
the carrier of i is non empty set
p " is Element of the carrier of G
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
f1 is set
i is non empty unital Group-like associative (O) (O) (O,G)
(O,G,i) is Element of bool the carrier of G
the carrier of i is non empty set
f1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of f1 is non empty set
(O,G,f1) is Element of bool the carrier of G
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is set
s1 is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative (O) (O,s1)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is Element of the carrier of G
s19 is Element of the carrier of G
s2 * s19 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s2,s19) is Element of the carrier of G
s29 is Element of the carrier of s1
p is Element of the carrier of s1
s29 * p is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (s29,p) is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
1_ s1 is non being_of_order_0 Element of the carrier of s1
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative Subgroup of G
1_ s2 is non being_of_order_0 Element of the carrier of s2
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is Element of the carrier of G
s2 " is Element of the carrier of G
s19 is Element of the carrier of s1
s19 " is Element of the carrier of s1
s19 * (s19 ") is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (s19,(s19 ")) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
s29 is Element of the carrier of G
s2 * s29 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s2,s29) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is Element of the carrier of G
s19 is Element of the carrier of G
s2 * s19 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s2,s19) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is Element of the carrier of G
s2 " is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is Element of O
(O,G,s1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s1) | the carrier of G is Relation-like the carrier of G -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the carrier of s1 is non empty set
s19 is Element of O
(O,G,s19) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,s1,s19) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,s19) | the carrier of G is Relation-like the carrier of G -defined the carrier of s1 -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of s1, the carrier of s1:]
the carrier of s2 is non empty set
(O,s2,s19) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s2:] is non empty set
(O,s2,s19) | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of s2 -defined the carrier of s2 -valued Function-like Element of bool [: the carrier of s2, the carrier of s2:]
((O,s2,s19) | the carrier of s1) | the carrier of G is Relation-like the carrier of G -defined the carrier of s1 -defined the carrier of s2 -defined the carrier of s2 -valued Function-like Element of bool [: the carrier of s2, the carrier of s2:]
the carrier of s1 /\ the carrier of G is set
(O,s2,s19) | ( the carrier of s1 /\ the carrier of G) is Relation-like the carrier of s2 -defined the carrier of s1 /\ the carrier of G -defined the carrier of s2 -defined the carrier of s2 -valued Function-like Element of bool [: the carrier of s2, the carrier of s2:]
(O,s2,s19) | the carrier of G is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of s2 -valued Function-like Element of bool [: the carrier of s2, the carrier of s2:]
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the carrier of s2 is non empty set
s19 is set
s29 is Element of the carrier of s1
p is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is non empty unital Group-like associative (O) (O,G)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
1_ s1 is non being_of_order_0 Element of the carrier of s1
the carrier of s1 is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
the carrier of (O,s1) is non empty set
{(1_ s1)} is non empty trivial finite 1 -element set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is non empty unital Group-like associative (O) (O,G)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
(O,G,s2) * (O,G,s1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s1) ) } is set
s29 is Element of the carrier of G
p is Element of the carrier of G
f1 is Element of the carrier of G
p * f1 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (p,f1) is Element of the carrier of G
s19 is Element of O
(O,G,s19) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s19) . p is Element of the carrier of G
(O,G,s19) . f1 is Element of the carrier of G
((O,G,s19) . p) * ((O,G,s19) . f1) is Element of the carrier of G
the multF of G . (((O,G,s19) . p),((O,G,s19) . f1)) is Element of the carrier of G
(O,G,s19) . s29 is Element of the carrier of G
s19 is Element of the carrier of G
s29 is Element of the carrier of G
p is Element of the carrier of G
s29 * p is Element of the carrier of G
the multF of G . (s29,p) is Element of the carrier of G
f1 is Element of the carrier of G
i is Element of the carrier of G
f1 * i is Element of the carrier of G
the multF of G . (f1,i) is Element of the carrier of G
f1 " is Element of the carrier of G
i " is Element of the carrier of G
s19 " is Element of the carrier of G
(i ") * (f1 ") is Element of the carrier of G
the multF of G . ((i "),(f1 ")) is Element of the carrier of G
s19 is Element of the carrier of G
s29 is Element of the carrier of G
p is Element of the carrier of G
f1 is Element of the carrier of G
p * f1 is Element of the carrier of G
the multF of G . (p,f1) is Element of the carrier of G
i is Element of the carrier of G
s199 is Element of the carrier of G
i * s199 is Element of the carrier of G
the multF of G . (i,s199) is Element of the carrier of G
f1 * i is Element of the carrier of G
the multF of G . (f1,i) is Element of the carrier of G
s199 is Element of the carrier of G
f2 is Element of the carrier of G
s199 * f2 is Element of the carrier of G
the multF of G . (s199,f2) is Element of the carrier of G
f2 * s199 is Element of the carrier of G
the multF of G . (f2,s199) is Element of the carrier of G
p * s199 is Element of the carrier of G
the multF of G . (p,s199) is Element of the carrier of G
s19 * s29 is Element of the carrier of G
the multF of G . (s19,s29) is Element of the carrier of G
(p * f1) * i is Element of the carrier of G
the multF of G . ((p * f1),i) is Element of the carrier of G
((p * f1) * i) * s199 is Element of the carrier of G
the multF of G . (((p * f1) * i),s199) is Element of the carrier of G
p * (f1 * i) is Element of the carrier of G
the multF of G . (p,(f1 * i)) is Element of the carrier of G
(p * (f1 * i)) * s199 is Element of the carrier of G
the multF of G . ((p * (f1 * i)),s199) is Element of the carrier of G
(p * s199) * f2 is Element of the carrier of G
the multF of G . ((p * s199),f2) is Element of the carrier of G
((p * s199) * f2) * s199 is Element of the carrier of G
the multF of G . (((p * s199) * f2),s199) is Element of the carrier of G
(p * s199) * (f2 * s199) is Element of the carrier of G
the multF of G . ((p * s199),(f2 * s199)) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
the carrier of s1 /\ the carrier of s2 is set
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
(O,G,s2) is Element of bool the carrier of G
s19 is non empty unital Group-like associative (O) (O,G)
the carrier of s19 is non empty set
(O,G,s1) /\ (O,G,s2) is Element of bool the carrier of G
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s1,s1) is non empty set
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s1) /\ (O,G,s1) is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O,G)
(O,G,(O,G,s1,s2),s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,(O,G,s2,s19)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,(O,G,s1,s2),s19) is non empty set
the carrier of G is non empty set
(O,G,(O,G,s1,s2)) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of (O,G,s1,s2) is non empty set
(O,G,s19) is Element of bool the carrier of G
the carrier of s19 is non empty set
(O,G,(O,G,s1,s2)) /\ (O,G,s19) is Element of bool the carrier of G
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) /\ (O,G,s2) is Element of bool the carrier of G
((O,G,s1) /\ (O,G,s2)) /\ (O,G,s19) is Element of bool the carrier of G
(O,G,s2) /\ (O,G,s19) is Element of bool the carrier of G
(O,G,s1) /\ ((O,G,s2) /\ (O,G,s19)) is Element of bool the carrier of G
(O,G,(O,G,s2,s19)) is Element of bool the carrier of G
the carrier of (O,G,s2,s19) is non empty set
(O,G,s1) /\ (O,G,(O,G,s2,s19)) is Element of bool the carrier of G
the carrier of (O,G,s1,(O,G,s2,s19)) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
the carrier of s1 is non empty set
the carrier of (O,G,s2,s1) is non empty set
the carrier of G is non empty set
(O,G,s2) is Element of bool the carrier of G
bool the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
(O,G,s2) /\ (O,G,s1) is Element of bool the carrier of G
the carrier of s2 is non empty set
the carrier of G is non empty set
(O,G,s2) is Element of bool the carrier of G
bool the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,s2) /\ (O,G,s1) is Element of bool the carrier of G
the carrier of s2 /\ the carrier of s1 is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,(O,G),s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is set
union (O,G,s1) is set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
the carrier of s2 is non empty set
the Element of the carrier of s2 is Element of the carrier of s2
s29 is set
f1 is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
f1 * (1_ G) is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (f1,(1_ G)) is Element of the carrier of G
p is Element of the carrier of G
p " is Element of the carrier of G
(p ") * p is Element of the carrier of G
the multF of G . ((p "),p) is Element of the carrier of G
f1 * ((p ") * p) is Element of the carrier of G
the multF of G . (f1,((p ") * p)) is Element of the carrier of G
f1 * (p ") is Element of the carrier of G
the multF of G . (f1,(p ")) is Element of the carrier of G
(f1 * (p ")) * p is Element of the carrier of G
the multF of G . ((f1 * (p ")),p) is Element of the carrier of G
(f1 * (p ")) * s2 is Element of bool the carrier of G
bool the carrier of G is non empty set
carr s2 is Element of bool the carrier of G
(f1 * (p ")) * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,(f1 * (p "))) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,(f1 * (p "))) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,(f1 * (p "))) & b₂ in carr s2 ) } is set
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
union (Left_Cosets s2) is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
p is non empty strict unital Group-like associative normal Subgroup of G
carr p is Element of bool the carrier of G
the carrier of p is non empty set
f1 is non empty strict unital Group-like associative normal Subgroup of G
carr f1 is Element of bool the carrier of G
the carrier of f1 is non empty set
(carr p) * (carr f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in carr f1 ) } is set
(carr f1) * (carr p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr f1 & b₂ in carr p ) } is set
f2 is non empty strict unital Group-like associative Subgroup of G
the carrier of f2 is non empty set
H1 is Element of the carrier of G
j is Element of the carrier of G
j is Element of the carrier of G
j * j is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (j,j) is Element of the carrier of G
p is Element of O
(O,G,p) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,p) . j is Element of the carrier of G
(O,G,p) . j is Element of the carrier of G
((O,G,p) . j) * ((O,G,p) . j) is Element of the carrier of G
the multF of G . (((O,G,p) . j),((O,G,p) . j)) is Element of the carrier of G
(O,G,p) . H1 is Element of the carrier of G
p is Element of the carrier of G
p " is Element of the carrier of G
p is Element of the carrier of G
H1 is Element of the carrier of G
p * H1 is Element of the carrier of G
the multF of G . (p,H1) is Element of the carrier of G
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
H1 is Element of the carrier of G
H1 * f2 is Element of bool the carrier of G
carr f2 is Element of bool the carrier of G
H1 * (carr f2) is Element of bool the carrier of G
K350( the carrier of G,H1) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H1) * (carr f2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H1) & b₂ in carr f2 ) } is set
H1 * p is Element of bool the carrier of G
H1 * (carr p) is Element of bool the carrier of G
K350( the carrier of G,H1) * (carr p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H1) & b₂ in carr p ) } is set
(H1 * p) * (carr f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H1 * p & b₂ in carr f1 ) } is set
p * H1 is Element of bool the carrier of G
(carr p) * H1 is Element of bool the carrier of G
(carr p) * K350( the carrier of G,H1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in K350( the carrier of G,H1) ) } is set
(p * H1) * (carr f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in p * H1 & b₂ in carr f1 ) } is set
H1 * f1 is Element of bool the carrier of G
H1 * (carr f1) is Element of bool the carrier of G
K350( the carrier of G,H1) * (carr f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H1) & b₂ in carr f1 ) } is set
(carr p) * (H1 * f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in H1 * f1 ) } is set
f1 * H1 is Element of bool the carrier of G
(carr f1) * H1 is Element of bool the carrier of G
(carr f1) * K350( the carrier of G,H1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr f1 & b₂ in K350( the carrier of G,H1) ) } is set
(carr p) * (f1 * H1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in f1 * H1 ) } is set
f2 * H1 is Element of bool the carrier of G
(carr f2) * H1 is Element of bool the carrier of G
(carr f2) * K350( the carrier of G,H1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr f2 & b₂ in K350( the carrier of G,H1) ) } is set
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
multMagma(# the carrier of p, the multF of p #) is non empty strict multMagma
H1 is non empty strict unital Group-like associative Subgroup of G
O is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom O is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len O) - G is V31() V32() integer ext-real set
((len O) - G) + 1 is V31() V32() integer ext-real set
0 - ((len O) - G) is V31() V32() integer ext-real set
G - (len O) is V31() V32() integer ext-real set
(len O) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 - 1 is V31() V32() integer ext-real set
(s1 - 1) - G is V31() V32() integer ext-real set
((len O) - G) - 1 is V31() V32() integer ext-real set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 - (G + 1) is V31() V32() integer ext-real set
(len O) - (G + 1) is V31() V32() integer ext-real set
G + 0 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is non empty unital Group-like associative multMagma
the carrier of O is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[: the carrier of O, the carrier of G:] is Relation-like non empty set
bool [: the carrier of O, the carrier of G:] is non empty set
s1 is Relation-like NAT -defined the carrier of O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of O
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 |^ s19 is Relation-like NAT -defined the carrier of O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of O
Product (s1 |^ s19) is Element of the carrier of O
s2 |^ s19 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s2 |^ s19) is Element of the carrier of G
s29 is Relation-like the carrier of O -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of O, the carrier of G:]
s29 . (Product (s1 |^ s19)) is Element of the carrier of G
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is non empty unital Group-like associative multMagma
the carrier of f1 is non empty set
i is non empty unital Group-like associative multMagma
the carrier of i is non empty set
[: the carrier of f1, the carrier of i:] is Relation-like non empty set
bool [: the carrier of f1, the carrier of i:] is non empty set
s199 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 |^ f2 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
Product (s199 |^ f2) is Element of the carrier of f1
s199 |^ f2 is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
Product (s199 |^ f2) is Element of the carrier of i
p is Relation-like the carrier of f1 -defined the carrier of i -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of f1, the carrier of i:]
p . (Product (s199 |^ f2)) is Element of the carrier of i
H1 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
j is Element of the carrier of f1
<*j*> is Relation-like NAT -defined the carrier of f1 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of f1
1 -tuples_on the carrier of f1 is FinSequenceSet of the carrier of f1
[1,j] is set
{1,j} is non empty finite set
{{1,j},{1}} is non empty finite V39() set
{[1,j]} is Relation-like Function-like constant non empty trivial finite 1 -element set
H1 ^ <*j*> is Relation-like NAT -defined the carrier of f1 -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
len H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*j*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len H1) + (len <*j*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len H1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
H2 is Element of the carrier of i
<*H2*> is Relation-like NAT -defined the carrier of i -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of i
1 -tuples_on the carrier of i is FinSequenceSet of the carrier of i
[1,H2] is set
{1,H2} is non empty finite set
{{1,H2},{1}} is non empty finite V39() set
{[1,H2]} is Relation-like Function-like constant non empty trivial finite 1 -element set
j ^ <*H2*> is Relation-like NAT -defined the carrier of i -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom f2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H2 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s299 is V31() V32() integer V46() ext-real Element of INT
<*s299*> is Relation-like NAT -defined INT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() increasing V196() V197() V198() Element of 1 -tuples_on INT
1 -tuples_on INT is FinSequenceSet of INT
[1,s299] is set
{1,s299} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{1,s299},{1}} is non empty finite V39() set
{[1,s299]} is Relation-like Function-like constant non empty trivial finite 1 -element set
H2 ^ <*s299*> is Relation-like NAT -defined INT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s299*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len H2) + (len <*s299*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len H2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*H2*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len j) + (len <*H2*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + p is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom H1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom j is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j . j is set
s199 . j is set
s199 . j is set
p . (s199 . j) is set
H1 . j is set
p . (H1 . j) is set
s199 . (p + 1) is set
(j ^ <*H2*>) . ((len j) + 1) is set
s199 . (p + 1) is set
(H1 ^ <*j*>) . ((len H1) + 1) is set
j |^ H2 is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
<*H2*> |^ <*s299*> is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
(j |^ H2) ^ (<*H2*> |^ <*s299*>) is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
Product ((j |^ H2) ^ (<*H2*> |^ <*s299*>)) is Element of the carrier of i
Product (j |^ H2) is Element of the carrier of i
Product (<*H2*> |^ <*s299*>) is Element of the carrier of i
(Product (j |^ H2)) * (Product (<*H2*> |^ <*s299*>)) is Element of the carrier of i
the multF of i is Relation-like [: the carrier of i, the carrier of i:] -defined the carrier of i -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of i, the carrier of i:], the carrier of i:]
[: the carrier of i, the carrier of i:] is Relation-like non empty set
[:[: the carrier of i, the carrier of i:], the carrier of i:] is Relation-like non empty set
bool [:[: the carrier of i, the carrier of i:], the carrier of i:] is non empty set
the multF of i . ((Product (j |^ H2)),(Product (<*H2*> |^ <*s299*>))) is Element of the carrier of i
<*j*> |^ <*s299*> is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
Product (<*j*> |^ <*s299*>) is Element of the carrier of f1
p . (Product (<*j*> |^ <*s299*>)) is Element of the carrier of i
@ s299 is V31() V32() integer V46() ext-real Element of INT
<*(@ s299)*> is Relation-like NAT -defined INT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() increasing V196() V197() V198() Element of 1 -tuples_on INT
[1,(@ s299)] is set
{1,(@ s299)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{1,(@ s299)},{1}} is non empty finite V39() set
{[1,(@ s299)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*j*> |^ <*(@ s299)*> is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
Product (<*j*> |^ <*(@ s299)*>) is Element of the carrier of f1
p . (Product (<*j*> |^ <*(@ s299)*>)) is Element of the carrier of i
j |^ s299 is Element of the carrier of f1
<*(j |^ s299)*> is Relation-like NAT -defined the carrier of f1 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of f1
[1,(j |^ s299)] is set
{1,(j |^ s299)} is non empty finite set
{{1,(j |^ s299)},{1}} is non empty finite V39() set
{[1,(j |^ s299)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Product <*(j |^ s299)*> is Element of the carrier of f1
p . (Product <*(j |^ s299)*>) is Element of the carrier of i
p . (j |^ s299) is Element of the carrier of i
p . j is Element of the carrier of i
(p . j) |^ s299 is Element of the carrier of i
H2 |^ s299 is Element of the carrier of i
<*(H2 |^ s299)*> is Relation-like NAT -defined the carrier of i -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of i
[1,(H2 |^ s299)] is set
{1,(H2 |^ s299)} is non empty finite set
{{1,(H2 |^ s299)},{1}} is non empty finite V39() set
{[1,(H2 |^ s299)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Product <*(H2 |^ s299)*> is Element of the carrier of i
<*H2*> |^ <*(@ s299)*> is Relation-like NAT -defined the carrier of i -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of i
Product (<*H2*> |^ <*(@ s299)*>) is Element of the carrier of i
H1 |^ H2 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
(H1 |^ H2) ^ (<*j*> |^ <*s299*>) is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
Product ((H1 |^ H2) ^ (<*j*> |^ <*s299*>)) is Element of the carrier of f1
Product (H1 |^ H2) is Element of the carrier of f1
(Product (H1 |^ H2)) * (Product (<*j*> |^ <*s299*>)) is Element of the carrier of f1
the multF of f1 is Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
[: the carrier of f1, the carrier of f1:] is Relation-like non empty set
[:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is Relation-like non empty set
bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
the multF of f1 . ((Product (H1 |^ H2)),(Product (<*j*> |^ <*s299*>))) is Element of the carrier of f1
p . (Product (H1 |^ H2)) is Element of the carrier of i
(p . (Product (H1 |^ H2))) * (p . (Product (<*j*> |^ <*s299*>))) is Element of the carrier of i
the multF of i . ((p . (Product (H1 |^ H2))),(p . (Product (<*j*> |^ <*s299*>)))) is Element of the carrier of i
p is non empty unital Group-like associative multMagma
the carrier of p is non empty set
f1 is non empty unital Group-like associative multMagma
the carrier of f1 is non empty set
[: the carrier of p, the carrier of f1:] is Relation-like non empty set
bool [: the carrier of p, the carrier of f1:] is non empty set
i is Relation-like NAT -defined the carrier of p -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of p
dom i is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i |^ s199 is Relation-like NAT -defined the carrier of p -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of p
Product (i |^ s199) is Element of the carrier of p
s199 |^ s199 is Relation-like NAT -defined the carrier of f1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of f1
Product (s199 |^ s199) is Element of the carrier of f1
f2 is Relation-like the carrier of p -defined the carrier of f1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of p, the carrier of f1:]
f2 . (Product (i |^ s199)) is Element of the carrier of f1
len (s199 |^ s199) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*> the carrier of f1 is Relation-like non-empty empty-yielding NAT -defined the carrier of f1 -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of the carrier of f1
[:NAT, the carrier of f1:] is Relation-like non empty non trivial non finite set
1_ f1 is non being_of_order_0 Element of the carrier of f1
len (i |^ s199) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*> the carrier of p is Relation-like non-empty empty-yielding NAT -defined the carrier of p -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of the carrier of p
[:NAT, the carrier of p:] is Relation-like non empty non trivial non finite set
1_ p is non being_of_order_0 Element of the carrier of p
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O, the carrier of G, the of G,s1) is Element of bool the carrier of G
s2 is Element of the carrier of G
the carrier of (O,G,s1) is non empty set
{ b₁ where b₁ is Element of the carrier of G : S₁[b₁] } is set
f1 is Element of the carrier of G
p is Element of bool the carrier of G
i is Element of the carrier of G
s199 is Element of the carrier of G
p is Element of bool the carrier of G
f2 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ f2 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ f2) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
f2 is Element of bool the carrier of G
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ s199) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
p is Element of the carrier of G
j is Element of bool the carrier of G
j is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
H1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
H1 |^ j is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (H1 |^ j) is Element of the carrier of G
len H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng H1 is finite Element of bool the carrier of G
j is Element of bool the carrier of G
H1 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
p is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
p |^ H1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (p |^ H1) is Element of the carrier of G
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng p is finite Element of bool the carrier of G
s199 ^ p is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len (s199 ^ p) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s199) + (len H1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 ^ H1 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len (s199 ^ H1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng (s199 ^ p) is finite Element of bool the carrier of G
(rng s199) \/ (rng p) is finite Element of bool the carrier of G
f1 * i is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (f1,i) is Element of the carrier of G
(s199 |^ s199) ^ (p |^ H1) is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product ((s199 |^ s199) ^ (p |^ H1)) is Element of the carrier of G
(s199 ^ p) |^ (s199 ^ H1) is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product ((s199 ^ p) |^ (s199 ^ H1)) is Element of the carrier of G
i is Element of the carrier of G
s199 is Element of the carrier of G
p is Element of bool the carrier of G
f2 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ f2 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ f2) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
f2 is Element of bool the carrier of G
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ s199) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
f1 is Element of O
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len s199) is finite len s199 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s199 ) } is set
H1 is set
rng p is finite set
j is set
p . j is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p . j is set
s199 . j is set
(O,G,f1) . (s199 . j) is set
the of G . f1 is Relation-like Function-like set
s299 is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
dom s299 is non empty Element of bool the carrier of G
s299 .: f2 is Element of bool the carrier of G
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
(O,G,f1) . i is Element of the carrier of G
H1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
H1 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (H1 |^ s199) is Element of the carrier of G
f1 is Element of the carrier of G
i is Element of the carrier of G
f2 is Element of bool the carrier of G
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ s199) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
s199 is Element of bool the carrier of G
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
i is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
i |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (i |^ s199) is Element of the carrier of G
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng i is finite Element of bool the carrier of G
f2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom f2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len s199) is finite len s199 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s199 ) } is set
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng f2 is finite set
p is set
H1 is set
f2 . H1 is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len i) - j is V31() V32() integer ext-real set
((len i) - j) + 1 is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom i is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i . (((len i) - j) + 1) is set
Seg (len i) is finite len i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len i ) } is set
dom i is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len (i |^ s199) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s199) - H1 is V31() V32() integer ext-real set
((len s199) - H1) + 1 is V31() V32() integer ext-real set
s199 . (((len s199) - H1) + 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 . j is V31() V32() integer ext-real set
rng s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered V210() V211() V212() Element of bool INT
bool INT is non empty non trivial non finite set
j is V31() V32() integer V46() ext-real Element of INT
H2 is V31() V32() integer ext-real set
- H2 is V31() V32() integer ext-real set
s299 is set
H1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom H1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng H1 is finite set
j is set
j is set
H1 . j is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s199) - H2 is V31() V32() integer ext-real set
((len s199) - H2) + 1 is V31() V32() integer ext-real set
s199 . (((len s199) - H2) + 1) is V31() V32() integer ext-real set
s299 is V31() V32() integer ext-real set
- s299 is V31() V32() integer ext-real set
j is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
j is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
j |^ j is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len (j |^ j) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (j |^ j) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom j is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j /. s299 is V31() V32() integer V46() ext-real Element of INT
@ (j /. s299) is V31() V32() integer V46() ext-real Element of INT
(len (i |^ s199)) - s299 is V31() V32() integer ext-real set
((len (i |^ s199)) - s299) + 1 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 /. H1 is V31() V32() integer V46() ext-real Element of INT
@ (s199 /. H1) is V31() V32() integer V46() ext-real Element of INT
(j |^ j) /. s299 is Element of the carrier of G
(j |^ j) . s299 is set
j /. s299 is Element of the carrier of G
j . s299 is set
i . H1 is set
dom j is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s199 . H1 is V31() V32() integer ext-real set
j . s299 is V31() V32() integer ext-real set
H2 is V31() V32() integer ext-real set
- H2 is V31() V32() integer ext-real set
(j /. s299) |^ (- H2) is Element of the carrier of G
i /. H1 is Element of the carrier of G
(i /. H1) |^ H2 is Element of the carrier of G
((i /. H1) |^ H2) " is Element of the carrier of G
((j |^ j) /. s299) " is Element of the carrier of G
(i |^ s199) . (((len (i |^ s199)) - s299) + 1) is set
(Product (i |^ s199)) " is Element of the carrier of G
Product (j |^ j) is Element of the carrier of G
f1 " is Element of the carrier of G
len {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of NAT
<*> the carrier of G is Relation-like non-empty empty-yielding NAT -defined the carrier of G -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of the carrier of G
[:NAT, the carrier of G:] is Relation-like non empty non trivial non finite set
rng (<*> the carrier of G) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered with_non-empty_elements ext-real non positive non negative V191() V192() V193() V194() increasing V196() V197() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
Product (<*> the carrier of G) is Element of the carrier of G
(<*> the carrier of G) |^ (<*> INT) is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
f1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of f1 is non empty set
s199 is set
f2 is Element of the carrier of G
<*f2*> is Relation-like NAT -defined the carrier of G -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of G
1 -tuples_on the carrier of G is FinSequenceSet of the carrier of G
[1,f2] is set
{1,f2} is non empty finite set
{{1,f2},{1}} is non empty finite V39() set
{[1,f2]} is Relation-like Function-like constant non empty trivial finite 1 -element set
rng <*f2*> is non empty trivial finite 1 -element Element of bool the carrier of G
{f2} is non empty trivial finite 1 -element set
s199 is V31() V32() integer ext-real set
@ s199 is V31() V32() integer V46() ext-real Element of INT
<*(@ s199)*> is Relation-like NAT -defined INT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() increasing V196() V197() V198() Element of 1 -tuples_on INT
1 -tuples_on INT is FinSequenceSet of INT
[1,(@ s199)] is set
{1,(@ s199)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{1,(@ s199)},{1}} is non empty finite V39() set
{[1,(@ s199)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*f2*> |^ <*(@ s199)*> is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (<*f2*> |^ <*(@ s199)*>) is Element of the carrier of G
f2 |^ 1 is Element of the carrier of G
<*(f2 |^ 1)*> is Relation-like NAT -defined the carrier of G -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of G
[1,(f2 |^ 1)] is set
{1,(f2 |^ 1)} is non empty finite set
{{1,(f2 |^ 1)},{1}} is non empty finite V39() set
{[1,(f2 |^ 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Product <*(f2 |^ 1)*> is Element of the carrier of G
len <*f2*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len <*(@ s199)*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is Element of the carrier of G
f2 is Element of bool the carrier of G
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s199 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s199 |^ s199) is Element of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng s199 is finite Element of bool the carrier of G
i is Element of bool the carrier of G
p is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng p is finite Element of bool the carrier of G
p |^ f1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (p |^ f1) is Element of the carrier of G
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
s199 is Element of O
f2 is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
the of G . s199 is Relation-like Function-like set
p is set
s199 is Element of bool the carrier of G
f2 .: s199 is Element of bool the carrier of G
dom f2 is non empty Element of bool the carrier of G
H1 is set
f2 . H1 is set
j is Element of the carrier of G
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
(O,G,s199) . j is Element of the carrier of G
f2 . j is Element of the carrier of G
s199 is non empty unital Group-like associative Subgroup of G
the carrier of s199 is non empty set
carr s199 is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
(O,s2) is non empty unital Group-like associative (O) (O) (O,s2) (O,s2)
the carrier of (O,G) is non empty set
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Element of O
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s2 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s2,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
s2 . s1 is Relation-like Function-like Element of Funcs (G,G)
len <*s1*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
<*s1*> . 1 is set
s2 . (<*s1*> . 1) is Relation-like Function-like set
s19 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
s19 . (len <*s1*>) is Relation-like Function-like set
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 . 1 is Relation-like Function-like set
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Element of O
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s2 ^ <*s1*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s19,(s2 ^ <*s1*>)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s19,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s2) * (O,G,s19,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
len (s2 ^ <*s1*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s1*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s2) + (len <*s1*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s2 ^ <*s1*>) . 1 is set
s19 . ((s2 ^ <*s1*>) . 1) is Relation-like Function-like set
p is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
p . (len (s2 ^ <*s1*>)) is Relation-like Function-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p . 1 is Relation-like Function-like set
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
p | (Seg (len s2)) is Relation-like NAT -defined Seg (len s2) -defined NAT -defined Funcs (G,G) -valued Function-like finite FinSubsequence-like Element of bool [:NAT,(Funcs (G,G)):]
[:NAT,(Funcs (G,G)):] is Relation-like non empty non trivial non finite set
bool [:NAT,(Funcs (G,G)):] is non empty non trivial non finite set
f1 is Relation-like NAT -defined Funcs (G,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of Funcs (G,G)
f1 . (len s2) is Relation-like Function-like set
s19 . s1 is Relation-like Function-like Element of Funcs (G,G)
(s2 ^ <*s1*>) . ((len s2) + 1) is set
s19 . ((s2 ^ <*s1*>) . ((len s2) + 1)) is Relation-like Function-like set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p . s199 is Relation-like Function-like set
(s2 ^ <*s1*>) . (s199 + 1) is set
s19 . ((s2 ^ <*s1*>) . (s199 + 1)) is Relation-like Function-like set
p . (s199 + 1) is Relation-like Function-like set
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 * f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f1 . s199 is Relation-like Function-like set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s2 . (s199 + 1) is set
s19 . (s2 . (s199 + 1)) is Relation-like Function-like set
f1 . (s199 + 1) is Relation-like Function-like set
p * H1 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p . (len s2) is Relation-like Function-like set
p . ((len s2) + 1) is Relation-like Function-like set
Seg (len p) is finite len p -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len p ) } is set
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom p) /\ (Seg (len s2)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng f1 is functional finite Element of bool (Funcs (G,G))
bool (Funcs (G,G)) is non empty set
s199 is Relation-like Function-like set
dom s199 is set
rng s199 is set
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 . 1 is Relation-like Function-like set
s2 . 1 is set
s19 . (s2 . 1) is Relation-like Function-like set
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 * f2 is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
<*> O is Relation-like non-empty empty-yielding NAT -defined O -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of O
[:NAT,O:] is Relation-like non empty non trivial non finite set
(<*> O) ^ <*s1*> is Relation-like NAT -defined O -valued Function-like non empty finite {} + 1 -element FinSequence-like FinSubsequence-like FinSequence of O
{} + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s19,((<*> O) ^ <*s1*>)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
(id G) * (O,G,s19,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Element of O
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
<*s1*> ^ s2 is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s19,(<*s1*> ^ s2)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s199 is Element of O
<*s199*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
[1,s199] is set
{1,s199} is non empty finite set
{{1,s199},{1}} is non empty finite V39() set
{[1,s199]} is Relation-like Function-like constant non empty trivial finite 1 -element set
i ^ <*s199*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s199*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len i) + (len <*s199*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
<*s1*> ^ i is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
<*s1*> ^ f1 is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(<*s1*> ^ f1)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(<*s1*> ^ i) ^ <*s199*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,((<*s1*> ^ i) ^ <*s199*>)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,(<*s1*> ^ i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,(<*s1*> ^ i)) * (O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
((O,G,s19,<*s1*>) * (O,G,s19,i)) * (O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,i) * (O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * ((O,G,s19,i) * (O,G,s19,<*s199*>)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*s1*> ^ f1 is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(<*s1*> ^ f1)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*s1*> ^ p is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(<*s1*> ^ p)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
<*> O is Relation-like non-empty empty-yielding NAT -defined O -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of O
[:NAT,O:] is Relation-like non empty non trivial non finite set
<*s1*> ^ (<*> O) is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(<*s1*> ^ (<*> O))) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s1*>) * (id G) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s1 ^ s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s19,(s1 ^ s2)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s1) * (O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s199 is Element of O
<*s199*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
[1,s199] is set
{1,s199} is non empty finite set
{{1,s199},{1}} is non empty finite V39() set
{[1,s199]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s199 ^ <*s199*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
<*s199*> ^ i is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s199*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s199) + (len <*s199*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 ^ i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(f1 ^ i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 ^ (<*s199*> ^ i) is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(s199 ^ (<*s199*> ^ i))) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,(<*s199*> ^ i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s199) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s199) * (O,G,s19,(<*s199*> ^ i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,<*s199*>) * (O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s199) * ((O,G,s19,<*s199*>) * (O,G,s19,i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s199) * (O,G,s19,<*s199*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
((O,G,s19,s199) * (O,G,s19,<*s199*>)) * (O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) * (O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
f1 ^ i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(f1 ^ i)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) * (O,G,s19,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
p ^ f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(p ^ f1)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) * (O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
<*> O is Relation-like non-empty empty-yielding NAT -defined O -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of O
[:NAT,O:] is Relation-like non empty non trivial non finite set
(<*> O) ^ f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,((<*> O) ^ f1)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
(id G) * (O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
O is set
G is set
bool G is non empty set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s2 is Element of bool G
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s19,s1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s19,s1) .: s2 is Element of bool G
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
i is Element of O
<*i*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V39() set
{[1,i]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f1 ^ <*i*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*i*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len f1) + (len <*i*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len f1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s19,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) .: s2 is Element of bool G
(O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,s199) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,f1) * (O,G,s19,s199) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) .: s2 is Element of bool G
(O,G,s19,s199) .: s2 is Element of bool G
(O,G,s19,f1) .: ((O,G,s19,s199) .: s2) is Element of bool G
s19 . i is Relation-like Function-like set
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) .: s2 is Element of bool G
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s19,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,p) .: s2 is Element of bool G
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
O is set
G is non empty set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
bool G is non empty set
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
s2 is Element of bool G
(O,G,s1,s2) is Element of bool G
s19 is Element of G
{ b₁ where b₁ is Element of G : S₁[b₁] } is set
f1 is Element of bool G
i is set
p is Element of bool G
s199 is Element of G
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s199) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like non empty set
bool [:G,G:] is non empty set
f2 is Element of s2
(O,G,s1,s199) . f2 is set
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s199) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like non empty set
bool [:G,G:] is non empty set
f2 is Element of s2
(O,G,s1,s199) . f2 is set
dom (O,G,s1,s199) is non empty Element of bool G
(O,G,s1,s199) .: f1 is Element of bool G
f1 is Element of O
i is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
s1 . f1 is Relation-like Function-like set
i .: p is Element of bool G
s199 is Element of O
<*s199*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,s199] is set
{1,s199} is non empty finite set
{{1,s199},{1}} is non empty finite V39() set
{[1,s199]} is Relation-like Function-like constant non empty trivial finite 1 -element set
f2 is set
i .: p is Element of bool G
dom i is non empty Element of bool G
p is set
i . p is set
rng i is non empty Element of bool G
j is Element of G
j is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,j) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
H2 is Element of s2
(O,G,s1,j) . H2 is set
j is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,j) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
H2 is Element of s2
(O,G,s1,j) . H2 is set
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s199 ^ j is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
dom (O,G,s1,j) is non empty Element of bool G
s299 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s299) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
(O,G,s1,s299) . H2 is set
(O,G,s1,s199) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
(O,G,s1,s199) * (O,G,s1,j) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
((O,G,s1,s199) * (O,G,s1,j)) . H2 is set
(O,G,s1,s199) . ((O,G,s1,j) . H2) is set
H1 is Element of G
<*> O is Relation-like non-empty empty-yielding NAT -defined O -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of O
i is set
s199 is Element of G
len (<*> O) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V46() ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of NAT
(O,G,s1,(<*> O)) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
(O,G,s1,(<*> O)) . i is set
id G is Relation-like G -defined G -valued Function-like one-to-one non empty total quasi_total Element of bool [:G,G:]
(id G) . i is set
s199 is Element of s2
(O,G,s1,(<*> O)) . s199 is set
f1 is Element of G
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,i) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
s199 is Element of s2
(O,G,s1,i) . s199 is set
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,i) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
s199 is Element of s2
(O,G,s1,i) . s199 is set
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s199) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
f2 is Element of s2
(O,G,s1,s199) . f2 is set
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,f1) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
i is Element of s2
(O,G,s1,f1) . i is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
s2 is Element of bool the carrier of G
(O,G,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,(O,G,s2)) is non empty unital Group-like associative (O) (O) (O,(O,G,s2)) (O,(O,G,s2))
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element set
1_ (O,G,s2) is non being_of_order_0 Element of the carrier of (O,G,s2)
the carrier of (O,G,s2) is non empty set
{(1_ (O,G,s2))} is non empty trivial finite 1 -element set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the of G,s2) is Element of bool the carrier of G
s29 is Element of the carrier of G
(O, the carrier of G, the of G,s1) is Element of bool the carrier of G
i is Element of bool the carrier of G
p is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng p is finite Element of bool the carrier of G
p |^ f1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (p |^ f1) is Element of the carrier of G
s199 is set
s199 is Element of the carrier of G
f2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O, the carrier of G, the of G,f2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
p is Element of s1
(O, the carrier of G, the of G,f2) . p is set
F₁() is set
F₂() is non empty unital Group-like associative (F₁()) (F₁())
the carrier of F₂() is non empty set
bool the carrier of F₂() is non empty set
{ b₁ where b₁ is Element of bool the carrier of F₂() : ex b₂ being non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂()) st
( b₁ = the carrier of b₂ & P₁[b₂] ) } is set
meet { b₁ where b₁ is Element of bool the carrier of F₂() : ex b₂ being non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂()) st
( b₁ = the carrier of b₂ & P₁[b₂] ) } is set
G is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
G is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
(F₁(),F₂(),G) is Element of bool the carrier of F₂()
the carrier of G is non empty set
s2 is Element of the carrier of F₂()
s1 is Element of bool the carrier of F₂()
s19 is set
s29 is Element of bool the carrier of F₂()
p is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
the carrier of p is non empty set
p is non empty unital Group-like associative (F₁()) (F₁(),F₂())
the carrier of p is non empty set
s2 " is Element of the carrier of F₂()
s2 is Element of the carrier of F₂()
s19 is Element of the carrier of F₂()
s29 is set
p is Element of bool the carrier of F₂()
f1 is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
the carrier of f1 is non empty set
f1 is non empty unital Group-like associative (F₁()) (F₁(),F₂())
the carrier of f1 is non empty set
s2 * s19 is Element of the carrier of F₂()
the multF of F₂() is Relation-like [: the carrier of F₂(), the carrier of F₂():] -defined the carrier of F₂() -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of F₂(), the carrier of F₂():], the carrier of F₂():]
[: the carrier of F₂(), the carrier of F₂():] is Relation-like non empty set
[:[: the carrier of F₂(), the carrier of F₂():], the carrier of F₂():] is Relation-like non empty set
bool [:[: the carrier of F₂(), the carrier of F₂():], the carrier of F₂():] is non empty set
the multF of F₂() . (s2,s19) is Element of the carrier of F₂()
s19 is Element of the carrier of F₂()
s29 is set
p is Element of bool the carrier of F₂()
f1 is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
the carrier of f1 is non empty set
f1 is non empty unital Group-like associative (F₁()) (F₁(),F₂())
the carrier of f1 is non empty set
s2 is Element of F₁()
(F₁(),F₂(),s2) is Relation-like the carrier of F₂() -defined the carrier of F₂() -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of F₂(), the carrier of F₂():]
bool [: the carrier of F₂(), the carrier of F₂():] is non empty set
(F₁(),F₂(),s2) . s19 is Element of the carrier of F₂()
s2 is set
s19 is Element of bool the carrier of F₂()
s29 is non empty unital Group-like associative (F₁()) (F₁()) (F₁(),F₂())
the carrier of s29 is non empty set
s29 is non empty unital Group-like associative (F₁()) (F₁(),F₂())
the carrier of s29 is non empty set
1_ F₂() is non being_of_order_0 Element of the carrier of F₂()
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s1) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & s1 c= (O,G,b₂) ) } is set
meet { b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & s1 c= (O,G,b₂) ) } is set
s19 is set
s29 is Element of bool the carrier of G
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
(O,G,p) is Element of bool the carrier of G
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
the carrier of (O,G) is non empty set
(O,G,(O,G)) is Element of bool the carrier of G
{ b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & S₁[b₂] ) } is set
meet { b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & S₁[b₂] ) } is set
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
(O,G,s29) is Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
s2 is Element of bool the carrier of G
gr s2 is non empty strict unital Group-like associative Subgroup of G
the carrier of (gr s2) is non empty set
(O,G,s2) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative Subgroup of G
the carrier of s29 is non empty set
the carrier of (O,G,s1) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1,s2) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
(O,G,s2,s1) is Element of bool the carrier of G
(O,G,s2) * (O,G,s1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s1) ) } is set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative normal Subgroup of G
carr s29 is Element of bool the carrier of G
the carrier of s29 is non empty set
s19 is non empty strict unital Group-like associative normal Subgroup of G
carr s19 is Element of bool the carrier of G
the carrier of s19 is non empty set
(carr s29) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s29 & b₂ in carr s19 ) } is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is Element of bool the carrier of G
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
(O,G,(O,G,s1,s2)) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative Subgroup of G
s29 is non empty unital Group-like associative Subgroup of G
s19 "\/" s29 is non empty strict unital Group-like associative Subgroup of G
carr s19 is Element of bool the carrier of G
the carrier of s19 is non empty set
carr s29 is Element of bool the carrier of G
the carrier of s29 is non empty set
(carr s19) \/ (carr s29) is Element of bool the carrier of G
gr ((carr s19) \/ (carr s29)) is non empty strict unital Group-like associative Subgroup of G
the carrier of (s19 "\/" s29) is non empty set
p is Element of bool the carrier of G
s19 * s29 is Element of bool the carrier of G
(carr s19) * (carr s29) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s19 & b₂ in carr s29 ) } is set
gr (s19 * s29) is non empty strict unital Group-like associative Subgroup of G
the carrier of (gr (s19 * s29)) is non empty set
(O,G,p) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(s19 * s29)) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
(O,G,s2,s1) is Element of bool the carrier of G
(O,G,s2) * (O,G,s1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s1) ) } is set
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s1,s2) is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s19 is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & (O,G,s1) \/ (O,G,s2) c= (O,G,b₂) ) } is set
p is Element of the carrier of G
i is Element of the carrier of G
s199 is Element of the carrier of G
i * s199 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (i,s199) is Element of the carrier of G
s199 is set
f2 is Element of bool the carrier of G
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
(O,G,p) is Element of bool the carrier of G
p is non empty unital Group-like associative (O) (O) (O,G)
the carrier of p is non empty set
(O,G,p) is Element of bool the carrier of G
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
meet { b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & (O,G,s1) \/ (O,G,s2) c= (O,G,b₂) ) } is set
the carrier of (O,G,((O,G,s1) \/ (O,G,s2))) is non empty set
s29 is set
1_ G is non being_of_order_0 Element of the carrier of G
p is Element of the carrier of G
p * (1_ G) is Element of the carrier of G
the multF of G . (p,(1_ G)) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
p is Element of the carrier of G
(1_ G) * p is Element of the carrier of G
the multF of G . ((1_ G),p) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s1,s2) is non empty set
(O,G,s1,s2) is Element of bool the carrier of G
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
(O,G,s2,s1) is Element of bool the carrier of G
(O,G,s2) * (O,G,s1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s1) ) } is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s1) & b₂ in (O,G,s2) ) } is set
the carrier of (O,G,s1,s2) is non empty set
(O,G,s1,s2) is Element of bool the carrier of G
s19 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of s19 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G),s1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,(O,G)) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of (O,G) is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,(O,G)) \/ (O,G,s1) is Element of bool the carrier of G
(O,G,((O,G,(O,G)) \/ (O,G,s1))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element set
{(1_ G)} \/ (O,G,s1) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
(O,G,(O,G),s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of (O,G) is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of s1 is non empty set
(O,G,(O,G)) \/ (O,G,s1) is Element of bool the carrier of G
(O,G,((O,G,(O,G)) \/ (O,G,s1))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G) \/ (O,G,s1) is non empty set
[#] the carrier of G is non empty non proper Element of bool the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,((O,G,s1) \/ (O,G,s2))) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2) \/ (O,G,s1) is Element of bool the carrier of G
(O,G,((O,G,s2) \/ (O,G,s1))) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s2)) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s1) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19) is Element of bool the carrier of G
the carrier of s19 is non empty set
(O,G,(O,G,s19)) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s19) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s1 is non empty set
(O,G,s19) is Element of bool the carrier of G
the carrier of s19 is non empty set
(O,G,s1) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,s1) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s2) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,s2) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,s1)
s19 is non empty unital Group-like associative (O) (O,s1)
(O,s1,s2,s19) is non empty unital Group-like associative (O) (O) (O,s1)
p is non empty unital Group-like associative (O) (O,G)
f1 is non empty unital Group-like associative (O) (O,G)
(O,G,p,f1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,s1,s2,s19) is non empty set
the carrier of s1 is non empty set
(O,s1,s2) is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
the carrier of s2 is non empty set
(O,s1,s19) is Element of bool the carrier of s1
the carrier of s19 is non empty set
(O,s1,s2) /\ (O,s1,s19) is Element of bool the carrier of s1
s29 is non empty unital Group-like associative (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
p is non empty strict unital Group-like associative Subgroup of s2
s19 is non empty unital Group-like associative (O) (O,s2)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty unital Group-like associative normal Subgroup of G
f1 is non empty unital Group-like associative Subgroup of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s1,s2) is non empty set
the carrier of s1 is non empty set
the carrier of s2 is non empty set
the carrier of s1 /\ the carrier of s2 is set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
(O,G,s2,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
p is non empty strict unital Group-like associative Subgroup of s2
s19 is non empty unital Group-like associative (O) (O,s2)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
the carrier of s2 is non empty set
f1 is Element of the carrier of s2
f1 * p is Element of bool the carrier of s2
bool the carrier of s2 is non empty set
carr p is Element of bool the carrier of s2
the carrier of p is non empty set
f1 * (carr p) is Element of bool the carrier of s2
K350( the carrier of s2,f1) is non empty trivial finite 1 -element Element of bool the carrier of s2
K350( the carrier of s2,f1) * (carr p) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in K350( the carrier of s2,f1) & b₂ in carr p ) } is set
p * f1 is Element of bool the carrier of s2
(carr p) * f1 is Element of bool the carrier of s2
(carr p) * K350( the carrier of s2,f1) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in carr p & b₂ in K350( the carrier of s2,f1) ) } is set
i is set
s199 is Element of the carrier of s2
f1 * s199 is Element of the carrier of s2
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the multF of s2 . (f1,s199) is Element of the carrier of s2
the carrier of G is non empty set
f2 is Element of the carrier of G
f2 " is Element of the carrier of G
f1 " is Element of the carrier of s2
(O,G,s2) is Element of bool the carrier of G
bool the carrier of G is non empty set
(O,G,s1) is Element of bool the carrier of G
(O,G,s2) /\ (O,G,s1) is Element of bool the carrier of G
s29 is non empty unital Group-like associative normal Subgroup of G
carr s29 is Element of bool the carrier of G
the carrier of s29 is non empty set
s199 is Element of the carrier of G
f2 * s199 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (f2,s199) is Element of the carrier of G
f2 * s29 is Element of bool the carrier of G
f2 * (carr s29) is Element of bool the carrier of G
K350( the carrier of G,f2) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,f2) * (carr s29) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,f2) & b₂ in carr s29 ) } is set
s29 * f2 is Element of bool the carrier of G
(carr s29) * f2 is Element of bool the carrier of G
(carr s29) * K350( the carrier of G,f2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s29 & b₂ in K350( the carrier of G,f2) ) } is set
H1 is Element of the carrier of G
H1 * f2 is Element of the carrier of G
the multF of G . (H1,f2) is Element of the carrier of G
j is Element of the carrier of G
j * (f2 ") is Element of the carrier of G
the multF of G . (j,(f2 ")) is Element of the carrier of G
p is Element of the carrier of s2
p * (f1 ") is Element of the carrier of s2
the multF of s2 . (p,(f1 ")) is Element of the carrier of s2
j is Element of the carrier of s2
j * f1 is Element of the carrier of s2
the multF of s2 . (j,f1) is Element of the carrier of s2
O is set
G is non empty unital Group-like associative (O) (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element set
s2 is set
{s2} is non empty trivial finite 1 -element set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of (O,G,s1) is non empty set
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is non empty strict multMagma
1_ (O,G,s1) is non being_of_order_0 Element of the carrier of (O,G,s1)
s2 is non empty unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
1_ (G ./. s2) is non being_of_order_0 Element of the carrier of (G ./. s2)
the carrier of (G ./. s2) is non empty set
s29 is Element of the carrier of (O,G,s1)
s19 is Element of the carrier of (O,G,s1)
s29 * s19 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (s29,s19) is Element of the carrier of (O,G,s1)
p is Element of the carrier of (G ./. s2)
p * (1_ (G ./. s2)) is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) is Relation-like [: the carrier of (G ./. s2), the carrier of (G ./. s2):] -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):]
[: the carrier of (G ./. s2), the carrier of (G ./. s2):] is Relation-like non empty set
[:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is non empty set
the multF of (G ./. s2) . (p,(1_ (G ./. s2))) is Element of the carrier of (G ./. s2)
s19 * s29 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (s19,s29) is Element of the carrier of (O,G,s1)
(1_ (G ./. s2)) * p is Element of the carrier of (G ./. s2)
the multF of (G ./. s2) . ((1_ (G ./. s2)),p) is Element of the carrier of (G ./. s2)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
1_ (O,G,s1) is non being_of_order_0 Element of the carrier of (O,G,s1)
the carrier of (O,G,s1) is non empty set
(O,G,s1) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
1_ (G ./. s2) is non being_of_order_0 Element of the carrier of (G ./. s2)
the carrier of (G ./. s2) is non empty set
carr s2 is Element of bool the carrier of G
the carrier of s2 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
p is non empty unital Group-like associative (O) (O,s2) (O,s2)
(O,s2,p) is non empty unital Group-like associative (O) (O)
(O,s2,p) is set
(O,s2,p) is Relation-like [:(O,s2,p),(O,s2,p):] -defined (O,s2,p) -valued Function-like quasi_total Element of bool [:[:(O,s2,p),(O,s2,p):],(O,s2,p):]
[:(O,s2,p),(O,s2,p):] is Relation-like set
[:[:(O,s2,p),(O,s2,p):],(O,s2,p):] is Relation-like set
bool [:[:(O,s2,p),(O,s2,p):],(O,s2,p):] is non empty set
(O,s2,p) is Relation-like O -defined Funcs ((O,s2,p),(O,s2,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s2,p),(O,s2,p))):]
Funcs ((O,s2,p),(O,s2,p)) is functional non empty set
[:O,(Funcs ((O,s2,p),(O,s2,p))):] is Relation-like set
bool [:O,(Funcs ((O,s2,p),(O,s2,p))):] is non empty set
(O,(O,s2,p),(O,s2,p),(O,s2,p)) is (O) (O)
s29 is non empty unital Group-like associative normal Subgroup of G
s19 is non empty unital Group-like associative normal Subgroup of G
(s29,s19) `*` is non empty strict unital Group-like associative normal Subgroup of s29
the carrier of p is non empty set
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
multMagma(# the carrier of p, the multF of p #) is non empty strict multMagma
the multF of s2 || the carrier of s1 is set
the multF of s2 | [: the carrier of s1, the carrier of s1:] is Relation-like [: the carrier of s1, the carrier of s1:] -defined [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like set
i is non empty unital Group-like associative normal Subgroup of s2
G ./. s19 is non empty strict unital Group-like associative multMagma
Left_Cosets s19 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s19 is Relation-like [:(Left_Cosets s19),(Left_Cosets s19):] -defined Left_Cosets s19 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):]
[:(Left_Cosets s19),(Left_Cosets s19):] is Relation-like non empty set
[:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):] is Relation-like non empty set
bool [:[:(Left_Cosets s19),(Left_Cosets s19):],(Left_Cosets s19):] is non empty set
multMagma(# (Left_Cosets s19),(CosOp s19) #) is non empty strict multMagma
s29 ./. ((s29,s19) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((s29,s19) `*`) is non empty Element of bool (bool the carrier of s29)
the carrier of s29 is non empty set
bool the carrier of s29 is non empty set
bool (bool the carrier of s29) is non empty set
CosOp ((s29,s19) `*`) is Relation-like [:(Left_Cosets ((s29,s19) `*`)),(Left_Cosets ((s29,s19) `*`)):] -defined Left_Cosets ((s29,s19) `*`) -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets ((s29,s19) `*`)),(Left_Cosets ((s29,s19) `*`)):],(Left_Cosets ((s29,s19) `*`)):]
[:(Left_Cosets ((s29,s19) `*`)),(Left_Cosets ((s29,s19) `*`)):] is Relation-like non empty set
[:[:(Left_Cosets ((s29,s19) `*`)),(Left_Cosets ((s29,s19) `*`)):],(Left_Cosets ((s29,s19) `*`)):] is Relation-like non empty set
bool [:[:(Left_Cosets ((s29,s19) `*`)),(Left_Cosets ((s29,s19) `*`)):],(Left_Cosets ((s29,s19) `*`)):] is non empty set
multMagma(# (Left_Cosets ((s29,s19) `*`)),(CosOp ((s29,s19) `*`)) #) is non empty strict multMagma
s199 is set
Left_Cosets i is non empty Element of bool (bool the carrier of s2)
bool the carrier of s2 is non empty set
bool (bool the carrier of s2) is non empty set
f2 is Element of the carrier of s2
f2 * i is Element of bool the carrier of s2
carr i is Element of bool the carrier of s2
the carrier of i is non empty set
f2 * (carr i) is Element of bool the carrier of s2
K350( the carrier of s2,f2) is non empty trivial finite 1 -element Element of bool the carrier of s2
K350( the carrier of s2,f2) * (carr i) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in K350( the carrier of s2,f2) & b₂ in carr i ) } is set
i * f2 is Element of bool the carrier of s2
(carr i) * f2 is Element of bool the carrier of s2
(carr i) * K350( the carrier of s2,f2) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in carr i & b₂ in K350( the carrier of s2,f2) ) } is set
{f2} is non empty trivial finite 1 -element set
p is Element of the carrier of s29
{p} is non empty trivial finite 1 -element set
j is set
H1 is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in H1 & b₂ in carr i ) } is set
H2 is Element of the carrier of s2
s299 is Element of the carrier of s2
H2 * s299 is Element of the carrier of s2
the multF of s2 . (H2,s299) is Element of the carrier of s2
H2 is Element of the carrier of s29
H1 is Element of the carrier of s29
H2 * H1 is Element of the carrier of s29
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
the multF of s29 . (H2,H1) is Element of the carrier of s29
j is Element of bool the carrier of s29
f1 is non empty unital Group-like associative normal Subgroup of s29
carr f1 is Element of bool the carrier of s29
the carrier of f1 is non empty set
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in j & b₂ in carr f1 ) } is set
H2 is Element of the carrier of s29
s299 is Element of the carrier of s29
H2 * s299 is Element of the carrier of s29
the multF of s29 . (H2,s299) is Element of the carrier of s29
H2 is Element of the carrier of s2
H1 is Element of the carrier of s2
H2 * H1 is Element of the carrier of s2
the multF of s2 . (H2,H1) is Element of the carrier of s2
p * f1 is Element of bool the carrier of s29
p * (carr f1) is Element of bool the carrier of s29
K350( the carrier of s29,p) is non empty trivial finite 1 -element Element of bool the carrier of s29
K350( the carrier of s29,p) * (carr f1) is Element of bool the carrier of s29
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in K350( the carrier of s29,p) & b₂ in carr f1 ) } is set
Left_Cosets f1 is non empty Element of bool (bool the carrier of s29)
f2 is Element of the carrier of s29
f2 * f1 is Element of bool the carrier of s29
f2 * (carr f1) is Element of bool the carrier of s29
K350( the carrier of s29,f2) is non empty trivial finite 1 -element Element of bool the carrier of s29
K350( the carrier of s29,f2) * (carr f1) is Element of bool the carrier of s29
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in K350( the carrier of s29,f2) & b₂ in carr f1 ) } is set
f1 * f2 is Element of bool the carrier of s29
(carr f1) * f2 is Element of bool the carrier of s29
(carr f1) * K350( the carrier of s29,f2) is Element of bool the carrier of s29
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in carr f1 & b₂ in K350( the carrier of s29,f2) ) } is set
{f2} is non empty trivial finite 1 -element set
p is Element of the carrier of s2
{p} is non empty trivial finite 1 -element set
j is set
j is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in j & b₂ in carr i ) } is set
H2 is Element of the carrier of s2
s299 is Element of the carrier of s2
H2 * s299 is Element of the carrier of s2
the multF of s2 . (H2,s299) is Element of the carrier of s2
H2 is Element of the carrier of s29
H1 is Element of the carrier of s29
H2 * H1 is Element of the carrier of s29
the multF of s29 . (H2,H1) is Element of the carrier of s29
H1 is Element of bool the carrier of s29
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in H1 & b₂ in carr f1 ) } is set
H2 is Element of the carrier of s29
s299 is Element of the carrier of s29
H2 * s299 is Element of the carrier of s29
the multF of s29 . (H2,s299) is Element of the carrier of s29
H2 is Element of the carrier of s2
H1 is Element of the carrier of s2
H2 * H1 is Element of the carrier of s2
the multF of s2 . (H2,H1) is Element of the carrier of s2
p * i is Element of bool the carrier of s2
p * (carr i) is Element of bool the carrier of s2
K350( the carrier of s2,p) is non empty trivial finite 1 -element Element of bool the carrier of s2
K350( the carrier of s2,p) * (carr i) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in K350( the carrier of s2,p) & b₂ in carr i ) } is set
s199 is non empty unital Group-like associative normal Subgroup of G ./. s19
the carrier of s199 is non empty set
the carrier of (O,s2,p) is non empty set
s199 is non empty strict unital Group-like associative Subgroup of (O,G,s1)
the multF of (O,s2,p) is Relation-like [: the carrier of (O,s2,p), the carrier of (O,s2,p):] -defined the carrier of (O,s2,p) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,s2,p), the carrier of (O,s2,p):], the carrier of (O,s2,p):]
[: the carrier of (O,s2,p), the carrier of (O,s2,p):] is Relation-like non empty set
[:[: the carrier of (O,s2,p), the carrier of (O,s2,p):], the carrier of (O,s2,p):] is Relation-like non empty set
bool [:[: the carrier of (O,s2,p), the carrier of (O,s2,p):], the carrier of (O,s2,p):] is non empty set
multMagma(# the carrier of (O,s2,p), the multF of (O,s2,p) #) is non empty strict multMagma
the carrier of (O,G,s1) is non empty set
the carrier of (G ./. s19) is non empty set
f2 is Element of the carrier of (O,G,s1)
H1 is set
carr s199 is Element of bool the carrier of (O,G,s1)
bool the carrier of (O,G,s1) is non empty set
the carrier of s199 is non empty set
f2 * (carr s199) is Element of bool the carrier of (O,G,s1)
K350( the carrier of (O,G,s1),f2) is non empty trivial finite 1 -element Element of bool the carrier of (O,G,s1)
K350( the carrier of (O,G,s1),f2) * (carr s199) is Element of bool the carrier of (O,G,s1)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (O,G,s1) : ( b₁ in K350( the carrier of (O,G,s1),f2) & b₂ in carr s199 ) } is set
j is Element of the carrier of (O,G,s1)
f2 * j is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
the multF of (O,G,s1) . (f2,j) is Element of the carrier of (O,G,s1)
p is Element of the carrier of (G ./. s19)
j is Element of the carrier of (G ./. s19)
p * j is Element of the carrier of (G ./. s19)
the multF of (G ./. s19) is Relation-like [: the carrier of (G ./. s19), the carrier of (G ./. s19):] -defined the carrier of (G ./. s19) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):]
[: the carrier of (G ./. s19), the carrier of (G ./. s19):] is Relation-like non empty set
[:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s19), the carrier of (G ./. s19):], the carrier of (G ./. s19):] is non empty set
the multF of (G ./. s19) . (p,j) is Element of the carrier of (G ./. s19)
p * s199 is Element of bool the carrier of (G ./. s19)
bool the carrier of (G ./. s19) is non empty set
carr s199 is Element of bool the carrier of (G ./. s19)
p * (carr s199) is Element of bool the carrier of (G ./. s19)
K350( the carrier of (G ./. s19),p) is non empty trivial finite 1 -element Element of bool the carrier of (G ./. s19)
K350( the carrier of (G ./. s19),p) * (carr s199) is Element of bool the carrier of (G ./. s19)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (G ./. s19) : ( b₁ in K350( the carrier of (G ./. s19),p) & b₂ in carr s199 ) } is set
s199 * p is Element of bool the carrier of (G ./. s19)
(carr s199) * p is Element of bool the carrier of (G ./. s19)
(carr s199) * K350( the carrier of (G ./. s19),p) is Element of bool the carrier of (G ./. s19)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (G ./. s19) : ( b₁ in carr s199 & b₂ in K350( the carrier of (G ./. s19),p) ) } is set
H2 is Element of the carrier of (G ./. s19)
s299 is Element of the carrier of (G ./. s19)
s299 * p is Element of the carrier of (G ./. s19)
the multF of (G ./. s19) . (s299,p) is Element of the carrier of (G ./. s19)
H1 is Element of the carrier of (O,G,s1)
H1 * f2 is Element of the carrier of (O,G,s1)
the multF of (O,G,s1) . (H1,f2) is Element of the carrier of (O,G,s1)
(carr s199) * f2 is Element of bool the carrier of (O,G,s1)
(carr s199) * K350( the carrier of (O,G,s1),f2) is Element of bool the carrier of (O,G,s1)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (O,G,s1) : ( b₁ in carr s199 & b₂ in K350( the carrier of (O,G,s1),f2) ) } is set
f2 * s199 is Element of bool the carrier of (O,G,s1)
s199 * f2 is Element of bool the carrier of (O,G,s1)
s199 is Element of O
f2 is set
p is set
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
id the carrier of (O,s2,p) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,s2,p) -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of (O,s2,p), the carrier of (O,s2,p):]
bool [: the carrier of (O,s2,p), the carrier of (O,s2,p):] is non empty set
id the carrier of (O,G,s1) is Relation-like the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):] is non empty set
(id the carrier of (O,G,s1)) | the carrier of (O,s2,p) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
f2 is set
p is set
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
(O,(O,s2,p),s199) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,s2,p) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s2,p), the carrier of (O,s2,p):]
(O,(O,G,s1),s199) is Relation-like the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
(O,(O,G,s1),s199) | the carrier of (O,s2,p) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
s199 is Element of O
the of (O,G,s1) is Relation-like O -defined Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):]
Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) is functional non empty set
[:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is non empty set
the of (O,G,s1) . s199 is Relation-like Function-like set
Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) is functional non empty FUNCTION_DOMAIN of the carrier of (O,G,s1), the carrier of (O,G,s1)
f2 is Relation-like Function-like set
dom f2 is set
rng f2 is set
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,G,s1) : ex b₃, b₄ being Element of the carrier of G st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,G,s199) . b₃ ) } is set
the of (O,s2,p) is Relation-like O -defined Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p))):]
Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p)) is functional non empty set
[:O,(Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p))):] is non empty set
the of (O,s2,p) . s199 is Relation-like Function-like set
Funcs ( the carrier of (O,s2,p), the carrier of (O,s2,p)) is functional non empty FUNCTION_DOMAIN of the carrier of (O,s2,p), the carrier of (O,s2,p)
p is Relation-like Function-like set
dom p is set
rng p is set
(dom f2) /\ the carrier of (O,s2,p) is set
(O,s2,s199) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
bool [: the carrier of s2, the carrier of s2:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,s2,p) : ex b₃, b₄ being Element of the carrier of s2 st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,s2,s199) . b₃ ) } is set
H1 is set
p . H1 is set
[H1,(p . H1)] is set
{H1,(p . H1)} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,(p . H1)},{H1}} is non empty finite V39() set
j is Element of (O,s2,p)
j is Element of (O,s2,p)
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
f2 . H1 is set
[H1,(f2 . H1)] is set
{H1,(f2 . H1)} is non empty finite set
{{H1,(f2 . H1)},{H1}} is non empty finite V39() set
H2 is Element of (O,G,s1)
s299 is Element of (O,G,s1)
[H2,s299] is set
{H2,s299} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,s299},{H2}} is non empty finite V39() set
H is Element of the carrier of s2
K is Element of the carrier of s2
(O,s2,s199) . H is Element of the carrier of s2
H is Element of the carrier of s2
K is Element of the carrier of s2
(O,s2,s199) . H is Element of the carrier of s2
H2 is Element of Left_Cosets i
K * i is Element of bool the carrier of s2
K * (carr i) is Element of bool the carrier of s2
K350( the carrier of s2,K) is non empty trivial finite 1 -element Element of bool the carrier of s2
K350( the carrier of s2,K) * (carr i) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in K350( the carrier of s2,K) & b₂ in carr i ) } is set
i9 is Element of the carrier of G
j9 is Element of the carrier of G
(O,G,s199) . i9 is Element of the carrier of G
i9 is Element of the carrier of G
j9 is Element of the carrier of G
(O,G,s199) . i9 is Element of the carrier of G
s299 is Element of Left_Cosets s19
i9 * s19 is Element of bool the carrier of G
carr s19 is Element of bool the carrier of G
the carrier of s19 is non empty set
i9 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,i9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,i9) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,i9) & b₂ in carr s19 ) } is set
JK is set
JK is Element of the carrier of s2
K * JK is Element of the carrier of s2
the multF of s2 . (K,JK) is Element of the carrier of s2
K9 is Element of the carrier of G
JH is Element of the carrier of G
K9 * JH is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (K9,JH) is Element of the carrier of G
K9 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,K9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,K9) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,K9) & b₂ in carr s19 ) } is set
JK is Element of the carrier of G
K9 * JK is Element of the carrier of G
the multF of G . (K9,JK) is Element of the carrier of G
JH is Element of the carrier of s2
K * JH is Element of the carrier of s2
the multF of s2 . (K,JH) is Element of the carrier of s2
K9 * s19 is Element of bool the carrier of G
JK is set
H * (carr i) is Element of bool the carrier of s2
K350( the carrier of s2,H) is non empty trivial finite 1 -element Element of bool the carrier of s2
K350( the carrier of s2,H) * (carr i) is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in K350( the carrier of s2,H) & b₂ in carr i ) } is set
JK is Element of the carrier of s2
H * JK is Element of the carrier of s2
the multF of s2 . (H,JK) is Element of the carrier of s2
H9 is Element of the carrier of G
JH is Element of the carrier of G
H9 * JH is Element of the carrier of G
the multF of G . (H9,JH) is Element of the carrier of G
H9 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,H9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H9) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H9) & b₂ in carr s19 ) } is set
JK is Element of the carrier of G
H9 * JK is Element of the carrier of G
the multF of G . (H9,JK) is Element of the carrier of G
JH is Element of the carrier of s2
H * JH is Element of the carrier of s2
the multF of s2 . (H,JH) is Element of the carrier of s2
H * i is Element of bool the carrier of s2
H9 * s19 is Element of bool the carrier of G
H1 is Element of Left_Cosets i
i9 " is Element of the carrier of G
(i9 ") * H9 is Element of the carrier of G
the multF of G . ((i9 "),H9) is Element of the carrier of G
(O,G,s199) . ((i9 ") * H9) is Element of the carrier of G
j9 " is Element of the carrier of G
(O,G,s199) . (i9 ") is Element of the carrier of G
(O,G,s199) | the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
((O,G,s199) | the carrier of s2) . H is set
(O,G,s199) . H9 is Element of the carrier of G
(j9 ") * K9 is Element of the carrier of G
the multF of G . ((j9 "),K9) is Element of the carrier of G
i is Element of Left_Cosets s19
j9 * s19 is Element of bool the carrier of G
j9 * (carr s19) is Element of bool the carrier of G
K350( the carrier of G,j9) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,j9) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,j9) & b₂ in carr s19 ) } is set
(O,(O,s2,p),s199) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,s2,p) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s2,p), the carrier of (O,s2,p):]
bool [: the carrier of (O,s2,p), the carrier of (O,s2,p):] is non empty set
f2 | the carrier of (O,s2,p) is Relation-like Function-like set
(O,(O,G,s1),s199) is Relation-like the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):] is non empty set
(O,(O,G,s1),s199) | the carrier of (O,s2,p) is Relation-like the carrier of (O,s2,p) -defined the carrier of (O,G,s1) -defined the carrier of (O,G,s1) -valued Function-like Element of bool [: the carrier of (O,G,s1), the carrier of (O,G,s1):]
s199 is Element of O
[:(Left_Cosets i),(Left_Cosets i):] is Relation-like non empty set
[:[:(Left_Cosets i),(Left_Cosets i):],(Left_Cosets i):] is Relation-like non empty set
bool [:[:(Left_Cosets i),(Left_Cosets i):],(Left_Cosets i):] is non empty set
CosOp f1 is Relation-like [:(Left_Cosets f1),(Left_Cosets f1):] -defined Left_Cosets f1 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets f1),(Left_Cosets f1):],(Left_Cosets f1):]
[:(Left_Cosets f1),(Left_Cosets f1):] is Relation-like non empty set
[:[:(Left_Cosets f1),(Left_Cosets f1):],(Left_Cosets f1):] is Relation-like non empty set
bool [:[:(Left_Cosets f1),(Left_Cosets f1):],(Left_Cosets f1):] is non empty set
f2 is Element of Left_Cosets i
p is Element of Left_Cosets i
j is Element of bool the carrier of s2
H2 is Element of bool the carrier of s2
H2 is set
j * H2 is Element of bool the carrier of s2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s2 : ( b₁ in j & b₂ in H2 ) } is set
s299 is Element of the carrier of s2
i is Element of the carrier of s2
s299 * i is Element of the carrier of s2
the multF of s2 . (s299,i) is Element of the carrier of s2
j is Element of the carrier of s29
H is Element of the carrier of s29
j * H is Element of the carrier of s29
the multF of s29 . (j,H) is Element of the carrier of s29
s299 is Element of bool the carrier of s29
H1 is Element of bool the carrier of s29
s299 * H1 is Element of bool the carrier of s29
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s29 : ( b₁ in s299 & b₂ in H1 ) } is set
s299 is Element of the carrier of s29
i is Element of the carrier of s29
s299 * i is Element of the carrier of s29
the multF of s29 . (s299,i) is Element of the carrier of s29
j is Element of the carrier of s2
H is Element of the carrier of s2
j * H is Element of the carrier of s2
the multF of s2 . (j,H) is Element of the carrier of s2
s199 is Relation-like [:(Left_Cosets i),(Left_Cosets i):] -defined Left_Cosets i -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets i),(Left_Cosets i):],(Left_Cosets i):]
s199 . (f2,p) is Element of Left_Cosets i
H1 is Element of Left_Cosets f1
j is Element of Left_Cosets f1
s199 . (H1,j) is set
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
CosOp i is Relation-like [:(Left_Cosets i),(Left_Cosets i):] -defined Left_Cosets i -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets i),(Left_Cosets i):],(Left_Cosets i):]
the multF of (G ./. s19) || the carrier of s199 is set
the multF of (G ./. s19) | [: the carrier of s199, the carrier of s199:] is Relation-like [: the carrier of s199, the carrier of s199:] -defined [: the carrier of (G ./. s19), the carrier of (G ./. s19):] -defined the carrier of (G ./. s19) -valued Function-like set
the multF of (O,G,s1) || the carrier of (O,s2,p) is set
the multF of (O,G,s1) | [: the carrier of (O,s2,p), the carrier of (O,s2,p):] is Relation-like [: the carrier of (O,s2,p), the carrier of (O,s2,p):] -defined [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
1_ G is non being_of_order_0 Element of the carrier of G
s2 . (1_ G) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s2 is Element of the carrier of G
s2 " is Element of the carrier of G
s19 . (s2 ") is Element of the carrier of s1
s19 . s2 is Element of the carrier of s1
(s19 . s2) " is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
1_ s1 is non being_of_order_0 Element of the carrier of s1
s2 is Element of the carrier of G
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,s19) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s19 . s2 is Element of the carrier of s1
the carrier of (O,G,s1,s19) is non empty set
{ b₁ where b₁ is Element of the carrier of G : s19 . b₁ = 1_ s1 } is set
s29 is Element of the carrier of G
s19 . s29 is Element of the carrier of s1
{ b₁ where b₁ is Element of the carrier of G : s19 . b₁ = 1_ s1 } is set
the carrier of (O,G,s1,s19) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
(O,G,s1) is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
the carrier of G is non empty set
the carrier of (O,G,s1) is non empty set
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
(O,G,(O,G,s1),(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
the carrier of (G ./. s2) is non empty set
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
1_ (O,G,s1) is non being_of_order_0 Element of the carrier of (O,G,s1)
1_ (G ./. s2) is non being_of_order_0 Element of the carrier of (G ./. s2)
the carrier of (O,G,(O,G,s1),(O,G,s1)) is non empty set
{ b₁ where b₁ is Element of the carrier of G : (O,G,s1) . b₁ = 1_ (O,G,s1) } is set
{ b₁ where b₁ is Element of the carrier of G : (nat_hom s2) . b₁ = 1_ (G ./. s2) } is set
Ker (nat_hom s2) is non empty strict unital Group-like associative normal Subgroup of G
the carrier of (Ker (nat_hom s2)) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
rng s2 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of (O,G,s1,s2) is non empty set
s2 .: the carrier of G is Element of bool the carrier of s1
dom s2 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
s2 .: (dom s2) is Element of bool the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
(O,G,s1) is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
the carrier of G is non empty set
the carrier of (O,G,s1) is non empty set
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
(O,G,(O,G,s1),(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,(O,G,s1))
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is non empty strict multMagma
the carrier of (O,G,(O,G,s1),(O,G,s1)) is non empty set
(O,G,s1) .: the carrier of G is Element of bool the carrier of (O,G,s1)
bool the carrier of (O,G,s1) is non empty set
the carrier of (G ./. s2) is non empty set
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
(nat_hom s2) .: the carrier of G is Element of bool the carrier of (G ./. s2)
bool the carrier of (G ./. s2) is non empty set
Image (nat_hom s2) is non empty strict unital Group-like associative Subgroup of G ./. s2
the carrier of (Image (nat_hom s2)) is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,(O,G,s1))
the carrier of s19 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,s1)
rng s2 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
s19 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s19 is non empty set
the carrier of (O,G,s1,s2) is non empty set
rng s2 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
s19 is set
s2 .: the carrier of G is Element of bool the carrier of s1
dom s2 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
s29 is set
s2 . s29 is set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s19 is Element of the carrier of s1
rng s2 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
dom s2 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
s29 is set
s2 . s29 is set
p is Element of the carrier of G
s2 . p is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of (O,G,s1) is non empty set
(O,G,s1) is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
the carrier of G is non empty set
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
(O,G,(O,G,s1),(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,(O,G,s1))
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,(O,G)) is non empty unital Group-like associative (O) (O)
(O,G,(O,G)) is set
(O,G,(O,G)) is Relation-like [:(O,G,(O,G)),(O,G,(O,G)):] -defined (O,G,(O,G)) -valued Function-like quasi_total Element of bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):]
[:(O,G,(O,G)),(O,G,(O,G)):] is Relation-like set
[:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is Relation-like set
bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is non empty set
(O,G,(O,G)) is Relation-like O -defined Funcs ((O,G,(O,G)),(O,G,(O,G))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):]
Funcs ((O,G,(O,G)),(O,G,(O,G))) is functional non empty set
[:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is Relation-like set
bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is non empty set
(O,(O,G,(O,G)),(O,G,(O,G)),(O,G,(O,G))) is (O) (O)
the carrier of (O,G,(O,G)) is non empty set
(O,G,(O,G)) is Relation-like the carrier of G -defined the carrier of (O,G,(O,G)) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,(O,G))) Element of bool [: the carrier of G, the carrier of (O,G,(O,G)):]
[: the carrier of G, the carrier of (O,G,(O,G)):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,(O,G)):] is non empty set
the carrier of (O,G) is non empty set
the multF of (O,G) is Relation-like [: the carrier of (O,G), the carrier of (O,G):] -defined the carrier of (O,G) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):]
[: the carrier of (O,G), the carrier of (O,G):] is Relation-like non empty set
[:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is Relation-like non empty set
bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is non empty set
multMagma(# the carrier of (O,G), the multF of (O,G) #) is non empty strict multMagma
s1 is non empty strict unital Group-like associative normal Subgroup of G
the carrier of s1 is non empty set
s19 is non empty unital Group-like associative multMagma
1_ s19 is non being_of_order_0 Element of the carrier of s19
the carrier of s19 is non empty set
{(1_ s19)} is non empty trivial finite 1 -element set
(1). s19 is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of s19
s19 ./. ((1). s19) is non empty strict unital Group-like associative multMagma
Left_Cosets ((1). s19) is non empty Element of bool (bool the carrier of s19)
bool the carrier of s19 is non empty set
bool (bool the carrier of s19) is non empty set
CosOp ((1). s19) is Relation-like [:(Left_Cosets ((1). s19)),(Left_Cosets ((1). s19)):] -defined Left_Cosets ((1). s19) -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets ((1). s19)),(Left_Cosets ((1). s19)):],(Left_Cosets ((1). s19)):]
[:(Left_Cosets ((1). s19)),(Left_Cosets ((1). s19)):] is Relation-like non empty set
[:[:(Left_Cosets ((1). s19)),(Left_Cosets ((1). s19)):],(Left_Cosets ((1). s19)):] is Relation-like non empty set
bool [:[:(Left_Cosets ((1). s19)),(Left_Cosets ((1). s19)):],(Left_Cosets ((1). s19)):] is non empty set
multMagma(# (Left_Cosets ((1). s19)),(CosOp ((1). s19)) #) is non empty strict multMagma
the carrier of (s19 ./. ((1). s19)) is non empty set
nat_hom ((1). s19) is Relation-like the carrier of s19 -defined the carrier of (s19 ./. ((1). s19)) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of (s19 ./. ((1). s19)):]
[: the carrier of s19, the carrier of (s19 ./. ((1). s19)):] is Relation-like non empty set
bool [: the carrier of s19, the carrier of (s19 ./. ((1). s19)):] is non empty set
nat_hom s1 is Relation-like the carrier of G -defined the carrier of (G ./. s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s1):]
G ./. s1 is non empty strict unital Group-like associative multMagma
Left_Cosets s1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s1 is Relation-like [:(Left_Cosets s1),(Left_Cosets s1):] -defined Left_Cosets s1 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s1),(Left_Cosets s1):],(Left_Cosets s1):]
[:(Left_Cosets s1),(Left_Cosets s1):] is Relation-like non empty set
[:[:(Left_Cosets s1),(Left_Cosets s1):],(Left_Cosets s1):] is Relation-like non empty set
bool [:[:(Left_Cosets s1),(Left_Cosets s1):],(Left_Cosets s1):] is non empty set
multMagma(# (Left_Cosets s1),(CosOp s1) #) is non empty strict multMagma
the carrier of (G ./. s1) is non empty set
[: the carrier of G, the carrier of (G ./. s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s1):] is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
the carrier of s2 is non empty set
[: the carrier of s1, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s2:] is non empty set
s29 is Relation-like the carrier of s1 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,s2) Element of bool [: the carrier of s1, the carrier of s2:]
rng s29 is non empty Element of bool the carrier of s2
bool the carrier of s2 is non empty set
rng s19 is non empty Element of bool the carrier of s1
bool the carrier of s1 is non empty set
dom s29 is non empty Element of bool the carrier of s1
(O,G,s1,s2,s19,s29) is Relation-like the carrier of G -defined the carrier of G -defined the carrier of s2 -valued the carrier of s2 -valued Function-like non empty total quasi_total quasi_total unity-preserving multiplicative (O,G,s2) Element of bool [: the carrier of G, the carrier of s2:]
[: the carrier of G, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s2:] is non empty set
rng (O,G,s1,s2,s19,s29) is non empty Element of bool the carrier of s2
O is set
G is non empty unital Group-like associative (O) (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,(O,G)) is non empty unital Group-like associative (O) (O)
(O,G,(O,G)) is set
(O,G,(O,G)) is Relation-like [:(O,G,(O,G)),(O,G,(O,G)):] -defined (O,G,(O,G)) -valued Function-like quasi_total Element of bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):]
[:(O,G,(O,G)),(O,G,(O,G)):] is Relation-like set
[:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is Relation-like set
bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is non empty set
(O,G,(O,G)) is Relation-like O -defined Funcs ((O,G,(O,G)),(O,G,(O,G))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):]
Funcs ((O,G,(O,G)),(O,G,(O,G))) is functional non empty set
[:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is Relation-like set
bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is non empty set
(O,(O,G,(O,G)),(O,G,(O,G)),(O,G,(O,G))) is (O) (O)
the carrier of G is non empty set
the carrier of (O,G,(O,G)) is non empty set
(O,G,(O,G)) is Relation-like the carrier of G -defined the carrier of (O,G,(O,G)) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,(O,G))) Element of bool [: the carrier of G, the carrier of (O,G,(O,G)):]
[: the carrier of G, the carrier of (O,G,(O,G)):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,(O,G)):] is non empty set
O is set
G is non empty unital Group-like associative (O) (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(O,G,(O,G)) is non empty unital Group-like associative (O) (O)
(O,G,(O,G)) is set
(O,G,(O,G)) is Relation-like [:(O,G,(O,G)),(O,G,(O,G)):] -defined (O,G,(O,G)) -valued Function-like quasi_total Element of bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):]
[:(O,G,(O,G)),(O,G,(O,G)):] is Relation-like set
[:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is Relation-like set
bool [:[:(O,G,(O,G)),(O,G,(O,G)):],(O,G,(O,G)):] is non empty set
(O,G,(O,G)) is Relation-like O -defined Funcs ((O,G,(O,G)),(O,G,(O,G))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):]
Funcs ((O,G,(O,G)),(O,G,(O,G))) is functional non empty set
[:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is Relation-like set
bool [:O,(Funcs ((O,G,(O,G)),(O,G,(O,G)))):] is non empty set
(O,(O,G,(O,G)),(O,G,(O,G)),(O,G,(O,G))) is (O) (O)
the carrier of (O,G) is non empty set
the multF of (O,G) is Relation-like [: the carrier of (O,G), the carrier of (O,G):] -defined the carrier of (O,G) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):]
[: the carrier of (O,G), the carrier of (O,G):] is Relation-like non empty set
[:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is Relation-like non empty set
bool [:[: the carrier of (O,G), the carrier of (O,G):], the carrier of (O,G):] is non empty set
multMagma(# the carrier of (O,G), the multF of (O,G) #) is non empty strict multMagma
s2 is non empty strict unital Group-like associative normal Subgroup of G
s1 is non empty unital Group-like associative multMagma
(Omega). s1 is non empty strict unital Group-like associative normal Subgroup of s1
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
the carrier of (O,G,(O,G)) is non empty set
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
{ the carrier of G} is non empty trivial finite 1 -element set
O is set
G is non empty unital Group-like associative (O) (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O)
the carrier of G is non empty set
s2 is set
{s2} is non empty trivial finite 1 -element set
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
dom s19 is non empty Element of bool the carrier of G
bool the carrier of G is non empty set
(O,G,s1,s19) is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of (O,G,s1,s19) is non empty set
Im (s19,s2) is set
s19 .: {s2} is finite set
s19 . s2 is set
{(s19 . s2)} is non empty trivial finite 1 -element set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of s1, the carrier of G:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of G:] is non empty set
s2 is Relation-like the carrier of s1 -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,G) Element of bool [: the carrier of s1, the carrier of G:]
(O,s1,G,s2) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
(O,s1,(O,s1,G,s2)) is non empty unital Group-like associative (O) (O)
(O,s1,(O,s1,G,s2)) is set
(O,s1,(O,s1,G,s2)) is Relation-like [:(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)):] -defined (O,s1,(O,s1,G,s2)) -valued Function-like quasi_total Element of bool [:[:(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)):],(O,s1,(O,s1,G,s2)):]
[:(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)):] is Relation-like set
[:[:(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)):],(O,s1,(O,s1,G,s2)):] is Relation-like set
bool [:[:(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)):],(O,s1,(O,s1,G,s2)):] is non empty set
(O,s1,(O,s1,G,s2)) is Relation-like O -defined Funcs ((O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)))):]
Funcs ((O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2))) is functional non empty set
[:O,(Funcs ((O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)))):] is Relation-like set
bool [:O,(Funcs ((O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)))):] is non empty set
(O,(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2)),(O,s1,(O,s1,G,s2))) is (O) (O)
(O,s1,G,s2) is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative multMagma
the carrier of s19 is non empty set
s29 is non empty unital Group-like associative multMagma
the carrier of s29 is non empty set
[: the carrier of s19, the carrier of s29:] is Relation-like non empty set
bool [: the carrier of s19, the carrier of s29:] is non empty set
p is Relation-like the carrier of s19 -defined the carrier of s29 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s29:]
Ker p is non empty strict unital Group-like associative normal Subgroup of s19
s19 ./. (Ker p) is non empty strict unital Group-like associative multMagma
Left_Cosets (Ker p) is non empty Element of bool (bool the carrier of s19)
bool the carrier of s19 is non empty set
bool (bool the carrier of s19) is non empty set
CosOp (Ker p) is Relation-like [:(Left_Cosets (Ker p)),(Left_Cosets (Ker p)):] -defined Left_Cosets (Ker p) -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets (Ker p)),(Left_Cosets (Ker p)):],(Left_Cosets (Ker p)):]
[:(Left_Cosets (Ker p)),(Left_Cosets (Ker p)):] is Relation-like non empty set
[:[:(Left_Cosets (Ker p)),(Left_Cosets (Ker p)):],(Left_Cosets (Ker p)):] is Relation-like non empty set
bool [:[:(Left_Cosets (Ker p)),(Left_Cosets (Ker p)):],(Left_Cosets (Ker p)):] is non empty set
multMagma(# (Left_Cosets (Ker p)),(CosOp (Ker p)) #) is non empty strict multMagma
the carrier of (s19 ./. (Ker p)) is non empty set
Image p is non empty strict unital Group-like associative Subgroup of s29
the carrier of (Image p) is non empty set
[: the carrier of (s19 ./. (Ker p)), the carrier of (Image p):] is Relation-like non empty set
bool [: the carrier of (s19 ./. (Ker p)), the carrier of (Image p):] is non empty set
nat_hom (Ker p) is Relation-like the carrier of s19 -defined the carrier of (s19 ./. (Ker p)) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of (s19 ./. (Ker p)):]
[: the carrier of s19, the carrier of (s19 ./. (Ker p)):] is Relation-like non empty set
bool [: the carrier of s19, the carrier of (s19 ./. (Ker p)):] is non empty set
f1 is Relation-like the carrier of (s19 ./. (Ker p)) -defined the carrier of (Image p) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (s19 ./. (Ker p)), the carrier of (Image p):]
f1 * (nat_hom (Ker p)) is Relation-like the carrier of s19 -defined the carrier of s19 -defined the carrier of (Image p) -valued the carrier of (Image p) -valued Function-like non empty total total quasi_total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of (Image p):]
[: the carrier of s19, the carrier of (Image p):] is Relation-like non empty set
bool [: the carrier of s19, the carrier of (Image p):] is non empty set
p .: the carrier of s19 is Element of bool the carrier of s29
bool the carrier of s29 is non empty set
the carrier of (O,s1,G,s2) is non empty set
i is set
the carrier of (O,s1,G,s2) is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{ b₁ where b₁ is Element of the carrier of s1 : s2 . b₁ = 1_ G } is set
the carrier of (Ker p) is non empty set
1_ s29 is non being_of_order_0 Element of the carrier of s29
{ b₁ where b₁ is Element of the carrier of s19 : p . b₁ = 1_ s29 } is set
the multF of (Ker p) is Relation-like [: the carrier of (Ker p), the carrier of (Ker p):] -defined the carrier of (Ker p) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (Ker p), the carrier of (Ker p):], the carrier of (Ker p):]
[: the carrier of (Ker p), the carrier of (Ker p):] is Relation-like non empty set
[:[: the carrier of (Ker p), the carrier of (Ker p):], the carrier of (Ker p):] is Relation-like non empty set
bool [:[: the carrier of (Ker p), the carrier of (Ker p):], the carrier of (Ker p):] is non empty set
multMagma(# the carrier of (Ker p), the multF of (Ker p) #) is non empty strict multMagma
the multF of (O,s1,G,s2) is Relation-like [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):]
[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] is Relation-like non empty set
[:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):] is Relation-like non empty set
bool [:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):] is non empty set
multMagma(# the carrier of (O,s1,G,s2), the multF of (O,s1,G,s2) #) is non empty strict multMagma
the carrier of (O,s1,(O,s1,G,s2)) is non empty set
[: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,G,s2):] is Relation-like non empty set
bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,G,s2):] is non empty set
the multF of (Image p) is Relation-like [: the carrier of (Image p), the carrier of (Image p):] -defined the carrier of (Image p) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (Image p), the carrier of (Image p):], the carrier of (Image p):]
[: the carrier of (Image p), the carrier of (Image p):] is Relation-like non empty set
[:[: the carrier of (Image p), the carrier of (Image p):], the carrier of (Image p):] is Relation-like non empty set
bool [:[: the carrier of (Image p), the carrier of (Image p):], the carrier of (Image p):] is non empty set
multMagma(# the carrier of (Image p), the multF of (Image p) #) is non empty strict multMagma
the multF of (O,s1,G,s2) is Relation-like [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):]
[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] is Relation-like non empty set
[:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):] is Relation-like non empty set
bool [:[: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):], the carrier of (O,s1,G,s2):] is non empty set
multMagma(# the carrier of (O,s1,G,s2), the multF of (O,s1,G,s2) #) is non empty strict multMagma
s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
i is Relation-like the carrier of (O,s1,(O,s1,G,s2)) -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,G,s2):]
s199 * s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
the multF of (O,s1,(O,s1,G,s2)) is Relation-like [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):] -defined the carrier of (O,s1,(O,s1,G,s2)) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):], the carrier of (O,s1,(O,s1,G,s2)):]
[: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):] is Relation-like non empty set
[:[: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):], the carrier of (O,s1,(O,s1,G,s2)):] is Relation-like non empty set
bool [:[: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):], the carrier of (O,s1,(O,s1,G,s2)):] is non empty set
the multF of (O,s1,(O,s1,G,s2)) . (s199,s199) is Element of the carrier of (O,s1,(O,s1,G,s2))
i . (s199 * s199) is Element of the carrier of (O,s1,G,s2)
p is Element of the carrier of (s19 ./. (Ker p))
f2 is Element of the carrier of (s19 ./. (Ker p))
p * f2 is Element of the carrier of (s19 ./. (Ker p))
the multF of (s19 ./. (Ker p)) is Relation-like [: the carrier of (s19 ./. (Ker p)), the carrier of (s19 ./. (Ker p)):] -defined the carrier of (s19 ./. (Ker p)) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (s19 ./. (Ker p)), the carrier of (s19 ./. (Ker p)):], the carrier of (s19 ./. (Ker p)):]
[: the carrier of (s19 ./. (Ker p)), the carrier of (s19 ./. (Ker p)):] is Relation-like non empty set
[:[: the carrier of (s19 ./. (Ker p)), the carrier of (s19 ./. (Ker p)):], the carrier of (s19 ./. (Ker p)):] is Relation-like non empty set
bool [:[: the carrier of (s19 ./. (Ker p)), the carrier of (s19 ./. (Ker p)):], the carrier of (s19 ./. (Ker p)):] is non empty set
the multF of (s19 ./. (Ker p)) . (p,f2) is Element of the carrier of (s19 ./. (Ker p))
f1 . (p * f2) is Element of the carrier of (Image p)
f1 . p is Element of the carrier of (Image p)
f1 . f2 is Element of the carrier of (Image p)
(f1 . p) * (f1 . f2) is Element of the carrier of (Image p)
the multF of (Image p) . ((f1 . p),(f1 . f2)) is Element of the carrier of (Image p)
i . s199 is Element of the carrier of (O,s1,G,s2)
i . s199 is Element of the carrier of (O,s1,G,s2)
(i . s199) * (i . s199) is Element of the carrier of (O,s1,G,s2)
the multF of (O,s1,G,s2) . ((i . s199),(i . s199)) is Element of the carrier of (O,s1,G,s2)
s199 is Element of O
(O,(O,s1,(O,s1,G,s2)),s199) is Relation-like the carrier of (O,s1,(O,s1,G,s2)) -defined the carrier of (O,s1,(O,s1,G,s2)) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):]
bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):] is non empty set
s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
(O,(O,s1,(O,s1,G,s2)),s199) . s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
i . ((O,(O,s1,(O,s1,G,s2)),s199) . s199) is Element of the carrier of (O,s1,G,s2)
id the carrier of (O,s1,(O,s1,G,s2)) is Relation-like the carrier of (O,s1,(O,s1,G,s2)) -defined the carrier of (O,s1,(O,s1,G,s2)) -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):]
(id the carrier of (O,s1,(O,s1,G,s2))) . s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
i . ((id the carrier of (O,s1,(O,s1,G,s2))) . s199) is Element of the carrier of (O,s1,G,s2)
i . s199 is Element of the carrier of (O,s1,G,s2)
id the carrier of (O,s1,G,s2) is Relation-like the carrier of (O,s1,G,s2) -defined the carrier of (O,s1,G,s2) -valued Function-like one-to-one non empty total quasi_total Element of bool [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):]
bool [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] is non empty set
(id the carrier of (O,s1,G,s2)) . (i . s199) is Element of the carrier of (O,s1,G,s2)
(O,(O,s1,G,s2),s199) is Relation-like the carrier of (O,s1,G,s2) -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):]
(O,(O,s1,G,s2),s199) . (i . s199) is Element of the carrier of (O,s1,G,s2)
the of (O,s1,(O,s1,G,s2)) is Relation-like O -defined Funcs ( the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)))):]
Funcs ( the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2))) is functional non empty set
[:O,(Funcs ( the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)))):] is non empty set
s199 is Element of O
the of (O,s1,(O,s1,G,s2)) . s199 is Relation-like Function-like set
(O,s1,s199) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
{ [b₁,b₂] where b₁, b₂ is Element of (O,s1,(O,s1,G,s2)) : ex b₃, b₄ being Element of the carrier of s1 st
( b₃ in b₁ & b₄ in b₂ & b₄ = (O,s1,s199) . b₃ ) } is set
(O,(O,s1,(O,s1,G,s2)),s199) is Relation-like the carrier of (O,s1,(O,s1,G,s2)) -defined the carrier of (O,s1,(O,s1,G,s2)) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):]
bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,(O,s1,G,s2)):] is non empty set
f2 is non empty unital Group-like associative multMagma
the carrier of f2 is non empty set
[: the carrier of f2, the carrier of f2:] is Relation-like non empty set
bool [: the carrier of f2, the carrier of f2:] is non empty set
s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
H1 is Relation-like the carrier of f2 -defined the carrier of f2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of f2, the carrier of f2:]
dom H1 is non empty Element of bool the carrier of f2
bool the carrier of f2 is non empty set
H1 . s199 is set
[s199,(H1 . s199)] is set
{s199,(H1 . s199)} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,(H1 . s199)},{s199}} is non empty finite V39() set
j is Element of (O,s1,(O,s1,G,s2))
j is Element of (O,s1,(O,s1,G,s2))
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
H2 is Element of Left_Cosets (Ker p)
s299 is Element of Left_Cosets (Ker p)
H1 is Element of the carrier of s1
H2 is Element of the carrier of s1
(O,s1,s199) . H1 is Element of the carrier of s1
H1 is Element of the carrier of s19
H2 is Element of the carrier of s19
(O,s1,s199) . H1 is set
H1 * (Ker p) is Element of bool the carrier of s19
carr (Ker p) is Element of bool the carrier of s19
H1 * (carr (Ker p)) is Element of bool the carrier of s19
K350( the carrier of s19,H1) is non empty trivial finite 1 -element Element of bool the carrier of s19
K350( the carrier of s19,H1) * (carr (Ker p)) is Element of bool the carrier of s19
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s19 : ( b₁ in K350( the carrier of s19,H1) & b₂ in carr (Ker p) ) } is set
dom (nat_hom (Ker p)) is non empty Element of bool the carrier of s19
(O,(O,s1,G,s2),s199) is Relation-like the carrier of (O,s1,G,s2) -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):]
bool [: the carrier of (O,s1,G,s2), the carrier of (O,s1,G,s2):] is non empty set
i . s199 is Element of the carrier of (O,s1,G,s2)
(O,(O,s1,G,s2),s199) . (i . s199) is Element of the carrier of (O,s1,G,s2)
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s199) | the carrier of (O,s1,G,s2) is Relation-like the carrier of G -defined the carrier of (O,s1,G,s2) -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
((O,G,s199) | the carrier of (O,s1,G,s2)) . (i . s199) is set
(O,G,s199) . (i . s199) is set
f1 . (H1 * (Ker p)) is set
(O,G,s199) . (f1 . (H1 * (Ker p))) is set
H2 * (Ker p) is Element of bool the carrier of s19
H2 * (carr (Ker p)) is Element of bool the carrier of s19
K350( the carrier of s19,H2) is non empty trivial finite 1 -element Element of bool the carrier of s19
K350( the carrier of s19,H2) * (carr (Ker p)) is Element of bool the carrier of s19
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s19 : ( b₁ in K350( the carrier of s19,H2) & b₂ in carr (Ker p) ) } is set
p . H2 is Element of the carrier of s29
p . H1 is Element of the carrier of s29
(O,G,s199) . (p . H1) is set
(nat_hom (Ker p)) . H2 is Element of the carrier of (s19 ./. (Ker p))
f1 . ((nat_hom (Ker p)) . H2) is Element of the carrier of (Image p)
(f1 * (nat_hom (Ker p))) . H1 is Element of the carrier of (Image p)
(O,G,s199) . ((f1 * (nat_hom (Ker p))) . H1) is set
(nat_hom (Ker p)) . H1 is Element of the carrier of (s19 ./. (Ker p))
f1 . ((nat_hom (Ker p)) . H1) is Element of the carrier of (Image p)
(O,G,s199) . (f1 . ((nat_hom (Ker p)) . H1)) is set
f1 . (H2 * (Ker p)) is set
(O,(O,s1,(O,s1,G,s2)),s199) . s199 is Element of the carrier of (O,s1,(O,s1,G,s2))
i . ((O,(O,s1,(O,s1,G,s2)),s199) . s199) is Element of the carrier of (O,s1,G,s2)
i . (H1 . s199) is set
s199 is Relation-like the carrier of (O,s1,(O,s1,G,s2)) -defined the carrier of (O,s1,G,s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,(O,s1,(O,s1,G,s2)),(O,s1,G,s2)) Element of bool [: the carrier of (O,s1,(O,s1,G,s2)), the carrier of (O,s1,G,s2):]
O is set
G is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s2,s19),s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,(O,G,s19,s1)) is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O,s19) (O,s19)
s29 is non empty unital Group-like associative (O) (O,s19)
(O,s19,p,s29) is non empty unital Group-like associative (O) (O) (O,s19)
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
[#] G is non proper Element of bool G
bool G is non empty set
O is set
G is set
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
{(id G)} is functional non empty trivial finite 1 -element set
[:O,{(id G)}:] is Relation-like set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s2 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
s1 is Element of O
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(O,G,s2,<*s1*>) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
s2 . s1 is Relation-like Function-like Element of Funcs (G,G)
O is non empty set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like non empty set
bool [:O,(Funcs (G,G)):] is non empty set
s19 is Relation-like O -defined Funcs (G,G) -valued Function-like non empty total quasi_total Element of bool [:O,(Funcs (G,G)):]
s1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s1 ^ s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s19,(s1 ^ s2)) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s19,s1) * (O,G,s19,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
bool G is non empty set
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
s19 is Element of bool G
s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s2) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
(O,G,s1,s2) .: s19 is Element of bool G
O is set
G is non empty set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
bool G is non empty set
s2 is Element of bool G
s19 is Element of G
s1 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
(O,G,s1,s2) is Element of bool G
s29 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s1,s29) is Relation-like G -defined G -valued Function-like non empty total quasi_total Element of bool [:G,G:]
[:G,G:] is Relation-like non empty set
bool [:G,G:] is non empty set
p is Element of s2
(O,G,s1,s29) . p is set
O is set
G is non empty strict unital Group-like associative multMagma
s1 is non empty (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s2 is non empty unital Group-like associative (O) (O) (O)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
O is set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,s1)
s19 is Element of the carrier of s1
G is Element of O
(O,s1,G) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,G) . s19 is Element of the carrier of s1
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s2 is non empty set
s19 .: the carrier of s2 is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
bool the carrier of G is non empty set
s1 is Element of bool the carrier of G
s2 is Element of bool the carrier of G
gr s2 is non empty strict unital Group-like associative Subgroup of G
the carrier of (gr s2) is non empty set
(O,G,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s2 is non empty unital Group-like associative normal Subgroup of G
s1 is non empty unital Group-like associative (O) (O,G) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of (O,G,s1) is non empty set
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
multMagma(# the carrier of (O,G,s1), the multF of (O,G,s1) #) is non empty strict multMagma
1_ (G ./. s2) is non being_of_order_0 Element of the carrier of (G ./. s2)
the carrier of (G ./. s2) is non empty set
1_ (O,G,s1) is non being_of_order_0 Element of the carrier of (O,G,s1)
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
(O, the carrier of s1, the multF of s1, the of s1) is (O) (O)
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
(O, the carrier of s2, the multF of s2, the of s2) is (O) (O)
dom the of s2 is Element of bool O
bool O is non empty set
dom the of s1 is Element of bool O
p is set
f1 is Element of O
(O,s1,f1) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
the of s1 . f1 is Relation-like Function-like set
the carrier of G is non empty set
(O,G,f1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,f1) | the carrier of s2 is Relation-like the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
(O,s2,f1) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
bool [: the carrier of s2, the carrier of s2:] is non empty set
the of s1 . p is Relation-like Function-like set
the of s2 . p is Relation-like Function-like set
s19 is non empty unital Group-like associative Subgroup of G
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty unital Group-like associative Subgroup of G
the carrier of s29 is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
(O, the carrier of s1, the multF of s1, the of s1) is (O) (O)
s2 is non empty unital Group-like associative (O) (O,s1) (O,s1)
(O,s1,s2) is non empty unital Group-like associative (O) (O)
(O,s1,s2) is set
(O,s1,s2) is Relation-like [:(O,s1,s2),(O,s1,s2):] -defined (O,s1,s2) -valued Function-like quasi_total Element of bool [:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):]
[:(O,s1,s2),(O,s1,s2):] is Relation-like set
[:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):] is Relation-like set
bool [:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):] is non empty set
(O,s1,s2) is Relation-like O -defined Funcs ((O,s1,s2),(O,s1,s2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s1,s2),(O,s1,s2))):]
Funcs ((O,s1,s2),(O,s1,s2)) is functional non empty set
[:O,(Funcs ((O,s1,s2),(O,s1,s2))):] is Relation-like set
bool [:O,(Funcs ((O,s1,s2),(O,s1,s2))):] is non empty set
(O,(O,s1,s2),(O,s1,s2),(O,s1,s2)) is (O) (O)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the of s2 is Relation-like O -defined Funcs ( the carrier of s2, the carrier of s2) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):]
Funcs ( the carrier of s2, the carrier of s2) is functional non empty set
[:O,(Funcs ( the carrier of s2, the carrier of s2)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s2, the carrier of s2)):] is non empty set
(O, the carrier of s2, the multF of s2, the of s2) is (O) (O)
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
p is non empty unital Group-like associative normal Subgroup of s1
Left_Cosets p is non empty Element of bool (bool the carrier of s1)
bool the carrier of s1 is non empty set
bool (bool the carrier of s1) is non empty set
s1 ./. p is non empty strict unital Group-like associative multMagma
CosOp p is Relation-like [:(Left_Cosets p),(Left_Cosets p):] -defined Left_Cosets p -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets p),(Left_Cosets p):],(Left_Cosets p):]
[:(Left_Cosets p),(Left_Cosets p):] is Relation-like non empty set
[:[:(Left_Cosets p),(Left_Cosets p):],(Left_Cosets p):] is Relation-like non empty set
bool [:[:(Left_Cosets p),(Left_Cosets p):],(Left_Cosets p):] is non empty set
multMagma(# (Left_Cosets p),(CosOp p) #) is non empty strict multMagma
the carrier of (s1 ./. p) is non empty set
f1 is set
{f1} is non empty trivial finite 1 -element set
union {f1} is set
the carrier of p is non empty set
the carrier of s1 \ the carrier of p is Element of bool the carrier of s1
i is set
s199 is Element of the carrier of s1
s199 * p is Element of bool the carrier of s1
carr p is Element of bool the carrier of s1
s199 * (carr p) is Element of bool the carrier of s1
K350( the carrier of s1,s199) is non empty trivial finite 1 -element Element of bool the carrier of s1
K350( the carrier of s1,s199) * (carr p) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,s199) & b₂ in carr p ) } is set
1_ s1 is non being_of_order_0 Element of the carrier of s1
s199 is Element of the carrier of s1
s199 * s199 is Element of the carrier of s1
the multF of s1 . (s199,s199) is Element of the carrier of s1
s199 " is Element of the carrier of s1
(s199 ") " is Element of the carrier of s1
s19 is non empty unital Group-like associative (O) (O,G)
the carrier of s19 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is non empty unital Group-like associative (O) (O,s1) (O,s1)
the carrier of s2 is non empty set
(O,s1,s2) is non empty unital Group-like associative (O) (O)
(O,s1,s2) is set
(O,s1,s2) is Relation-like [:(O,s1,s2),(O,s1,s2):] -defined (O,s1,s2) -valued Function-like quasi_total Element of bool [:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):]
[:(O,s1,s2),(O,s1,s2):] is Relation-like set
[:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):] is Relation-like set
bool [:[:(O,s1,s2),(O,s1,s2):],(O,s1,s2):] is non empty set
(O,s1,s2) is Relation-like O -defined Funcs ((O,s1,s2),(O,s1,s2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s1,s2),(O,s1,s2))):]
Funcs ((O,s1,s2),(O,s1,s2)) is functional non empty set
[:O,(Funcs ((O,s1,s2),(O,s1,s2))):] is Relation-like set
bool [:O,(Funcs ((O,s1,s2),(O,s1,s2))):] is non empty set
(O,(O,s1,s2),(O,s1,s2),(O,s1,s2)) is (O) (O)
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s29 is set
s19 is non empty strict unital Group-like associative normal Subgroup of s1
Left_Cosets s19 is non empty Element of bool (bool the carrier of s1)
bool the carrier of s1 is non empty set
bool (bool the carrier of s1) is non empty set
p is Element of bool the carrier of s1
f1 is Element of the carrier of s1
f1 * s19 is Element of bool the carrier of s1
carr s19 is Element of bool the carrier of s1
the carrier of s19 is non empty set
f1 * (carr s19) is Element of bool the carrier of s1
K350( the carrier of s1,f1) is non empty trivial finite 1 -element Element of bool the carrier of s1
K350( the carrier of s1,f1) * (carr s19) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,f1) & b₂ in carr s19 ) } is set
[#] the carrier of s1 is non empty non proper Element of bool the carrier of s1
f1 * ([#] the carrier of s1) is Element of bool the carrier of s1
K350( the carrier of s1,f1) * ([#] the carrier of s1) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,f1) & b₂ in [#] the carrier of s1 ) } is set
1_ s1 is non being_of_order_0 Element of the carrier of s1
(1_ s1) * ([#] the carrier of s1) is Element of bool the carrier of s1
K350( the carrier of s1,(1_ s1)) is non empty trivial finite 1 -element Element of bool the carrier of s1
K350( the carrier of s1,(1_ s1)) * ([#] the carrier of s1) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,(1_ s1)) & b₂ in [#] the carrier of s1 ) } is set
(1_ s1) * s19 is Element of bool the carrier of s1
(1_ s1) * (carr s19) is Element of bool the carrier of s1
K350( the carrier of s1,(1_ s1)) * (carr s19) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,(1_ s1)) & b₂ in carr s19 ) } is set
{ the carrier of s1} is non empty trivial finite 1 -element set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s19 is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
p is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,p) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
p " the carrier of s19 is Element of bool the carrier of G
bool the carrier of G is non empty set
s29 is non empty strict unital Group-like associative Subgroup of s1
{ b₁ where b₁ is Element of the carrier of G : p . b₁ in s29 } is set
1_ s1 is non being_of_order_0 Element of the carrier of s1
1_ G is non being_of_order_0 Element of the carrier of G
p . (1_ G) is Element of the carrier of s1
s199 is set
i is non empty set
s199 is Element of the carrier of G
p . s199 is Element of the carrier of s1
s199 is Element of the carrier of G
s199 is Element of bool the carrier of G
f2 is Element of the carrier of G
p is Element of the carrier of G
p . p is Element of the carrier of s1
H1 is Element of the carrier of G
p . H1 is Element of the carrier of s1
H1 * p is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H1,p) is Element of the carrier of G
p . (H1 * p) is Element of the carrier of s1
(p . H1) * (p . p) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((p . H1),(p . p)) is Element of the carrier of s1
s199 * f2 is Element of the carrier of G
the multF of G . (s199,f2) is Element of the carrier of G
f2 is Element of the carrier of G
p is Element of the carrier of G
p . p is Element of the carrier of s1
the carrier of s29 is non empty set
s199 is Element of O
(O,s1,s199) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
p . f2 is Element of the carrier of s1
(O,s1,s199) . (p . f2) is Element of the carrier of s1
(O,G,s199) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,s199) . f2 is Element of the carrier of G
p . ((O,G,s199) . f2) is Element of the carrier of s1
s199 is Element of the carrier of G
f2 is Element of the carrier of G
p . f2 is Element of the carrier of s1
(p . f2) " is Element of the carrier of s1
f2 " is Element of the carrier of G
p . (f2 ") is Element of the carrier of s1
s199 " is Element of the carrier of G
s199 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s199 is non empty set
p is Element of the carrier of G
p . p is Element of the carrier of s1
H1 is Element of the carrier of G
p . H1 is Element of the carrier of s1
dom p is non empty Element of bool the carrier of G
[p,(p . p)] is set
{p,(p . p)} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,(p . p)},{p}} is non empty finite V39() set
f2 is Relation-like the carrier of G -defined the carrier of s1 -valued Element of bool [: the carrier of G, the carrier of s1:]
H1 is Element of the carrier of s1
[p,H1] is set
{p,H1} is non empty finite set
{{p,H1},{p}} is non empty finite V39() set
the multF of s199 is Relation-like [: the carrier of s199, the carrier of s199:] -defined the carrier of s199 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:]
[: the carrier of s199, the carrier of s199:] is Relation-like non empty set
[:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is Relation-like non empty set
bool [:[: the carrier of s199, the carrier of s199:], the carrier of s199:] is non empty set
multMagma(# the carrier of s199, the multF of s199 #) is non empty strict multMagma
p is Element of the carrier of G
p . p is Element of the carrier of s1
p is Element of the carrier of G
p . p is Element of the carrier of s1
(p . p) " is Element of the carrier of s1
s29 |^ ((p . p) ") is non empty strict unital Group-like associative Subgroup of s1
H1 is set
f2 is non empty strict unital Group-like associative Subgroup of G
p * f2 is Element of bool the carrier of G
carr f2 is Element of bool the carrier of G
the carrier of f2 is non empty set
p * (carr f2) is Element of bool the carrier of G
K350( the carrier of G,p) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,p) * (carr f2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,p) & b₂ in carr f2 ) } is set
j is Element of the carrier of G
p * j is Element of the carrier of G
the multF of G . (p,j) is Element of the carrier of G
p " is Element of the carrier of G
(p * j) * (p ") is Element of the carrier of G
the multF of G . ((p * j),(p ")) is Element of the carrier of G
p . ((p * j) * (p ")) is Element of the carrier of s1
p . (p * j) is Element of the carrier of s1
p . (p ") is Element of the carrier of s1
(p . (p * j)) * (p . (p ")) is Element of the carrier of s1
the multF of s1 . ((p . (p * j)),(p . (p "))) is Element of the carrier of s1
p . j is Element of the carrier of s1
(p . p) * (p . j) is Element of the carrier of s1
the multF of s1 . ((p . p),(p . j)) is Element of the carrier of s1
((p . p) * (p . j)) * (p . (p ")) is Element of the carrier of s1
the multF of s1 . (((p . p) * (p . j)),(p . (p "))) is Element of the carrier of s1
((p . p) * (p . j)) * ((p . p) ") is Element of the carrier of s1
the multF of s1 . (((p . p) * (p . j)),((p . p) ")) is Element of the carrier of s1
((p . p) ") " is Element of the carrier of s1
(((p . p) ") ") * (p . j) is Element of the carrier of s1
the multF of s1 . ((((p . p) ") "),(p . j)) is Element of the carrier of s1
((((p . p) ") ") * (p . j)) * ((p . p) ") is Element of the carrier of s1
the multF of s1 . (((((p . p) ") ") * (p . j)),((p . p) ")) is Element of the carrier of s1
(p . j) |^ ((p . p) ") is Element of the carrier of s1
H2 is Element of the carrier of G
p . H2 is Element of the carrier of s1
((p * j) * (p ")) * p is Element of the carrier of G
the multF of G . (((p * j) * (p ")),p) is Element of the carrier of G
(p ") * p is Element of the carrier of G
the multF of G . ((p "),p) is Element of the carrier of G
(p * j) * ((p ") * p) is Element of the carrier of G
the multF of G . ((p * j),((p ") * p)) is Element of the carrier of G
(p * j) * (1_ G) is Element of the carrier of G
the multF of G . ((p * j),(1_ G)) is Element of the carrier of G
f2 * p is Element of bool the carrier of G
(carr f2) * p is Element of bool the carrier of G
(carr f2) * K350( the carrier of G,p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr f2 & b₂ in K350( the carrier of G,p) ) } is set
p is non empty strict unital Group-like associative Subgroup of G
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
s29 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of s29 is non empty set
(O,G,s29,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29) is Element of bool the carrier of G
bool the carrier of G is non empty set
(O,G,s2) is Element of bool the carrier of G
(O,G,s29) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s29) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of (O,G,s29,s2) is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
f1 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,f1) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
f1 .: the carrier of s29 is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
p is non empty strict unital Group-like associative Subgroup of G
{ (f1 . b₁) where b₁ is Element of the carrier of G : b₁ in p } is set
s19 is non empty strict unital Group-like associative Subgroup of G
p * s19 is Element of bool the carrier of G
carr p is Element of bool the carrier of G
the carrier of p is non empty set
carr s19 is Element of bool the carrier of G
the carrier of s19 is non empty set
(carr p) * (carr s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in carr s19 ) } is set
(O,G,s29,s2) is Element of bool the carrier of G
(O,G,s29) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s29) & b₂ in (O,G,s2) ) } is set
s19 * p is Element of bool the carrier of G
(carr s19) * (carr p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s19 & b₂ in carr p ) } is set
(O,G,s2,s29) is Element of bool the carrier of G
(O,G,s2) * (O,G,s29) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s29) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
f1 . (1_ G) is Element of the carrier of s1
s199 is set
s199 is non empty set
f2 is Element of the carrier of G
f1 . f2 is Element of the carrier of s1
f2 is Element of the carrier of s1
s199 is Element of bool the carrier of s1
p is Element of the carrier of s1
H1 is Element of the carrier of G
f1 . H1 is Element of the carrier of s1
j is Element of the carrier of G
f1 . j is Element of the carrier of s1
H1 * j is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H1,j) is Element of the carrier of G
f1 . (H1 * j) is Element of the carrier of s1
f2 * p is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (f2,p) is Element of the carrier of s1
p is Element of the carrier of s1
H1 is Element of the carrier of G
f1 . H1 is Element of the carrier of s1
f2 is Element of O
(O,G,f2) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,f2) . H1 is Element of the carrier of G
(O,s1,f2) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
(O,s1,f2) . p is Element of the carrier of s1
f1 . ((O,G,f2) . H1) is Element of the carrier of s1
f2 is Element of the carrier of s1
p is Element of the carrier of G
f1 . p is Element of the carrier of s1
p " is Element of the carrier of G
f2 " is Element of the carrier of s1
f1 . (p ") is Element of the carrier of s1
f2 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of f2 is non empty set
(carr p) * s19 is Element of bool the carrier of G
s19 * (carr p) is Element of bool the carrier of G
the multF of f2 is Relation-like [: the carrier of f2, the carrier of f2:] -defined the carrier of f2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of f2, the carrier of f2:], the carrier of f2:]
[: the carrier of f2, the carrier of f2:] is Relation-like non empty set
[:[: the carrier of f2, the carrier of f2:], the carrier of f2:] is Relation-like non empty set
bool [:[: the carrier of f2, the carrier of f2:], the carrier of f2:] is non empty set
multMagma(# the carrier of f2, the multF of f2 #) is non empty strict multMagma
f1 " the carrier of f2 is Element of bool the carrier of G
j is Element of the carrier of s1
j is Element of the carrier of G
f1 . j is Element of the carrier of s1
dom f1 is non empty Element of bool the carrier of G
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
H1 is Relation-like the carrier of G -defined the carrier of s1 -valued Element of bool [: the carrier of G, the carrier of s1:]
j is Element of the carrier of G
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
f1 . j is Element of the carrier of s1
H1 is set
p is non empty strict unital Group-like associative Subgroup of s1
the carrier of p is non empty set
f1 " the carrier of p is Element of bool the carrier of G
f1 . H1 is set
j is set
f1 . j is set
H2 is Element of the carrier of G
j is Element of the carrier of G
s299 is Element of the carrier of G
j * s299 is Element of the carrier of G
the multF of G . (j,s299) is Element of the carrier of G
f1 . H2 is Element of the carrier of s1
f1 . s299 is Element of the carrier of s1
(f1 . H2) * (f1 . s299) is Element of the carrier of s1
the multF of s1 . ((f1 . H2),(f1 . s299)) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
H1 is set
j is Element of the carrier of G
j is Element of the carrier of G
j * j is Element of the carrier of G
the multF of G . (j,j) is Element of the carrier of G
f1 . H1 is set
f1 . j is Element of the carrier of s1
f1 . j is Element of the carrier of s1
(f1 . j) * (f1 . j) is Element of the carrier of s1
the multF of s1 . ((f1 . j),(f1 . j)) is Element of the carrier of s1
(f1 . j) * (1_ s1) is Element of the carrier of s1
the multF of s1 . ((f1 . j),(1_ s1)) is Element of the carrier of s1
f1 " (f1 .: the carrier of s29) is Element of bool the carrier of G
j is set
H1 is Element of the carrier of s1
H1 * p is Element of bool the carrier of s1
carr p is Element of bool the carrier of s1
H1 * (carr p) is Element of bool the carrier of s1
K350( the carrier of s1,H1) is non empty trivial finite 1 -element Element of bool the carrier of s1
K350( the carrier of s1,H1) * (carr p) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in K350( the carrier of s1,H1) & b₂ in carr p ) } is set
j is Element of the carrier of s1
H1 * j is Element of the carrier of s1
the multF of s1 . (H1,j) is Element of the carrier of s1
H1 " is Element of the carrier of s1
(H1 * j) * (H1 ") is Element of the carrier of s1
the multF of s1 . ((H1 * j),(H1 ")) is Element of the carrier of s1
s299 is set
f1 . s299 is set
rng f1 is non empty Element of bool the carrier of s1
H1 is set
f1 . H1 is set
H2 is Element of the carrier of G
s299 is Element of the carrier of G
H2 * s299 is Element of the carrier of G
the multF of G . (H2,s299) is Element of the carrier of G
H2 " is Element of the carrier of G
(H2 * s299) * (H2 ") is Element of the carrier of G
the multF of G . ((H2 * s299),(H2 ")) is Element of the carrier of G
(H2 ") " is Element of the carrier of G
((H2 ") ") * s299 is Element of the carrier of G
the multF of G . (((H2 ") "),s299) is Element of the carrier of G
(((H2 ") ") * s299) * (H2 ") is Element of the carrier of G
the multF of G . ((((H2 ") ") * s299),(H2 ")) is Element of the carrier of G
s299 |^ (H2 ") is Element of the carrier of G
p |^ (H2 ") is non empty strict unital Group-like associative Subgroup of G
the carrier of (p |^ (H2 ")) is non empty set
f1 . H2 is Element of the carrier of s1
f1 . s299 is Element of the carrier of s1
(f1 . H2) * (f1 . s299) is Element of the carrier of s1
the multF of s1 . ((f1 . H2),(f1 . s299)) is Element of the carrier of s1
f1 . (H2 ") is Element of the carrier of s1
((f1 . H2) * (f1 . s299)) * (f1 . (H2 ")) is Element of the carrier of s1
the multF of s1 . (((f1 . H2) * (f1 . s299)),(f1 . (H2 "))) is Element of the carrier of s1
f1 . (H2 * s299) is Element of the carrier of s1
(f1 . (H2 * s299)) * (f1 . (H2 ")) is Element of the carrier of s1
the multF of s1 . ((f1 . (H2 * s299)),(f1 . (H2 "))) is Element of the carrier of s1
f1 . ((H2 * s299) * (H2 ")) is Element of the carrier of s1
f1 .: the carrier of p is Element of bool the carrier of s1
((H1 * j) * (H1 ")) * H1 is Element of the carrier of s1
the multF of s1 . (((H1 * j) * (H1 ")),H1) is Element of the carrier of s1
(H1 ") * H1 is Element of the carrier of s1
the multF of s1 . ((H1 "),H1) is Element of the carrier of s1
(H1 * j) * ((H1 ") * H1) is Element of the carrier of s1
the multF of s1 . ((H1 * j),((H1 ") * H1)) is Element of the carrier of s1
(H1 * j) * (1_ s1) is Element of the carrier of s1
the multF of s1 . ((H1 * j),(1_ s1)) is Element of the carrier of s1
p * H1 is Element of bool the carrier of s1
(carr p) * H1 is Element of bool the carrier of s1
(carr p) * K350( the carrier of s1,H1) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in carr p & b₂ in K350( the carrier of s1,H1) ) } is set
H1 is non empty strict unital Group-like associative Subgroup of s1
O is set
G is non empty unital Group-like associative (O) (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of (O,G,s1) is non empty set
(O,G,s1) is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
(O,(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,(O,G,s1)) (O,(O,G,s1))
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
the of (O,G,s1) is Relation-like O -defined Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):]
Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1)) is functional non empty set
[:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,G,s1), the carrier of (O,G,s1))):] is non empty set
(O, the carrier of (O,G,s1), the multF of (O,G,s1), the of (O,G,s1)) is (O) (O)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s19 is non empty unital Group-like associative (O) (O) (O,(O,G,s1))
the carrier of s19 is non empty set
(O,G,s1) " the carrier of s19 is Element of bool the carrier of G
bool the carrier of G is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative Subgroup of (O,G,s1)
the carrier of s29 is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
the multF of (O,G,s1) || the carrier of s29 is set
the multF of (O,G,s1) | [: the carrier of s29, the carrier of s29:] is Relation-like [: the carrier of s29, the carrier of s29:] -defined [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like set
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
the carrier of (G ./. s2) is non empty set
[: the carrier of (G ./. s2), the carrier of (G ./. s2):] is Relation-like non empty set
the multF of (G ./. s2) is Relation-like [: the carrier of (G ./. s2), the carrier of (G ./. s2):] -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):]
[:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is non empty set
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
p is non empty strict unital Group-like associative Subgroup of G ./. s2
the carrier of p is non empty set
(nat_hom s2) " the carrier of p is Element of bool the carrier of G
i is Element of the carrier of (G ./. s2)
(Omega). (G ./. s2) is non empty strict unital Group-like associative normal Subgroup of G ./. s2
multMagma(# the carrier of (G ./. s2), the multF of (G ./. s2) #) is non empty strict multMagma
s199 is Element of the carrier of G
s199 * s2 is Element of bool the carrier of G
carr s2 is Element of bool the carrier of G
the carrier of s2 is non empty set
s199 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,s199) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,s199) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,s199) & b₂ in carr s2 ) } is set
f1 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Element of bool [: the carrier of G, the carrier of (G ./. s2):]
s199 is Element of the carrier of (G ./. s2)
[s199,s199] is set
{s199,s199} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,s199},{s199}} is non empty finite V39() set
(nat_hom s2) . s199 is Element of the carrier of (G ./. s2)
O is set
G is non empty unital Group-like associative (O) (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,s1) is non empty unital Group-like associative (O) (O)
(O,G,s1) is set
(O,G,s1) is Relation-like [:(O,G,s1),(O,G,s1):] -defined (O,G,s1) -valued Function-like quasi_total Element of bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):]
[:(O,G,s1),(O,G,s1):] is Relation-like set
[:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is Relation-like set
bool [:[:(O,G,s1),(O,G,s1):],(O,G,s1):] is non empty set
(O,G,s1) is Relation-like O -defined Funcs ((O,G,s1),(O,G,s1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,G,s1),(O,G,s1))):]
Funcs ((O,G,s1),(O,G,s1)) is functional non empty set
[:O,(Funcs ((O,G,s1),(O,G,s1))):] is Relation-like set
bool [:O,(Funcs ((O,G,s1),(O,G,s1))):] is non empty set
(O,(O,G,s1),(O,G,s1),(O,G,s1)) is (O) (O)
the carrier of s1 is non empty set
the carrier of (O,G,s1) is non empty set
(O,G,s1) is Relation-like the carrier of G -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,(O,G,s1)) Element of bool [: the carrier of G, the carrier of (O,G,s1):]
[: the carrier of G, the carrier of (O,G,s1):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (O,G,s1):] is non empty set
(O,(O,G,s1)) is non empty unital Group-like associative (O) (O) (O,(O,G,s1)) (O,(O,G,s1))
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s19 is non empty unital Group-like associative (O) (O) (O,(O,G,s1))
the carrier of s19 is non empty set
(O,G,s1) " the carrier of s19 is Element of bool the carrier of G
bool the carrier of G is non empty set
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
multMagma(# the carrier of s19, the multF of s19 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative Subgroup of (O,G,s1)
the carrier of s29 is non empty set
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
the multF of (O,G,s1) is Relation-like [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):]
[: the carrier of (O,G,s1), the carrier of (O,G,s1):] is Relation-like non empty set
[:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is Relation-like non empty set
bool [:[: the carrier of (O,G,s1), the carrier of (O,G,s1):], the carrier of (O,G,s1):] is non empty set
the multF of (O,G,s1) || the carrier of s29 is set
the multF of (O,G,s1) | [: the carrier of s29, the carrier of s29:] is Relation-like [: the carrier of s29, the carrier of s29:] -defined [: the carrier of (O,G,s1), the carrier of (O,G,s1):] -defined the carrier of (O,G,s1) -valued Function-like set
s2 is non empty strict unital Group-like associative normal Subgroup of G
G ./. s2 is non empty strict unital Group-like associative multMagma
Left_Cosets s2 is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp s2 is Relation-like [:(Left_Cosets s2),(Left_Cosets s2):] -defined Left_Cosets s2 -valued Function-like non empty total quasi_total Element of bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):]
[:(Left_Cosets s2),(Left_Cosets s2):] is Relation-like non empty set
[:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is Relation-like non empty set
bool [:[:(Left_Cosets s2),(Left_Cosets s2):],(Left_Cosets s2):] is non empty set
multMagma(# (Left_Cosets s2),(CosOp s2) #) is non empty strict multMagma
the carrier of (G ./. s2) is non empty set
[: the carrier of (G ./. s2), the carrier of (G ./. s2):] is Relation-like non empty set
the multF of (G ./. s2) is Relation-like [: the carrier of (G ./. s2), the carrier of (G ./. s2):] -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):]
[:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is Relation-like non empty set
bool [:[: the carrier of (G ./. s2), the carrier of (G ./. s2):], the carrier of (G ./. s2):] is non empty set
the carrier of s2 is non empty set
nat_hom s2 is Relation-like the carrier of G -defined the carrier of (G ./. s2) -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of (G ./. s2):]
[: the carrier of G, the carrier of (G ./. s2):] is Relation-like non empty set
bool [: the carrier of G, the carrier of (G ./. s2):] is non empty set
p is non empty strict unital Group-like associative Subgroup of G ./. s2
the carrier of p is non empty set
(nat_hom s2) " the carrier of p is Element of bool the carrier of G
1_ (O,G,s1) is non being_of_order_0 Element of the carrier of (O,G,s1)
{(1_ (O,G,s1))} is non empty trivial finite 1 -element set
f1 is set
{f1} is non empty trivial finite 1 -element set
(nat_hom s2) " {f1} is Element of bool the carrier of G
rng (nat_hom s2) is non empty Element of bool the carrier of (G ./. s2)
bool the carrier of (G ./. s2) is non empty set
Image (nat_hom s2) is non empty strict unital Group-like associative Subgroup of G ./. s2
the carrier of (Image (nat_hom s2)) is non empty set
dom (nat_hom s2) is non empty Element of bool the carrier of G
i is set
(nat_hom s2) . i is set
(O,G,s1) is Element of bool the carrier of G
s199 is Element of the carrier of G
s199 * s2 is Element of bool the carrier of G
carr s2 is Element of bool the carrier of G
s199 * (carr s2) is Element of bool the carrier of G
K350( the carrier of G,s199) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,s199) * (carr s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,s199) & b₂ in carr s2 ) } is set
O is set
G is non empty unital Group-like associative (O) (O) (O)
s1 is non empty unital Group-like associative (O) (O) (O)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
(O, the carrier of s1, the multF of s1, the of s1) is (O) (O)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
s2 is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
s2 is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of G is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
the carrier of s2 is non empty set
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
multMagma(# the carrier of s1, the multF of s1 #) is non empty strict multMagma
s29 is non empty strict unital Group-like associative normal Subgroup of s1
p is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
{(1_ s1)} is non empty trivial finite 1 -element set
f1 is set
i is Element of the carrier of s1
s199 is Element of the carrier of G
s19 . s199 is Element of the carrier of s1
{ b₁ where b₁ is Element of the carrier of G : s19 . b₁ in s29 } is set
f2 is Element of the carrier of G
s19 . f2 is Element of the carrier of s1
1_ G is non being_of_order_0 Element of the carrier of G
s19 . (1_ G) is Element of the carrier of s1
H1 is set
p is non empty set
j is Element of the carrier of G
s19 . j is Element of the carrier of s1
bool the carrier of G is non empty set
j is Element of the carrier of G
H1 is Element of bool the carrier of G
j is Element of the carrier of G
H2 is Element of the carrier of G
s19 . H2 is Element of the carrier of s1
s299 is Element of the carrier of G
s19 . s299 is Element of the carrier of s1
s299 * H2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s299,H2) is Element of the carrier of G
s19 . (s299 * H2) is Element of the carrier of s1
(s19 . s299) * (s19 . H2) is Element of the carrier of s1
the multF of s1 . ((s19 . s299),(s19 . H2)) is Element of the carrier of s1
j * j is Element of the carrier of G
the multF of G . (j,j) is Element of the carrier of G
j is Element of the carrier of G
H2 is Element of the carrier of G
s19 . H2 is Element of the carrier of s1
the carrier of s29 is non empty set
j is Element of O
(O,s1,j) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
s19 . j is Element of the carrier of s1
(O,s1,j) . (s19 . j) is Element of the carrier of s1
(O,G,j) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,j) . j is Element of the carrier of G
s19 . ((O,G,j) . j) is Element of the carrier of s1
j is Element of the carrier of G
j is Element of the carrier of G
s19 . j is Element of the carrier of s1
(s19 . j) " is Element of the carrier of s1
j " is Element of the carrier of G
s19 . (j ") is Element of the carrier of s1
j " is Element of the carrier of G
j is non empty unital Group-like associative (O) (O) (O,G)
the carrier of j is non empty set
the multF of j is Relation-like [: the carrier of j, the carrier of j:] -defined the carrier of j -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of j, the carrier of j:], the carrier of j:]
[: the carrier of j, the carrier of j:] is Relation-like non empty set
[:[: the carrier of j, the carrier of j:], the carrier of j:] is Relation-like non empty set
bool [:[: the carrier of j, the carrier of j:], the carrier of j:] is non empty set
multMagma(# the carrier of j, the multF of j #) is non empty strict multMagma
H2 is Element of the carrier of G
s19 . H2 is Element of the carrier of s1
(s19 . H2) " is Element of the carrier of s1
s29 |^ ((s19 . H2) ") is non empty strict unital Group-like associative Subgroup of s1
s299 is set
j is non empty strict unital Group-like associative Subgroup of G
H2 * j is Element of bool the carrier of G
carr j is Element of bool the carrier of G
the carrier of j is non empty set
H2 * (carr j) is Element of bool the carrier of G
K350( the carrier of G,H2) is non empty trivial finite 1 -element Element of bool the carrier of G
K350( the carrier of G,H2) * (carr j) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K350( the carrier of G,H2) & b₂ in carr j ) } is set
H1 is Element of the carrier of G
H2 * H1 is Element of the carrier of G
the multF of G . (H2,H1) is Element of the carrier of G
H2 " is Element of the carrier of G
(H2 * H1) * (H2 ") is Element of the carrier of G
the multF of G . ((H2 * H1),(H2 ")) is Element of the carrier of G
s19 . ((H2 * H1) * (H2 ")) is Element of the carrier of s1
s19 . (H2 * H1) is Element of the carrier of s1
s19 . (H2 ") is Element of the carrier of s1
(s19 . (H2 * H1)) * (s19 . (H2 ")) is Element of the carrier of s1
the multF of s1 . ((s19 . (H2 * H1)),(s19 . (H2 "))) is Element of the carrier of s1
s19 . H1 is Element of the carrier of s1
(s19 . H2) * (s19 . H1) is Element of the carrier of s1
the multF of s1 . ((s19 . H2),(s19 . H1)) is Element of the carrier of s1
((s19 . H2) * (s19 . H1)) * (s19 . (H2 ")) is Element of the carrier of s1
the multF of s1 . (((s19 . H2) * (s19 . H1)),(s19 . (H2 "))) is Element of the carrier of s1
((s19 . H2) * (s19 . H1)) * ((s19 . H2) ") is Element of the carrier of s1
the multF of s1 . (((s19 . H2) * (s19 . H1)),((s19 . H2) ")) is Element of the carrier of s1
((s19 . H2) ") " is Element of the carrier of s1
(((s19 . H2) ") ") * (s19 . H1) is Element of the carrier of s1
the multF of s1 . ((((s19 . H2) ") "),(s19 . H1)) is Element of the carrier of s1
((((s19 . H2) ") ") * (s19 . H1)) * ((s19 . H2) ") is Element of the carrier of s1
the multF of s1 . (((((s19 . H2) ") ") * (s19 . H1)),((s19 . H2) ")) is Element of the carrier of s1
(s19 . H1) |^ ((s19 . H2) ") is Element of the carrier of s1
s299 is Element of the carrier of G
s19 . s299 is Element of the carrier of s1
((H2 * H1) * (H2 ")) * H2 is Element of the carrier of G
the multF of G . (((H2 * H1) * (H2 ")),H2) is Element of the carrier of G
(H2 ") * H2 is Element of the carrier of G
the multF of G . ((H2 "),H2) is Element of the carrier of G
(H2 * H1) * ((H2 ") * H2) is Element of the carrier of G
the multF of G . ((H2 * H1),((H2 ") * H2)) is Element of the carrier of G
(H2 * H1) * (1_ G) is Element of the carrier of G
the multF of G . ((H2 * H1),(1_ G)) is Element of the carrier of G
j * H2 is Element of bool the carrier of G
(carr j) * H2 is Element of bool the carrier of G
(carr j) * K350( the carrier of G,H2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr j & b₂ in K350( the carrier of G,H2) ) } is set
H2 is non empty strict unital Group-like associative Subgroup of G
{(1_ G)} is non empty trivial finite 1 -element set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
H2 is Element of the carrier of G
s19 . H2 is Element of the carrier of s1
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of s1 is non empty set
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
(O, the carrier of s1, the multF of s1, the of s1) is (O) (O)
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
the carrier of s1 is non empty set
s2 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
s29 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 |^ s29 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s2 |^ s29) is Element of the carrier of G
s19 |^ s29 is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (s19 |^ s29) is Element of the carrier of s1
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
s199 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 |^ s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (f1 |^ s199) is Element of the carrier of G
i |^ s199 is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (i |^ s199) is Element of the carrier of s1
s199 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
f2 is Element of the carrier of G
<*f2*> is Relation-like NAT -defined the carrier of G -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of G
1 -tuples_on the carrier of G is FinSequenceSet of the carrier of G
[1,f2] is set
{1,f2} is non empty finite set
{{1,f2},{1}} is non empty finite V39() set
{[1,f2]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s199 ^ <*f2*> is Relation-like NAT -defined the carrier of G -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*f2*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s199) + (len <*f2*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
H1 is Element of the carrier of s1
<*H1*> is Relation-like NAT -defined the carrier of s1 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of s1
1 -tuples_on the carrier of s1 is FinSequenceSet of the carrier of s1
[1,H1] is set
{1,H1} is non empty finite set
{{1,H1},{1}} is non empty finite V39() set
{[1,H1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
p ^ <*H1*> is Relation-like NAT -defined the carrier of s1 -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
j is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
j is V31() V32() integer V46() ext-real Element of INT
<*j*> is Relation-like NAT -defined INT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() increasing V196() V197() V198() Element of 1 -tuples_on INT
1 -tuples_on INT is FinSequenceSet of INT
[1,j] is set
{1,j} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{1,j},{1}} is non empty finite V39() set
{[1,j]} is Relation-like Function-like constant non empty trivial finite 1 -element set
j ^ <*j*> is Relation-like NAT -defined INT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*j*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len j) + (len <*j*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*H1*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len p) + (len <*H1*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i . (p + 1) is set
(len p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(p ^ <*H1*>) . ((len p) + 1) is set
f1 . (p + 1) is set
(s199 ^ <*f2*>) . ((len s199) + 1) is set
H1 is non empty unital Group-like associative Subgroup of G
the carrier of H1 is non empty set
f2 |^ j is Element of the carrier of G
H2 is Element of the carrier of H1
H2 |^ j is Element of the carrier of H1
H1 |^ j is Element of the carrier of s1
s199 |^ j is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
<*f2*> |^ <*j*> is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
(s199 |^ j) ^ (<*f2*> |^ <*j*>) is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product ((s199 |^ j) ^ (<*f2*> |^ <*j*>)) is Element of the carrier of G
Product (s199 |^ j) is Element of the carrier of G
Product (<*f2*> |^ <*j*>) is Element of the carrier of G
(Product (s199 |^ j)) * (Product (<*f2*> |^ <*j*>)) is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((Product (s199 |^ j)),(Product (<*f2*> |^ <*j*>))) is Element of the carrier of G
@ j is V31() V32() integer V46() ext-real Element of INT
<*(@ j)*> is Relation-like NAT -defined INT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() increasing V196() V197() V198() Element of 1 -tuples_on INT
[1,(@ j)] is set
{1,(@ j)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{1,(@ j)},{1}} is non empty finite V39() set
{[1,(@ j)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*f2*> |^ <*(@ j)*> is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (<*f2*> |^ <*(@ j)*>) is Element of the carrier of G
<*(f2 |^ j)*> is Relation-like NAT -defined the carrier of G -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of G
[1,(f2 |^ j)] is set
{1,(f2 |^ j)} is non empty finite set
{{1,(f2 |^ j)},{1}} is non empty finite V39() set
{[1,(f2 |^ j)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Product <*(f2 |^ j)*> is Element of the carrier of G
<*(H1 |^ j)*> is Relation-like NAT -defined the carrier of s1 -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of s1
[1,(H1 |^ j)] is set
{1,(H1 |^ j)} is non empty finite set
{{1,(H1 |^ j)},{1}} is non empty finite V39() set
{[1,(H1 |^ j)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Product <*(H1 |^ j)*> is Element of the carrier of s1
<*H1*> |^ <*(@ j)*> is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (<*H1*> |^ <*(@ j)*>) is Element of the carrier of s1
<*H1*> |^ <*j*> is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (<*H1*> |^ <*j*>) is Element of the carrier of s1
p |^ j is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (p |^ j) is Element of the carrier of s1
(Product (p |^ j)) * (Product (<*H1*> |^ <*j*>)) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((Product (p |^ j)),(Product (<*H1*> |^ <*j*>))) is Element of the carrier of s1
(p |^ j) ^ (<*H1*> |^ <*j*>) is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product ((p |^ j) ^ (<*H1*> |^ <*j*>)) is Element of the carrier of s1
(p ^ <*H1*>) |^ (j ^ <*j*>) is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product ((p ^ <*H1*>) |^ (j ^ <*j*>)) is Element of the carrier of s1
p is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
i is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p |^ i is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (p |^ i) is Element of the carrier of G
f1 |^ i is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (f1 |^ i) is Element of the carrier of s1
len (p |^ i) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*> the carrier of G is Relation-like non-empty empty-yielding NAT -defined the carrier of G -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of the carrier of G
[:NAT, the carrier of G:] is Relation-like non empty non trivial non finite set
1_ G is non being_of_order_0 Element of the carrier of G
len (f1 |^ i) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*> the carrier of s1 is Relation-like non-empty empty-yielding NAT -defined the carrier of s1 -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() FinSequence of the carrier of s1
[:NAT, the carrier of s1:] is Relation-like non empty non trivial non finite set
1_ s1 is non being_of_order_0 Element of the carrier of s1
O is set
G is set
Funcs (G,G) is functional non empty set
[:O,(Funcs (G,G)):] is Relation-like set
bool [:O,(Funcs (G,G)):] is non empty set
s1 is set
Funcs (s1,s1) is functional non empty set
[:O,(Funcs (s1,s1)):] is Relation-like set
bool [:O,(Funcs (s1,s1)):] is non empty set
[:G,G:] is Relation-like set
bool [:G,G:] is non empty set
[:s1,s1:] is Relation-like set
bool [:s1,s1:] is non empty set
s2 is Relation-like O -defined Funcs (G,G) -valued Function-like total quasi_total Element of bool [:O,(Funcs (G,G)):]
s19 is Relation-like O -defined Funcs (s1,s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs (s1,s1)):]
s29 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s2,s29) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,s1,s19,s29) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
(O,s1,s19,s29) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
p is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s2,p) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,s1,s19,p) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
(O,s1,s19,p) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
f1 is set
id G is Relation-like G -defined G -valued Function-like one-to-one total quasi_total Element of bool [:G,G:]
dom (id G) is Element of bool G
bool G is non empty set
(id G) . f1 is set
id s1 is Relation-like s1 -defined s1 -valued Function-like one-to-one total quasi_total Element of bool [:s1,s1:]
(id s1) . f1 is set
s1 /\ G is set
dom (id s1) is Element of bool s1
bool s1 is non empty set
(dom (id s1)) /\ G is Element of bool s1
(id s1) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
s199 is Element of O
<*s199*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,s199] is set
{1,s199} is non empty finite set
{{1,s199},{1}} is non empty finite V39() set
{[1,s199]} is Relation-like Function-like constant non empty trivial finite 1 -element set
i ^ <*s199*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s199*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len i) + (len <*s199*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
{s199} is non empty trivial finite 1 -element set
rng <*s199*> is non empty trivial finite 1 -element set
f2 is set
(O,G,s2,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
dom (O,G,s2,f1) is Element of bool G
bool G is non empty set
dom s2 is Element of bool O
bool O is non empty set
s2 . s199 is Relation-like Function-like set
rng s2 is functional Element of bool (Funcs (G,G))
bool (Funcs (G,G)) is non empty set
p is Relation-like Function-like set
dom p is set
rng p is set
s199 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O,G,s2,s199) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
dom s19 is Element of bool O
s19 . s199 is Relation-like Function-like set
rng s19 is functional Element of bool (Funcs (s1,s1))
bool (Funcs (s1,s1)) is non empty set
H1 is Relation-like Function-like set
dom H1 is set
rng H1 is set
(O,s1,s19,s199) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
p . f2 is set
(O,s1,s19,f1) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
(O,s1,s19,i) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
(O,s1,s19,i) * (O,s1,s19,s199) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
H1 * (O,s1,s19,i) is Relation-like s1 -valued Function-like set
(O,G,s2,i) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,G,s2,i) * (O,G,s2,s199) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
p * (O,G,s2,i) is Relation-like G -valued Function-like set
(O,G,s2,f1) . f2 is set
(O,G,s2,i) . (p . f2) is set
(O,s1,s19,i) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
((O,s1,s19,i) | G) . (p . f2) is set
(O,s1,s19,i) . (p . f2) is set
H1 | G is Relation-like Function-like set
(H1 | G) . f2 is set
(O,s1,s19,i) . ((H1 | G) . f2) is set
H1 . f2 is set
(O,s1,s19,i) . (H1 . f2) is set
(H1 * (O,s1,s19,i)) . f2 is set
(O,s1,s19,f1) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
((O,s1,s19,f1) | G) . f2 is set
dom ((O,s1,s19,f1) | G) is Element of bool G
s1 /\ G is set
s199 is Relation-like Function-like set
dom s199 is set
rng s199 is set
s199 is Relation-like Function-like set
dom s199 is set
rng s199 is set
f1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s2,f1) is Relation-like G -defined G -valued Function-like total quasi_total Element of bool [:G,G:]
(O,s1,s19,f1) is Relation-like s1 -defined s1 -valued Function-like total quasi_total Element of bool [:s1,s1:]
(O,s1,s19,f1) | G is Relation-like G -defined s1 -defined s1 -valued Function-like Element of bool [:s1,s1:]
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,s1)
s19 is non empty unital Group-like associative (O) (O) (O,s1)
(O,s1,s2,s19) is Element of bool the carrier of s1
the carrier of s1 is non empty set
bool the carrier of s1 is non empty set
(O,s1,s2) is Element of bool the carrier of s1
the carrier of s2 is non empty set
(O,s1,s19) is Element of bool the carrier of s1
the carrier of s19 is non empty set
(O,s1,s2) * (O,s1,s19) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in (O,s1,s2) & b₂ in (O,s1,s19) ) } is set
s29 is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,p) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
(O,G,s29) is Element of bool the carrier of G
the carrier of s29 is non empty set
(O,G,p) is Element of bool the carrier of G
the carrier of p is non empty set
(O,G,s29) * (O,G,p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s29) & b₂ in (O,G,p) ) } is set
s199 is set
s199 is Element of the carrier of G
f2 is Element of the carrier of G
s199 * f2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s199,f2) is Element of the carrier of G
p is Element of the carrier of s1
H1 is Element of the carrier of s1
p * H1 is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . (p,H1) is Element of the carrier of s1
s199 is set
s199 is Element of the carrier of s1
f2 is Element of the carrier of s1
s199 * f2 is Element of the carrier of s1
the multF of s1 . (s199,f2) is Element of the carrier of s1
p is Element of the carrier of G
H1 is Element of the carrier of G
p * H1 is Element of the carrier of G
the multF of G . (p,H1) is Element of the carrier of G
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,s1)
s19 is non empty unital Group-like associative (O) (O) (O,s1)
(O,s1,s2,s19) is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s1 is non empty set
(O,s1,s2) is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
the carrier of s2 is non empty set
(O,s1,s19) is Element of bool the carrier of s1
the carrier of s19 is non empty set
(O,s1,s2) \/ (O,s1,s19) is Element of bool the carrier of s1
(O,s1,((O,s1,s2) \/ (O,s1,s19))) is non empty unital Group-like associative (O) (O) (O,s1)
(O,s1,s2,s19) is Element of bool the carrier of s1
(O,s1,s2) * (O,s1,s19) is Element of bool the carrier of s1
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of s1 : ( b₁ in (O,s1,s2) & b₂ in (O,s1,s19) ) } is set
(O,s1,(O,s1,s2,s19)) is non empty unital Group-like associative (O) (O) (O,s1)
p is non empty unital Group-like associative (O) (O) (O,G)
f1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,p,f1) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,p) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of p is non empty set
(O,G,f1) is Element of bool the carrier of G
the carrier of f1 is non empty set
(O,G,p) \/ (O,G,f1) is Element of bool the carrier of G
(O,G,((O,G,p) \/ (O,G,f1))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,p,f1) is Element of bool the carrier of G
(O,G,p) * (O,G,f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,p) & b₂ in (O,G,f1) ) } is set
(O,G,(O,G,p,f1)) is non empty unital Group-like associative (O) (O) (O,G)
{ b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & (O,G,p,f1) c= (O,G,b₂) ) } is set
{ b₁ where b₁ is Element of bool the carrier of s1 : ex b₂ being non empty unital Group-like associative (O) (O) (O,s1) st
( b₁ = the carrier of b₂ & (O,s1,s2,s19) c= (O,s1,b₂) ) } is set
the carrier of (O,G,(O,G,p,f1)) is non empty set
meet { b₁ where b₁ is Element of bool the carrier of G : ex b₂ being non empty unital Group-like associative (O) (O) (O,G) st
( b₁ = the carrier of b₂ & (O,G,p,f1) c= (O,G,b₂) ) } is set
the carrier of (O,s1,(O,s1,s2,s19)) is non empty set
meet { b₁ where b₁ is Element of bool the carrier of s1 : ex b₂ being non empty unital Group-like associative (O) (O) (O,s1) st
( b₁ = the carrier of b₂ & (O,s1,s2,s19) c= (O,s1,b₂) ) } is set
f2 is set
p is Element of bool the carrier of s1
H1 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of H1 is non empty set
(O,s1,H1) is Element of bool the carrier of s1
H1 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of H1 is non empty set
(O,s1,H1) is Element of bool the carrier of s1
j is non empty unital Group-like associative (O) (O) (O,G)
j is non empty unital Group-like associative (O) (O) (O,G)
the carrier of j is non empty set
(O,G,j) is Element of bool the carrier of G
H1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of H1 is non empty set
(O,G,H1) is Element of bool the carrier of G
(O,G,s1) is Element of bool the carrier of G
p is set
(O,s1) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the of s1 is Relation-like O -defined Funcs ( the carrier of s1, the carrier of s1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):]
Funcs ( the carrier of s1, the carrier of s1) is functional non empty set
[:O,(Funcs ( the carrier of s1, the carrier of s1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of s1, the carrier of s1)):] is non empty set
(O, the carrier of s1, the multF of s1, the of s1) is (O) (O)
j is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of j is non empty set
(O,s1,j) is Element of bool the carrier of s1
H1 is set
j is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of j is non empty set
(O,s1,j) is Element of bool the carrier of s1
f2 is set
f2 is set
p is Element of the carrier of s1
(O, the carrier of s1, the of s1,(O,s1,s2,s19)) is Element of bool the carrier of s1
j is Element of bool the carrier of s1
H1 is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
len H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng H1 is finite Element of bool the carrier of s1
H1 |^ j is Relation-like NAT -defined the carrier of s1 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of s1
Product (H1 |^ j) is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
H2 is set
(1_ s1) * (1_ s1) is Element of the carrier of s1
the multF of s1 . ((1_ s1),(1_ s1)) is Element of the carrier of s1
s299 is Element of the carrier of s1
H1 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
(O, the carrier of s1, the of s1,H1) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
H2 is Element of (O,s1,s2,s19)
(O, the carrier of s1, the of s1,H1) . H2 is set
[: the carrier of G, the carrier of G:] is Relation-like non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
H is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total Element of bool [: the carrier of s1, the carrier of s1:]
j is Element of O
the of s1 . j is Relation-like Function-like set
K is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
the of G . j is Relation-like Function-like set
(O,s1,j) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
(O,G,j) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
K | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
dom the of s1 is Element of bool O
bool O is non empty set
K | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
(O, the carrier of G, the of G,H1) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total Element of bool [: the carrier of G, the carrier of G:]
(O, the carrier of G, the of G,H1) | the carrier of s1 is Relation-like the carrier of s1 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
i is Element of (O,G,p,f1)
(O, the carrier of G, the of G,H1) . i is set
s299 is Element of the carrier of G
(O, the carrier of G, the of G,(O,G,p,f1)) is Element of bool the carrier of G
s299 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s299 |^ j is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s299 |^ j) is Element of the carrier of G
H2 is Element of the carrier of G
s29 is non empty unital Group-like associative (O) (O,G)
the carrier of s29 is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s19) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s2) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s2 is non empty set
(O,G,s19) is Element of bool the carrier of G
the carrier of s19 is non empty set
(O,G,s2) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,s2) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O,s1)
p is non empty unital Group-like associative (O) (O,s1)
(O,s1,s29,p) is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s1 is non empty set
(O,s1,s29) is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
the carrier of s29 is non empty set
(O,s1,p) is Element of bool the carrier of s1
the carrier of p is non empty set
(O,s1,s29) \/ (O,s1,p) is Element of bool the carrier of s1
(O,s1,((O,s1,s29) \/ (O,s1,p))) is non empty unital Group-like associative (O) (O) (O,s1)
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O)
the carrier of s2 is non empty set
[: the carrier of s1, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of s1, the carrier of s2:] is non empty set
s19 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s29 is Relation-like the carrier of s1 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s1,s2) Element of bool [: the carrier of s1, the carrier of s2:]
(O,G,s1,s2,s19,s29) is Relation-like the carrier of G -defined the carrier of G -defined the carrier of s2 -valued the carrier of s2 -valued Function-like non empty total quasi_total quasi_total unity-preserving multiplicative (O,G,s2) Element of bool [: the carrier of G, the carrier of s2:]
[: the carrier of G, the carrier of s2:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s2:] is non empty set
(O,G,s2,(O,G,s1,s2,s19,s29)) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of (O,G,s2,(O,G,s1,s2,s19,s29)) is non empty set
(O,s1,s2,s29) is non empty unital Group-like associative (O) (O) (O,s1) (O,s1)
the carrier of (O,s1,s2,s29) is non empty set
s19 " the carrier of (O,s1,s2,s29) is Element of bool the carrier of G
bool the carrier of G is non empty set
p is set
s19 . p is set
1_ s2 is non being_of_order_0 Element of the carrier of s2
{ b₁ where b₁ is Element of the carrier of s1 : s29 . b₁ = 1_ s2 } is set
dom s19 is non empty Element of bool the carrier of G
(O,G,s1,s2,s19,s29) . p is set
f1 is Element of the carrier of s1
s29 . f1 is Element of the carrier of s2
{ b₁ where b₁ is Element of the carrier of G : (O,G,s1,s2,s19,s29) . b₁ = 1_ s2 } is set
p is set
f1 is Element of the carrier of G
(O,G,s1,s2,s19,s29) . f1 is Element of the carrier of s2
s19 . f1 is Element of the carrier of s1
i is Element of the carrier of s1
s29 . i is Element of the carrier of s2
s19 . p is set
O is set
G is non empty unital Group-like associative (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O,G)
the carrier of s2 is non empty set
s19 is non empty unital Group-like associative (O) (O,s1)
the carrier of s19 is non empty set
[: the carrier of s2, the carrier of s19:] is Relation-like non empty set
bool [: the carrier of s2, the carrier of s19:] is non empty set
s29 is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
s29 .: the carrier of s2 is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
s29 " the carrier of s19 is Element of bool the carrier of G
bool the carrier of G is non empty set
s29 | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of G, the carrier of s1:]
dom s29 is non empty Element of bool the carrier of G
dom (s29 | the carrier of s2) is Element of bool the carrier of G
f1 is set
s29 . f1 is set
f1 is set
rng (s29 | the carrier of s2) is Element of bool the carrier of s1
i is set
(s29 | the carrier of s2) . i is set
s29 . i is set
i is Element of the carrier of s2
s199 is Element of the carrier of s2
s199 is Element of the carrier of G
s29 . s199 is Element of the carrier of s1
f1 is Relation-like the carrier of s2 -defined the carrier of s19 -valued Function-like non empty total quasi_total Element of bool [: the carrier of s2, the carrier of s19:]
f1 . i is Element of the carrier of s19
f2 is Element of the carrier of G
s29 . f2 is Element of the carrier of s1
f1 . s199 is Element of the carrier of s19
i * s199 is Element of the carrier of s2
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
the multF of s2 . (i,s199) is Element of the carrier of s2
f1 . (i * s199) is Element of the carrier of s19
s29 . (i * s199) is set
s199 * f2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (s199,f2) is Element of the carrier of G
s29 . (s199 * f2) is Element of the carrier of s1
(s29 . s199) * (s29 . f2) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((s29 . s199),(s29 . f2)) is Element of the carrier of s1
(f1 . i) * (f1 . s199) is Element of the carrier of s19
the multF of s19 is Relation-like [: the carrier of s19, the carrier of s19:] -defined the carrier of s19 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:]
[: the carrier of s19, the carrier of s19:] is Relation-like non empty set
[:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is Relation-like non empty set
bool [:[: the carrier of s19, the carrier of s19:], the carrier of s19:] is non empty set
the multF of s19 . ((f1 . i),(f1 . s199)) is Element of the carrier of s19
s199 is Element of the carrier of s2
i is Element of O
(O,s2,i) is Relation-like the carrier of s2 -defined the carrier of s2 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s2, the carrier of s2:]
bool [: the carrier of s2, the carrier of s2:] is non empty set
(O,s2,i) . s199 is Element of the carrier of s2
f1 . ((O,s2,i) . s199) is Element of the carrier of s19
s29 . ((O,s2,i) . s199) is set
(O,G,i) is Relation-like the carrier of G -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
(O,G,i) | the carrier of s2 is Relation-like the carrier of G -defined the carrier of s2 -defined the carrier of G -defined the carrier of G -valued Function-like Element of bool [: the carrier of G, the carrier of G:]
((O,G,i) | the carrier of s2) . s199 is set
s29 . (((O,G,i) | the carrier of s2) . s199) is set
s199 is Element of the carrier of G
(O,G,i) . s199 is Element of the carrier of G
s29 . ((O,G,i) . s199) is Element of the carrier of s1
(O,s1,i) is Relation-like the carrier of s1 -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s1, the carrier of s1:]
bool [: the carrier of s1, the carrier of s1:] is non empty set
s29 . s199 is Element of the carrier of s1
(O,s1,i) . (s29 . s199) is Element of the carrier of s1
f1 . s199 is Element of the carrier of s19
(O,s1,i) . (f1 . s199) is set
(O,s1,i) | the carrier of s19 is Relation-like the carrier of s1 -defined the carrier of s19 -defined the carrier of s1 -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of s1, the carrier of s1:]
((O,s1,i) | the carrier of s19) . (f1 . s199) is set
(O,s19,i) is Relation-like the carrier of s19 -defined the carrier of s19 -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of s19, the carrier of s19:]
bool [: the carrier of s19, the carrier of s19:] is non empty set
(O,s19,i) . (f1 . s199) is Element of the carrier of s19
O is set
G is non empty unital Group-like associative (O) (O) (O)
the carrier of G is non empty set
s1 is non empty unital Group-like associative (O) (O) (O)
the carrier of s1 is non empty set
[: the carrier of G, the carrier of s1:] is Relation-like non empty set
bool [: the carrier of G, the carrier of s1:] is non empty set
s2 is non empty unital Group-like associative (O) (O) (O,G)
s19 is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s29,s2)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s19 is non empty set
(O,G,(O,G,s29,s2)) is Element of bool the carrier of G
the carrier of (O,G,s29,s2) is non empty set
(O,G,s19) \/ (O,G,(O,G,s29,s2)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s29,s2)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2) is Element of bool the carrier of G
the carrier of s2 is non empty set
(O,G,s19) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29) is Element of bool the carrier of G
the carrier of s29 is non empty set
(O,G,s29) \/ (O,G,s2) is Element of bool the carrier of G
(O,G,((O,G,s29) \/ (O,G,s2))) is non empty unital Group-like associative (O) (O) (O,G)
the multF of s29 is Relation-like [: the carrier of s29, the carrier of s29:] -defined the carrier of s29 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:]
[: the carrier of s29, the carrier of s29:] is Relation-like non empty set
[:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is Relation-like non empty set
bool [:[: the carrier of s29, the carrier of s29:], the carrier of s29:] is non empty set
multMagma(# the carrier of s29, the multF of s29 #) is non empty strict multMagma
i is Relation-like the carrier of G -defined the carrier of s1 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,G,s1) Element of bool [: the carrier of G, the carrier of s1:]
(O,G,s1,i) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
i .: the carrier of s29 is Element of bool the carrier of s1
bool the carrier of s1 is non empty set
the carrier of (O,G,s29,s2) is non empty set
s199 is non empty unital Group-like associative (O) (O) (O,s1)
the carrier of s199 is non empty set
i " the carrier of s199 is Element of bool the carrier of G
[: the carrier of (O,G,s29,s2), the carrier of s199:] is Relation-like non empty set
bool [: the carrier of (O,G,s29,s2), the carrier of s199:] is non empty set
i | the carrier of (O,G,s29,s2) is Relation-like the carrier of G -defined the carrier of (O,G,s29,s2) -defined the carrier of G -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of G, the carrier of s1:]
the multF of s2 is Relation-like [: the carrier of s2, the carrier of s2:] -defined the carrier of s2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:]
[: the carrier of s2, the carrier of s2:] is Relation-like non empty set
[:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is Relation-like non empty set
bool [:[: the carrier of s2, the carrier of s2:], the carrier of s2:] is non empty set
multMagma(# the carrier of s2, the multF of s2 #) is non empty strict multMagma
p is non empty strict unital Group-like associative normal Subgroup of G
f1 is non empty strict unital Group-like associative Subgroup of G
carr f1 is Element of bool the carrier of G
the carrier of f1 is non empty set
(carr f1) * p is Element of bool the carrier of G
carr p is Element of bool the carrier of G
the carrier of p is non empty set
(carr f1) * (carr p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr f1 & b₂ in carr p ) } is set
p * (carr f1) is Element of bool the carrier of G
(carr p) * (carr f1) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in carr f1 ) } is set
(O,G,s29,s2) is Element of bool the carrier of G
(O,G,s29) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s29) & b₂ in (O,G,s2) ) } is set
(O,G,s2,s29) is Element of bool the carrier of G
(O,G,s2) * (O,G,s29) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s29) ) } is set
H1 is set
dom i is non empty Element of bool the carrier of G
j is set
i . j is set
j is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
j * (1_ G) is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (j,(1_ G)) is Element of the carrier of G
H2 is set
i . j is Element of the carrier of s1
1_ s1 is non being_of_order_0 Element of the carrier of s1
(i . j) * (1_ s1) is Element of the carrier of s1
the multF of s1 is Relation-like [: the carrier of s1, the carrier of s1:] -defined the carrier of s1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:]
[: the carrier of s1, the carrier of s1:] is Relation-like non empty set
[:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is Relation-like non empty set
bool [:[: the carrier of s1, the carrier of s1:], the carrier of s1:] is non empty set
the multF of s1 . ((i . j),(1_ s1)) is Element of the carrier of s1
i . (1_ G) is Element of the carrier of s1
(i . j) * (i . (1_ G)) is Element of the carrier of s1
the multF of s1 . ((i . j),(i . (1_ G))) is Element of the carrier of s1
i . H2 is set
i .: (O,G,s29,s2) is Element of bool the carrier of s1
H1 is set
j is set
i . j is set
j is Element of the carrier of G
H2 is Element of the carrier of G
s299 is Element of the carrier of G
H2 * s299 is Element of the carrier of G
the multF of G . (H2,s299) is Element of the carrier of G
i . H2 is Element of the carrier of s1
i . s299 is Element of the carrier of s1
(i . H2) * (i . s299) is Element of the carrier of s1
the multF of s1 . ((i . H2),(i . s299)) is Element of the carrier of s1
(i . H2) * (1_ s1) is Element of the carrier of s1
the multF of s1 . ((i . H2),(1_ s1)) is Element of the carrier of s1
f2 is Relation-like the carrier of (O,G,s29,s2) -defined the carrier of s199 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,(O,G,s29,s2),s199) Element of bool [: the carrier of (O,G,s29,s2), the carrier of s199:]
f2 .: the carrier of (O,G,s29,s2) is Element of bool the carrier of s199
bool the carrier of s199 is non empty set
i .: the carrier of (O,G,s29,s2) is Element of bool the carrier of s1
H1 is set
i .: the carrier of s19 is Element of bool the carrier of s1
f2 " (i .: the carrier of s19) is Element of bool the carrier of (O,G,s29,s2)
bool the carrier of (O,G,s29,s2) is non empty set
i " (i .: the carrier of s19) is Element of bool the carrier of G
the carrier of (O,G,s29,s2) /\ (i " (i .: the carrier of s19)) is Element of bool the carrier of G
i . H1 is set
j is set
i . j is set
H2 is Element of the carrier of G
j is Element of the carrier of G
s299 is Element of the carrier of G
j * s299 is Element of the carrier of G
the multF of G . (j,s299) is Element of the carrier of G
i . H2 is Element of the carrier of s1
i . s299 is Element of the carrier of s1
(i . H2) * (i . s299) is Element of the carrier of s1
the multF of s1 . ((i . H2),(i . s299)) is Element of the carrier of s1
(O,G,s19,s2) is Element of bool the carrier of G
(O,G,s19) * (O,G,s2) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s19) & b₂ in (O,G,s2) ) } is set
[: the carrier of s29, the carrier of s199:] is Relation-like non empty set
bool [: the carrier of s29, the carrier of s199:] is non empty set
i | the carrier of s29 is Relation-like the carrier of G -defined the carrier of s29 -defined the carrier of G -defined the carrier of s1 -valued Function-like Element of bool [: the carrier of G, the carrier of s1:]
j is set
p is non empty unital Group-like associative (O) (O,s29)
the carrier of p is non empty set
(O,G,s29) /\ (O,G,s2) is Element of bool the carrier of G
j is Element of the carrier of s29
H2 is Element of the carrier of G
i . H2 is Element of the carrier of s1
i . j is set
1_ s199 is non being_of_order_0 Element of the carrier of s199
H1 is Relation-like the carrier of s29 -defined the carrier of s199 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s29,s199) Element of bool [: the carrier of s29, the carrier of s199:]
H1 . j is Element of the carrier of s199
{ b₁ where b₁ is Element of the carrier of s29 : H1 . b₁ = 1_ s199 } is set
s199 is non empty unital Group-like associative Subgroup of G
carr s199 is Element of bool the carrier of G
the carrier of s199 is non empty set
(carr s199) * p is Element of bool the carrier of G
(carr s199) * (carr p) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr s199 & b₂ in carr p ) } is set
p * (carr s199) is Element of bool the carrier of G
(carr p) * (carr s199) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr p & b₂ in carr s199 ) } is set
(O,G,s2,s19) is Element of bool the carrier of G
(O,G,s2) * (O,G,s19) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (O,G,s2) & b₂ in (O,G,s19) ) } is set
H2 is set
s299 is Element of the carrier of s29
H1 . s299 is Element of the carrier of s199
H1 is Element of the carrier of G
i . H1 is Element of the carrier of s1
(O,s29,s199,H1) is non empty unital Group-like associative (O) (O) (O,s29) (O,s29)
j is non empty unital Group-like associative (O) (O) (O,s29)
the carrier of j is non empty set
H1 .: the carrier of j is Element of bool the carrier of s199
(O,s29,j,p) is non empty unital Group-like associative (O) (O) (O,s29)
(O,s29,j) is Element of bool the carrier of s29
bool the carrier of s29 is non empty set
(O,s29,p) is Element of bool the carrier of s29
(O,s29,j) \/ (O,s29,p) is Element of bool the carrier of s29
(O,s29,((O,s29,j) \/ (O,s29,p))) is non empty unital Group-like associative (O) (O) (O,s29)
the carrier of (O,s29,j,p) is non empty set
H2 is non empty unital Group-like associative (O) (O) (O,s199)
the carrier of H2 is non empty set
H1 " the carrier of H2 is Element of bool the carrier of s29
s299 is non empty unital Group-like associative (O) (O) (O,s29)
the carrier of s299 is non empty set
H1 .: the carrier of s29 is Element of bool the carrier of s199
(O,s29,s199,H1) is non empty unital Group-like associative (O) (O) (O,s199)
rng H1 is non empty Element of bool the carrier of s199
H1 is non empty unital Group-like associative (O) (O,s199) (O,s199)
(O,s199,H1) is Relation-like the carrier of s199 -defined the carrier of (O,s199,H1) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,s199,(O,s199,H1)) Element of bool [: the carrier of s199, the carrier of (O,s199,H1):]
(O,s199,H1) is non empty unital Group-like associative (O) (O)
(O,s199,H1) is set
(O,s199,H1) is Relation-like [:(O,s199,H1),(O,s199,H1):] -defined (O,s199,H1) -valued Function-like quasi_total Element of bool [:[:(O,s199,H1),(O,s199,H1):],(O,s199,H1):]
[:(O,s199,H1),(O,s199,H1):] is Relation-like set
[:[:(O,s199,H1),(O,s199,H1):],(O,s199,H1):] is Relation-like set
bool [:[:(O,s199,H1),(O,s199,H1):],(O,s199,H1):] is non empty set
(O,s199,H1) is Relation-like O -defined Funcs ((O,s199,H1),(O,s199,H1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,H1),(O,s199,H1))):]
Funcs ((O,s199,H1),(O,s199,H1)) is functional non empty set
[:O,(Funcs ((O,s199,H1),(O,s199,H1))):] is Relation-like set
bool [:O,(Funcs ((O,s199,H1),(O,s199,H1))):] is non empty set
(O,(O,s199,H1),(O,s199,H1),(O,s199,H1)) is (O) (O)
the carrier of (O,s199,H1) is non empty set
[: the carrier of s199, the carrier of (O,s199,H1):] is Relation-like non empty set
bool [: the carrier of s199, the carrier of (O,s199,H1):] is non empty set
(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)) is Relation-like the carrier of (O,G,s29,s2) -defined the carrier of (O,G,s29,s2) -defined the carrier of (O,s199,H1) -valued the carrier of (O,s199,H1) -valued Function-like non empty total quasi_total quasi_total unity-preserving multiplicative (O,(O,G,s29,s2),(O,s199,H1)) Element of bool [: the carrier of (O,G,s29,s2), the carrier of (O,s199,H1):]
[: the carrier of (O,G,s29,s2), the carrier of (O,s199,H1):] is Relation-like non empty set
bool [: the carrier of (O,G,s29,s2), the carrier of (O,s199,H1):] is non empty set
i is set
j is Element of the carrier of G
H is Element of the carrier of G
j * H is Element of the carrier of G
the multF of G . (j,H) is Element of the carrier of G
i . i is set
i . j is Element of the carrier of s1
i . H is Element of the carrier of s1
(i . j) * (i . H) is Element of the carrier of s1
the multF of s1 . ((i . j),(i . H)) is Element of the carrier of s1
(i . j) * (1_ s1) is Element of the carrier of s1
the multF of s1 . ((i . j),(1_ s1)) is Element of the carrier of s1
the carrier of (O,G,s19,s2) is non empty set
H1 .: the carrier of s19 is Element of bool the carrier of s199
f2 " (H1 .: the carrier of s19) is Element of bool the carrier of (O,G,s29,s2)
the carrier of H1 is non empty set
f2 " the carrier of H1 is Element of bool the carrier of (O,G,s29,s2)
j is non empty unital Group-like associative (O) (O,(O,G,s29,s2))
the carrier of j is non empty set
(O,s199,(O,s199,H1),(O,s199,H1)) is non empty unital Group-like associative (O) (O) (O,s199) (O,s199)
the carrier of (O,s199,(O,s199,H1),(O,s199,H1)) is non empty set
f2 " the carrier of (O,s199,(O,s199,H1),(O,s199,H1)) is Element of bool the carrier of (O,G,s29,s2)
(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))) is non empty unital Group-like associative (O) (O) (O,(O,G,s29,s2)) (O,(O,G,s29,s2))
the carrier of (O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))) is non empty set
(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)) is Relation-like the carrier of s29 -defined the carrier of s29 -defined the carrier of (O,s199,H1) -valued the carrier of (O,s199,H1) -valued Function-like non empty total quasi_total quasi_total unity-preserving multiplicative (O,s29,(O,s199,H1)) Element of bool [: the carrier of s29, the carrier of (O,s199,H1):]
[: the carrier of s29, the carrier of (O,s199,H1):] is Relation-like non empty set
bool [: the carrier of s29, the carrier of (O,s199,H1):] is non empty set
j is non empty unital Group-like associative (O) (O) (O,(O,G,s29,s2)) (O,(O,G,s29,s2))
(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)) .: the carrier of (O,G,s29,s2) is Element of bool the carrier of (O,s199,H1)
bool the carrier of (O,s199,H1) is non empty set
(O,s199,H1) .: (f2 .: the carrier of (O,G,s29,s2)) is Element of bool the carrier of (O,s199,H1)
(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)) .: the carrier of s29 is Element of bool the carrier of (O,s199,H1)
H1 " the carrier of (O,s199,(O,s199,H1),(O,s199,H1)) is Element of bool the carrier of s29
H1 " the carrier of H1 is Element of bool the carrier of s29
H is non empty unital Group-like associative (O) (O) (O,s29) (O,s29)
(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))) is non empty unital Group-like associative (O) (O) (O,s29) (O,s29)
(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))) is non empty unital Group-like associative (O) (O) (O,(O,s199,H1))
the carrier of (O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))) is non empty set
(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))) is non empty unital Group-like associative (O) (O) (O,(O,s199,H1))
the carrier of (O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))) is non empty set
(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))) is non empty unital Group-like associative (O) (O)
(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))) is set
(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))) is Relation-like [:(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):] -defined (O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):],(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):]
[:(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):] is Relation-like set
[:[:(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):],(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):] is Relation-like set
bool [:[:(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):],(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))):] is non empty set
(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))) is Relation-like O -defined Funcs ((O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))))):]
Funcs ((O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))))) is functional non empty set
[:O,(Funcs ((O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))))):] is non empty set
(O,(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1)))),(O,(O,G,s29,s2),(O,(O,G,s29,s2),(O,s199,H1),(O,(O,G,s29,s2),s199,(O,s199,H1),f2,(O,s199,H1))))) is (O) (O)
(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))) is non empty unital Group-like associative (O) (O)
(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))) is set
(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))) is Relation-like [:(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):] -defined (O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))) -valued Function-like quasi_total Element of bool [:[:(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):],(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):]
[:(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):] is Relation-like set
[:[:(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):],(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):] is Relation-like set
bool [:[:(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):],(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))):] is non empty set
(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))) is Relation-like O -defined Funcs ((O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))))) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))))):]
Funcs ((O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))))) is functional non empty set
[:O,(Funcs ((O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))))):] is Relation-like set
bool [:O,(Funcs ((O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))))):] is non empty set
(O,(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1)))),(O,s29,(O,s29,(O,s199,H1),(O,s29,s199,(O,s199,H1),H1,(O,s199,H1))))) is (O) (O)
(O,(O,G,s29,s2),j) is non empty unital Group-like associative (O) (O)
(O,(O,G,s29,s2),j) is set
(O,(O,G,s29,s2),j) is Relation-like [:(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j):] -defined (O,(O,G,s29,s2),j) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j):],(O,(O,G,s29,s2),j):]
[:(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j):] is Relation-like set
[:[:(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j):],(O,(O,G,s29,s2),j):] is Relation-like set
bool [:[:(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j):],(O,(O,G,s29,s2),j):] is non empty set
(O,(O,G,s29,s2),j) is Relation-like O -defined Funcs ((O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j))):]
Funcs ((O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j)) is functional non empty set
[:O,(Funcs ((O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j))):] is non empty set
(O,(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j),(O,(O,G,s29,s2),j)) is (O) (O)
(O,s29,H) is non empty unital Group-like associative (O) (O)
(O,s29,H) is set
(O,s29,H) is Relation-like [:(O,s29,H),(O,s29,H):] -defined (O,s29,H) -valued Function-like quasi_total Element of bool [:[:(O,s29,H),(O,s29,H):],(O,s29,H):]
[:(O,s29,H),(O,s29,H):] is Relation-like set
[:[:(O,s29,H),(O,s29,H):],(O,s29,H):] is Relation-like set
bool [:[:(O,s29,H),(O,s29,H):],(O,s29,H):] is non empty set
(O,s29,H) is Relation-like O -defined Funcs ((O,s29,H),(O,s29,H)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s29,H),(O,s29,H))):]
Funcs ((O,s29,H),(O,s29,H)) is functional non empty set
[:O,(Funcs ((O,s29,H),(O,s29,H))):] is Relation-like set
bool [:O,(Funcs ((O,s29,H),(O,s29,H))):] is non empty set
(O,(O,s29,H),(O,s29,H),(O,s29,H)) is (O) (O)
i is non empty unital Group-like associative (O) (O) (O,(O,G,s29,s2)) (O,(O,G,s29,s2))
j is non empty unital Group-like associative (O) (O) (O,s29) (O,s29)
(O,(O,G,s29,s2),i) is non empty unital Group-like associative (O) (O)
(O,(O,G,s29,s2),i) is set
(O,(O,G,s29,s2),i) is Relation-like [:(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i):] -defined (O,(O,G,s29,s2),i) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i):],(O,(O,G,s29,s2),i):]
[:(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i):] is Relation-like set
[:[:(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i):],(O,(O,G,s29,s2),i):] is Relation-like set
bool [:[:(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i):],(O,(O,G,s29,s2),i):] is non empty set
(O,(O,G,s29,s2),i) is Relation-like O -defined Funcs ((O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i))):]
Funcs ((O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i)) is functional non empty set
[:O,(Funcs ((O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i))):] is non empty set
(O,(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i),(O,(O,G,s29,s2),i)) is (O) (O)
(O,s29,j) is non empty unital Group-like associative (O) (O)
(O,s29,j) is set
(O,s29,j) is Relation-like [:(O,s29,j),(O,s29,j):] -defined (O,s29,j) -valued Function-like quasi_total Element of bool [:[:(O,s29,j),(O,s29,j):],(O,s29,j):]
[:(O,s29,j),(O,s29,j):] is Relation-like set
[:[:(O,s29,j),(O,s29,j):],(O,s29,j):] is Relation-like set
bool [:[:(O,s29,j),(O,s29,j):],(O,s29,j):] is non empty set
(O,s29,j) is Relation-like O -defined Funcs ((O,s29,j),(O,s29,j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s29,j),(O,s29,j))):]
Funcs ((O,s29,j),(O,s29,j)) is functional non empty set
[:O,(Funcs ((O,s29,j),(O,s29,j))):] is Relation-like set
bool [:O,(Funcs ((O,s29,j),(O,s29,j))):] is non empty set
(O,(O,s29,j),(O,s29,j),(O,s29,j)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
s19 is non empty unital Group-like associative (O) (O) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s2)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s19) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s19 is non empty set
(O,G,(O,G,s1,s2)) is Element of bool the carrier of G
the carrier of (O,G,s1,s2) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s2)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s2)))) is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s29) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s29)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s29)) is Element of bool the carrier of G
the carrier of (O,G,s1,s29) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s29)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s29)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s19,s2),(O,G,s29,s1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s19,s2)) is Element of bool the carrier of G
the carrier of (O,G,s19,s2) is non empty set
(O,G,(O,G,s29,s1)) is Element of bool the carrier of G
the carrier of (O,G,s29,s1) is non empty set
(O,G,(O,G,s19,s2)) \/ (O,G,(O,G,s29,s1)) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s19,s2)) \/ (O,G,(O,G,s29,s1)))) is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O)
s199 is non empty unital Group-like associative (O) (O,(O,G,s19,(O,G,s1,s2))) (O,(O,G,s19,(O,G,s1,s2)))
(O,(O,G,s19,(O,G,s1,s2)),s199) is non empty unital Group-like associative (O) (O)
(O,(O,G,s19,(O,G,s1,s2)),s199) is set
(O,(O,G,s19,(O,G,s1,s2)),s199) is Relation-like [:(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199):] -defined (O,(O,G,s19,(O,G,s1,s2)),s199) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199):],(O,(O,G,s19,(O,G,s1,s2)),s199):]
[:(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199):] is Relation-like set
[:[:(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199):],(O,(O,G,s19,(O,G,s1,s2)),s199):] is Relation-like set
bool [:[:(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199):],(O,(O,G,s19,(O,G,s1,s2)),s199):] is non empty set
(O,(O,G,s19,(O,G,s1,s2)),s199) is Relation-like O -defined Funcs ((O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199))):]
Funcs ((O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199)) is functional non empty set
[:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199))):] is non empty set
(O,(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199),(O,(O,G,s19,(O,G,s1,s2)),s199)) is (O) (O)
f2 is non empty unital Group-like associative (O) (O,(O,G,s1,s2)) (O,(O,G,s1,s2))
(O,(O,G,s1,s2),f2) is non empty unital Group-like associative (O) (O)
(O,(O,G,s1,s2),f2) is set
(O,(O,G,s1,s2),f2) is Relation-like [:(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2):] -defined (O,(O,G,s1,s2),f2) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2):],(O,(O,G,s1,s2),f2):]
[:(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2):] is Relation-like set
[:[:(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2):],(O,(O,G,s1,s2),f2):] is Relation-like set
bool [:[:(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2):],(O,(O,G,s1,s2),f2):] is non empty set
(O,(O,G,s1,s2),f2) is Relation-like O -defined Funcs ((O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2))):]
Funcs ((O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2)) is functional non empty set
[:O,(Funcs ((O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2))):] is non empty set
(O,(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2),(O,(O,G,s1,s2),f2)) is (O) (O)
s199 is non empty unital Group-like associative (O) (O) (O,p)
p is non empty unital Group-like associative (O) (O,p) (O,p)
H1 is non empty unital Group-like associative (O) (O,p)
(O,p,s199,H1) is non empty unital Group-like associative (O) (O) (O,p)
(O,p,s199,H1) is non empty unital Group-like associative (O) (O) (O,p)
the carrier of p is non empty set
(O,p,s199) is Element of bool the carrier of p
bool the carrier of p is non empty set
the carrier of s199 is non empty set
(O,p,H1) is Element of bool the carrier of p
the carrier of H1 is non empty set
(O,p,s199) \/ (O,p,H1) is Element of bool the carrier of p
(O,p,((O,p,s199) \/ (O,p,H1))) is non empty unital Group-like associative (O) (O) (O,p)
(O,G,(O,G,s1,s2),s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s2)) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s1,s2)) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s2),s19) is non empty unital Group-like associative (O) (O) (O,G)
j is non empty unital Group-like associative (O) (O,p)
(O,p,j,(O,p,s199,H1)) is non empty unital Group-like associative (O) (O) (O,p)
(O,p,j) is Element of bool the carrier of p
the carrier of j is non empty set
(O,p,(O,p,s199,H1)) is Element of bool the carrier of p
the carrier of (O,p,s199,H1) is non empty set
(O,p,j) \/ (O,p,(O,p,s199,H1)) is Element of bool the carrier of p
(O,p,((O,p,j) \/ (O,p,(O,p,s199,H1)))) is non empty unital Group-like associative (O) (O) (O,p)
(O,G,(O,G,s1,s29),(O,G,(O,G,s1,s2),s19)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,(O,G,s1,s2),s19)) is Element of bool the carrier of G
the carrier of (O,G,(O,G,s1,s2),s19) is non empty set
(O,G,(O,G,s1,s29)) \/ (O,G,(O,G,(O,G,s1,s2),s19)) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s1,s29)) \/ (O,G,(O,G,(O,G,s1,s2),s19)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s19,s1),s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,(O,G,s19,s1),s2),(O,G,s29,s1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,(O,G,s19,s1),s2)) is Element of bool the carrier of G
the carrier of (O,G,(O,G,s19,s1),s2) is non empty set
(O,G,(O,G,(O,G,s19,s1),s2)) \/ (O,G,(O,G,s29,s1)) is Element of bool the carrier of G
(O,G,((O,G,(O,G,(O,G,s19,s1),s2)) \/ (O,G,(O,G,s29,s1)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,p,p) is non empty unital Group-like associative (O) (O)
(O,p,p) is set
(O,p,p) is Relation-like [:(O,p,p),(O,p,p):] -defined (O,p,p) -valued Function-like quasi_total Element of bool [:[:(O,p,p),(O,p,p):],(O,p,p):]
[:(O,p,p),(O,p,p):] is Relation-like set
[:[:(O,p,p),(O,p,p):],(O,p,p):] is Relation-like set
bool [:[:(O,p,p),(O,p,p):],(O,p,p):] is non empty set
(O,p,p) is Relation-like O -defined Funcs ((O,p,p),(O,p,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,p),(O,p,p))):]
Funcs ((O,p,p),(O,p,p)) is functional non empty set
[:O,(Funcs ((O,p,p),(O,p,p))):] is Relation-like set
bool [:O,(Funcs ((O,p,p),(O,p,p))):] is non empty set
(O,(O,p,p),(O,p,p),(O,p,p)) is (O) (O)
H2 is non empty unital Group-like associative (O) (O)
the carrier of H2 is non empty set
[: the carrier of p, the carrier of H2:] is Relation-like non empty set
bool [: the carrier of p, the carrier of H2:] is non empty set
(O,p,p) is Relation-like the carrier of p -defined the carrier of (O,p,p) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,p,(O,p,p)) Element of bool [: the carrier of p, the carrier of (O,p,p):]
the carrier of (O,p,p) is non empty set
[: the carrier of p, the carrier of (O,p,p):] is Relation-like non empty set
bool [: the carrier of p, the carrier of (O,p,p):] is non empty set
s299 is Relation-like the carrier of p -defined the carrier of H2 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,p,H2) Element of bool [: the carrier of p, the carrier of H2:]
(O,p,H2,s299) is non empty unital Group-like associative (O) (O) (O,p) (O,p)
(O,p,j,H1) is non empty unital Group-like associative (O) (O) (O,p)
(O,p,j) \/ (O,p,H1) is Element of bool the carrier of p
(O,p,((O,p,j) \/ (O,p,H1))) is non empty unital Group-like associative (O) (O) (O,p)
(O,G,(O,G,s1,s29),s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s29)) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s1,s29)) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s19 is non empty unital Group-like associative (O) (O) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s29) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s29)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s19) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s19 is non empty set
(O,G,(O,G,s1,s29)) is Element of bool the carrier of G
the carrier of (O,G,s1,s29) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s29)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s29)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s2)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s2)) is Element of bool the carrier of G
the carrier of (O,G,s1,s2) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s2)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s2)))) is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O)
i is non empty unital Group-like associative (O) (O,p) (O,p)
(O,p,i) is non empty unital Group-like associative (O) (O)
(O,p,i) is set
(O,p,i) is Relation-like [:(O,p,i),(O,p,i):] -defined (O,p,i) -valued Function-like quasi_total Element of bool [:[:(O,p,i),(O,p,i):],(O,p,i):]
[:(O,p,i),(O,p,i):] is Relation-like set
[:[:(O,p,i),(O,p,i):],(O,p,i):] is Relation-like set
bool [:[:(O,p,i),(O,p,i):],(O,p,i):] is non empty set
(O,p,i) is Relation-like O -defined Funcs ((O,p,i),(O,p,i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,i),(O,p,i))):]
Funcs ((O,p,i),(O,p,i)) is functional non empty set
[:O,(Funcs ((O,p,i),(O,p,i))):] is Relation-like set
bool [:O,(Funcs ((O,p,i),(O,p,i))):] is non empty set
(O,(O,p,i),(O,p,i),(O,p,i)) is (O) (O)
the carrier of p is non empty set
s199 is non empty unital Group-like associative (O) (O)
the carrier of s199 is non empty set
[: the carrier of p, the carrier of s199:] is Relation-like non empty set
bool [: the carrier of p, the carrier of s199:] is non empty set
(O,p,i) is Relation-like the carrier of p -defined the carrier of (O,p,i) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,p,(O,p,i)) Element of bool [: the carrier of p, the carrier of (O,p,i):]
the carrier of (O,p,i) is non empty set
[: the carrier of p, the carrier of (O,p,i):] is Relation-like non empty set
bool [: the carrier of p, the carrier of (O,p,i):] is non empty set
f1 is non empty unital Group-like associative (O) (O) (O,p)
s199 is non empty unital Group-like associative (O) (O,p)
f2 is Relation-like the carrier of p -defined the carrier of s199 -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,p,s199) Element of bool [: the carrier of p, the carrier of s199:]
(O,p,s199,f2) is non empty unital Group-like associative (O) (O) (O,p) (O,p)
(O,p,f1,s199) is non empty unital Group-like associative (O) (O) (O,p)
(O,p,f1) is Element of bool the carrier of p
bool the carrier of p is non empty set
the carrier of f1 is non empty set
(O,p,s199) is Element of bool the carrier of p
the carrier of s199 is non empty set
(O,p,f1) \/ (O,p,s199) is Element of bool the carrier of p
(O,p,((O,p,f1) \/ (O,p,s199))) is non empty unital Group-like associative (O) (O) (O,p)
(O,G,(O,G,s1,s2),s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s2)) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s1,s2)) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
H1 is non empty unital Group-like associative (O) (O) (O,p)
(O,p,H1,s199) is non empty unital Group-like associative (O) (O) (O,p)
(O,p,H1) is Element of bool the carrier of p
the carrier of H1 is non empty set
(O,p,H1) \/ (O,p,s199) is Element of bool the carrier of p
(O,p,((O,p,H1) \/ (O,p,s199))) is non empty unital Group-like associative (O) (O) (O,p)
(O,G,(O,G,s1,s29),s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s29)) \/ (O,G,s19) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s1,s29)) \/ (O,G,s19))) is non empty unital Group-like associative (O) (O) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
s19 is non empty unital Group-like associative (O) (O) (O,G)
s1 is non empty unital Group-like associative (O) (O) (O,G)
s2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s2)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s19) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s19 is non empty set
(O,G,(O,G,s1,s2)) is Element of bool the carrier of G
the carrier of (O,G,s1,s2) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s2)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s2)))) is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,(O,G,s2,s1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29) is Element of bool the carrier of G
the carrier of s29 is non empty set
(O,G,(O,G,s2,s1)) is Element of bool the carrier of G
the carrier of (O,G,s2,s1) is non empty set
(O,G,s29) \/ (O,G,(O,G,s2,s1)) is Element of bool the carrier of G
(O,G,((O,G,s29) \/ (O,G,(O,G,s2,s1)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s29) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s19,(O,G,s1,s29)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s1,s29)) is Element of bool the carrier of G
the carrier of (O,G,s1,s29) is non empty set
(O,G,s19) \/ (O,G,(O,G,s1,s29)) is Element of bool the carrier of G
(O,G,((O,G,s19) \/ (O,G,(O,G,s1,s29)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s2,s19) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,(O,G,s2,s19)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s2,s19)) is Element of bool the carrier of G
the carrier of (O,G,s2,s19) is non empty set
(O,G,s29) \/ (O,G,(O,G,s2,s19)) is Element of bool the carrier of G
(O,G,((O,G,s29) \/ (O,G,(O,G,s2,s19)))) is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O,(O,G,s19,(O,G,s1,s2))) (O,(O,G,s19,(O,G,s1,s2)))
(O,(O,G,s19,(O,G,s1,s2)),p) is non empty unital Group-like associative (O) (O)
(O,(O,G,s19,(O,G,s1,s2)),p) is set
(O,(O,G,s19,(O,G,s1,s2)),p) is Relation-like [:(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p):] -defined (O,(O,G,s19,(O,G,s1,s2)),p) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p):],(O,(O,G,s19,(O,G,s1,s2)),p):]
[:(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p):] is Relation-like set
[:[:(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p):],(O,(O,G,s19,(O,G,s1,s2)),p):] is Relation-like set
bool [:[:(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p):],(O,(O,G,s19,(O,G,s1,s2)),p):] is non empty set
(O,(O,G,s19,(O,G,s1,s2)),p) is Relation-like O -defined Funcs ((O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p))):]
Funcs ((O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p)) is functional non empty set
[:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p))):] is non empty set
(O,(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p),(O,(O,G,s19,(O,G,s1,s2)),p)) is (O) (O)
f1 is non empty unital Group-like associative (O) (O,(O,G,s29,(O,G,s2,s1))) (O,(O,G,s29,(O,G,s2,s1)))
(O,(O,G,s29,(O,G,s2,s1)),f1) is non empty unital Group-like associative (O) (O)
(O,(O,G,s29,(O,G,s2,s1)),f1) is set
(O,(O,G,s29,(O,G,s2,s1)),f1) is Relation-like [:(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1):] -defined (O,(O,G,s29,(O,G,s2,s1)),f1) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1):],(O,(O,G,s29,(O,G,s2,s1)),f1):]
[:(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1):] is Relation-like set
[:[:(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1):],(O,(O,G,s29,(O,G,s2,s1)),f1):] is Relation-like set
bool [:[:(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1):],(O,(O,G,s29,(O,G,s2,s1)),f1):] is non empty set
(O,(O,G,s29,(O,G,s2,s1)),f1) is Relation-like O -defined Funcs ((O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1))):]
Funcs ((O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1)) is functional non empty set
[:O,(Funcs ((O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1))):] is non empty set
(O,(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1),(O,(O,G,s29,(O,G,s2,s1)),f1)) is (O) (O)
(O,G,s19,s2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s29,s1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s19,s2),(O,G,s29,s1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s19,s2)) is Element of bool the carrier of G
the carrier of (O,G,s19,s2) is non empty set
(O,G,(O,G,s29,s1)) is Element of bool the carrier of G
the carrier of (O,G,s29,s1) is non empty set
(O,G,(O,G,s19,s2)) \/ (O,G,(O,G,s29,s1)) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s19,s2)) \/ (O,G,(O,G,s29,s1)))) is non empty unital Group-like associative (O) (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,(O,G,s1,s2)) (O,(O,G,s1,s2))
(O,G,(O,G,s29,s1),(O,G,s19,s2)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,s29,s1)) \/ (O,G,(O,G,s19,s2)) is Element of bool the carrier of G
(O,G,((O,G,(O,G,s29,s1)) \/ (O,G,(O,G,s19,s2)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,(O,G,s1,s2),s199) is non empty unital Group-like associative (O) (O)
(O,(O,G,s1,s2),s199) is set
(O,(O,G,s1,s2),s199) is Relation-like [:(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199):] -defined (O,(O,G,s1,s2),s199) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199):],(O,(O,G,s1,s2),s199):]
[:(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199):] is Relation-like set
[:[:(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199):],(O,(O,G,s1,s2),s199):] is Relation-like set
bool [:[:(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199):],(O,(O,G,s1,s2),s199):] is non empty set
(O,(O,G,s1,s2),s199) is Relation-like O -defined Funcs ((O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199))):]
Funcs ((O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199)) is functional non empty set
[:O,(Funcs ((O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199))):] is non empty set
(O,(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199),(O,(O,G,s1,s2),s199)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
<*(O,G),(O,G)*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(O,G)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(O,G)] is set
{1,(O,G)} is non empty finite set
{{1,(O,G)},{1}} is non empty finite V39() set
{[1,(O,G)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(O,G)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(O,G)] is set
{1,(O,G)} is non empty finite set
{{1,(O,G)},{1}} is non empty finite V39() set
{[1,(O,G)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*(O,G)*> ^ <*(O,G)*> is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
s1 is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 . (len s2) is set
s2 . 2 is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . s19 is set
s2 . (s19 + 1) is set
s29 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
f1 is non empty unital Group-like associative (O) (O,s29)
the carrier of f1 is non empty set
the multF of f1 is Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
[: the carrier of f1, the carrier of f1:] is Relation-like non empty set
[:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is Relation-like non empty set
bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
multMagma(# the carrier of f1, the multF of f1 #) is non empty strict multMagma
i is non empty strict unital Group-like associative Subgroup of s29
the carrier of i is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element set
s199 is non empty unital Group-like associative Subgroup of G
1_ s199 is non being_of_order_0 Element of the carrier of s199
the carrier of s199 is non empty set
{(1_ s199)} is non empty trivial finite 1 -element set
(1). s199 is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of s199
s29 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
s2 . 1 is set
O is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p is set
f1 is set
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Sgm f1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Seg (len s19) is finite len s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s19 ) } is set
Sgm (Seg (len s19)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
idseq (len s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s19 * (Sgm f1) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is non empty unital Group-like associative (O) (O)
(O,s1) is non empty set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg s19 is finite s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s19 ) } is set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s1) - 1) + 1 is V31() V32() integer ext-real set
0 + s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s29 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s29 is set
s1 . (s29 + 1) is set
f1 is Element of (O,G)
i is Element of (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
f2 is non empty unital Group-like associative (O) (O,s199) (O,s199)
(O,s199,f2) is non empty unital Group-like associative (O) (O)
(O,s199,f2) is set
(O,s199,f2) is Relation-like [:(O,s199,f2),(O,s199,f2):] -defined (O,s199,f2) -valued Function-like quasi_total Element of bool [:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):]
[:(O,s199,f2),(O,s199,f2):] is Relation-like set
[:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):] is Relation-like set
bool [:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):] is non empty set
(O,s199,f2) is Relation-like O -defined Funcs ((O,s199,f2),(O,s199,f2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,f2),(O,s199,f2))):]
Funcs ((O,s199,f2),(O,s199,f2)) is functional non empty set
[:O,(Funcs ((O,s199,f2),(O,s199,f2))):] is Relation-like set
bool [:O,(Funcs ((O,s199,f2),(O,s199,f2))):] is non empty set
(O,(O,s199,f2),(O,s199,f2),(O,s199,f2)) is (O) (O)
p is non empty unital Group-like associative (O) (O,G)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . j is set
H1 is non empty unital Group-like associative (O) (O,p) (O,p)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (j + 1) is set
(O,p,H1) is non empty unital Group-like associative (O) (O)
(O,p,H1) is set
(O,p,H1) is Relation-like [:(O,p,H1),(O,p,H1):] -defined (O,p,H1) -valued Function-like quasi_total Element of bool [:[:(O,p,H1),(O,p,H1):],(O,p,H1):]
[:(O,p,H1),(O,p,H1):] is Relation-like set
[:[:(O,p,H1),(O,p,H1):],(O,p,H1):] is Relation-like set
bool [:[:(O,p,H1),(O,p,H1):],(O,p,H1):] is non empty set
(O,p,H1) is Relation-like O -defined Funcs ((O,p,H1),(O,p,H1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,H1),(O,p,H1))):]
Funcs ((O,p,H1),(O,p,H1)) is functional non empty set
[:O,(Funcs ((O,p,H1),(O,p,H1))):] is Relation-like set
bool [:O,(Funcs ((O,p,H1),(O,p,H1))):] is non empty set
(O,(O,p,H1),(O,p,H1),(O,p,H1)) is (O) (O)
s29 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i is non empty unital Group-like associative (O) (O,G)
s1 . f1 is set
s199 is non empty unital Group-like associative (O) (O,i) (O,i)
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (f1 + 1) is set
p . f1 is set
(O,i,s199) is non empty unital Group-like associative (O) (O)
(O,i,s199) is set
(O,i,s199) is Relation-like [:(O,i,s199),(O,i,s199):] -defined (O,i,s199) -valued Function-like quasi_total Element of bool [:[:(O,i,s199),(O,i,s199):],(O,i,s199):]
[:(O,i,s199),(O,i,s199):] is Relation-like set
[:[:(O,i,s199),(O,i,s199):],(O,i,s199):] is Relation-like set
bool [:[:(O,i,s199),(O,i,s199):],(O,i,s199):] is non empty set
(O,i,s199) is Relation-like O -defined Funcs ((O,i,s199),(O,i,s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,i,s199),(O,i,s199))):]
Funcs ((O,i,s199),(O,i,s199)) is functional non empty set
[:O,(Funcs ((O,i,s199),(O,i,s199))):] is Relation-like set
bool [:O,(Funcs ((O,i,s199),(O,i,s199))):] is non empty set
(O,(O,i,s199),(O,i,s199),(O,i,s199)) is (O) (O)
s2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s19) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . s29 is set
s19 . s29 is set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
((len s1) - 1) + 1 is V31() V32() integer ext-real set
0 + s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s29 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s29 is set
s1 . (s29 + 1) is set
f1 is Element of (O,G)
i is Element of (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
f2 is non empty unital Group-like associative (O) (O,s199) (O,s199)
(O,s199,f2) is non empty unital Group-like associative (O) (O)
(O,s199,f2) is set
(O,s199,f2) is Relation-like [:(O,s199,f2),(O,s199,f2):] -defined (O,s199,f2) -valued Function-like quasi_total Element of bool [:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):]
[:(O,s199,f2),(O,s199,f2):] is Relation-like set
[:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):] is Relation-like set
bool [:[:(O,s199,f2),(O,s199,f2):],(O,s199,f2):] is non empty set
(O,s199,f2) is Relation-like O -defined Funcs ((O,s199,f2),(O,s199,f2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,f2),(O,s199,f2))):]
Funcs ((O,s199,f2),(O,s199,f2)) is functional non empty set
[:O,(Funcs ((O,s199,f2),(O,s199,f2))):] is Relation-like set
bool [:O,(Funcs ((O,s199,f2),(O,s199,f2))):] is non empty set
(O,(O,s199,f2),(O,s199,f2),(O,s199,f2)) is (O) (O)
p is non empty unital Group-like associative (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . s29 is set
f1 is non empty unital Group-like associative (O) (O,p) (O,p)
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s29 + 1) is set
s2 . s29 is set
(O,p,f1) is non empty unital Group-like associative (O) (O)
(O,p,f1) is set
(O,p,f1) is Relation-like [:(O,p,f1),(O,p,f1):] -defined (O,p,f1) -valued Function-like quasi_total Element of bool [:[:(O,p,f1),(O,p,f1):],(O,p,f1):]
[:(O,p,f1),(O,p,f1):] is Relation-like set
[:[:(O,p,f1),(O,p,f1):],(O,p,f1):] is Relation-like set
bool [:[:(O,p,f1),(O,p,f1):],(O,p,f1):] is non empty set
(O,p,f1) is Relation-like O -defined Funcs ((O,p,f1),(O,p,f1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,f1),(O,p,f1))):]
Funcs ((O,p,f1),(O,p,f1)) is functional non empty set
[:O,(Funcs ((O,p,f1),(O,p,f1))):] is Relation-like set
bool [:O,(Funcs ((O,p,f1),(O,p,f1))):] is non empty set
(O,(O,p,f1),(O,p,f1),(O,p,f1)) is (O) (O)
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . i is set
s199 is non empty unital Group-like associative (O) (O,s199) (O,s199)
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (i + 1) is set
s19 . i is set
(O,s199,s199) is non empty unital Group-like associative (O) (O)
(O,s199,s199) is set
(O,s199,s199) is Relation-like [:(O,s199,s199),(O,s199,s199):] -defined (O,s199,s199) -valued Function-like quasi_total Element of bool [:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):]
[:(O,s199,s199),(O,s199,s199):] is Relation-like set
[:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):] is Relation-like set
bool [:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):] is non empty set
(O,s199,s199) is Relation-like O -defined Funcs ((O,s199,s199),(O,s199,s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,s199),(O,s199,s199))):]
Funcs ((O,s199,s199),(O,s199,s199)) is functional non empty set
[:O,(Funcs ((O,s199,s199),(O,s199,s199))):] is Relation-like set
bool [:O,(Funcs ((O,s199,s199),(O,s199,s199))):] is non empty set
(O,(O,s199,s199),(O,s199,s199),(O,s199,s199)) is (O) (O)
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom G),(dom G):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom G),(dom G):] is non empty finite V39() set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s19 is set
s1 . (s19 + 1) is set
Del (s1,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s19} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {s19})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Del (s1,s19)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is non empty unital Group-like associative (O) (O,G)
s2 . p is set
i is non empty unital Group-like associative (O) (O,G)
s2 . (p + 1) is set
s1 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s2 . (len s2) is set
s1 . (1 + 1) is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 . 2 is set
s2 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s1 . (1 + 1) is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (s1,p) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{p} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {p} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {p}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {p})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
(Del (s1,p)) . 1 is set
s1 . 1 is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
1 + (len s2) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (f1 + 1) is non empty finite f1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= f1 + 1 ) } is set
i is non empty unital Group-like associative (O) (O,G)
s2 . p is set
s199 is non empty unital Group-like associative (O) (O,G)
s2 . (p + 1) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (s1,H1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{H1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {H1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {H1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {H1})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(Del (s1,H1)) . p is set
s1 . p is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (s1,j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{j} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {j} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {j}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {j})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(Del (s1,j)) . j is set
s1 . j is set
(Del (s1,H1)) . f1 is set
s1 . (f1 + 1) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (s1,H1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{H1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {H1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {H1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {H1})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(Del (s1,H1)) . p is set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (p + 1) is set
0 + f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (s1,j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{j} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {j} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {j}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {j})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
(Del (s1,j)) . j is set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (j + 1) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . (len s2) is set
s1 . (len s1) is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s1) - (len s2) is V31() V32() integer ext-real set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is set
Sgm s29 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s29) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
f1 is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p is finite set
Sgm p is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
dom (Sgm p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
card p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
Seg (card p) is finite card p -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= card p ) } is set
card (dom s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
Seg (card (dom s1)) is finite card (dom s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= card (dom s1) ) } is set
card (Seg (len s1)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
Seg (card (Seg (len s1))) is finite card (Seg (len s1)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= card (Seg (len s1)) ) } is set
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
(len s2) - (len s2) is V31() V32() integer ext-real set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s2) + f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is set
Sgm s29 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s29) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s19 is finite set
s19 \ s29 is finite Element of bool s19
bool s19 is non empty finite V39() set
rng (Sgm s29) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (s1 * (Sgm s29)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (Sgm s29) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
f1 is finite set
Sgm f1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
dom (Sgm f1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len (Sgm f1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
card f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
s19 \/ f1 is finite set
i is finite set
f1 \/ i is finite set
card (f1 \/ i) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
card i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
(card f1) + (card i) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
idseq (len s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s1) + s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
O is Relation-like set
rng O is set
G is Relation-like set
(rng O) |` G is Relation-like set
O ~ is Relation-like set
G ~ is Relation-like set
dom (O ~) is set
(G ~) | (dom (O ~)) is Relation-like set
s1 is set
s2 is set
[s1,s2] is set
{s1,s2} is non empty finite set
{s1} is non empty trivial finite 1 -element set
{{s1,s2},{s1}} is non empty finite V39() set
[s2,s1] is set
{s2,s1} is non empty finite set
{s2} is non empty trivial finite 1 -element set
{{s2,s1},{s2}} is non empty finite V39() set
s1 is set
s2 is set
[s1,s2] is set
{s1,s2} is non empty finite set
{s1} is non empty trivial finite 1 -element set
{{s1,s2},{s1}} is non empty finite V39() set
[s2,s1] is set
{s2,s1} is non empty finite set
{s2} is non empty trivial finite 1 -element set
{{s2,s1},{s2}} is non empty finite V39() set
O is set
G is Relation-like set
s1 is Relation-like set
s1 | O is Relation-like set
G * (s1 | O) is Relation-like set
O |` G is Relation-like set
(O |` G) * s1 is Relation-like set
s2 is set
s19 is set
s29 is set
[s19,s29] is set
{s19,s29} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,s29},{s19}} is non empty finite V39() set
p is set
[s19,p] is set
{s19,p} is non empty finite set
{{s19,p},{s19}} is non empty finite V39() set
[p,s29] is set
{p,s29} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,s29},{p}} is non empty finite V39() set
s2 is set
s19 is set
s29 is set
[s19,s29] is set
{s19,s29} is non empty finite set
{s19} is non empty trivial finite 1 -element set
{{s19,s29},{s19}} is non empty finite V39() set
p is set
[s19,p] is set
{s19,p} is non empty finite set
{{s19,p},{s19}} is non empty finite V39() set
[p,s29] is set
{p,s29} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,s29},{p}} is non empty finite V39() set
{0} is functional non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
NAT \ {0} is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() Element of bool REAL
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg O is finite O -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= O ) } is set
G is set
Sgm G is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 is Relation-like REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
dom s1 is complex-membered ext-real-membered real-membered Element of bool REAL
s1 | G is Relation-like REAL -defined G -defined REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
s1 .: G is complex-membered ext-real-membered real-membered Element of bool REAL
Sgm (s1 .: G) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm G) is Relation-like NAT -defined REAL -valued Function-like finite V191() V192() V193() Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial non finite V191() V192() V193() set
bool [:NAT,REAL:] is non empty non trivial non finite set
rng (Sgm G) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal 0 -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= 0 ) } is set
s29 is Relation-like Function-like set
s2 is finite set
s29 .: s2 is finite set
p is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
sup p is V31() V32() ext-real set
s1 .: (rng (Sgm G)) is finite complex-membered ext-real-membered real-membered V210() V211() V212() Element of bool REAL
s199 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng s199 is finite set
p is set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= i ) } is set
f2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
dom f2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len f2) is finite len f2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len f2 ) } is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm G) . p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (Sgm G) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(Sgm G) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
len (Sgm G) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (Sgm G)) is finite len (Sgm G) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Sgm G) ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is V31() V32() ext-real Element of REAL
G /\ (dom s1) is complex-membered ext-real-membered real-membered Element of bool REAL
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 . p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . ((Sgm G) . H1) is V31() V32() ext-real set
H is V31() V32() ext-real Element of REAL
s1 . ((Sgm G) . p) is V31() V32() ext-real set
rng f2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O is set
Sgm O is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (G + 1) is non empty finite G + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= G + 1 ) } is set
Seg G is finite G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= G ) } is set
s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
{s1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (G + 1)) \ {s1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (G + 1)) \ {s1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (G + 1)) \ {s1})) " is Relation-like Function-like set
(Sgm O) * ((Sgm ((Seg (G + 1)) \ {s1})) ") is Relation-like NAT -defined Function-like finite set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } is set
{ (b<sub>1</sub> - 1) where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not b<sub>1</sub> <= s1 ) } is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } \/ { (b<sub>1</sub> - 1) where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not b<sub>1</sub> <= s1 ) } is set
id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } is Relation-like { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } -defined { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } -valued Function-like one-to-one total quasi_total Element of bool [: { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } , { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } :]
[: { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } , { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } :] is Relation-like set
bool [: { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } , { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } :] is non empty set
rng (Sgm O) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
f1 is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i - 1 is V31() V32() integer ext-real set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 - 1 is V31() V32() integer ext-real set
s199 is V31() V32() integer ext-real set
(G + 1) - 1 is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is finite set
{ [(b<sub>1</sub> - 1),b<sub>1</sub>] where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in i & not b<sub>1</sub> <= s1 ) } is set
s199 is set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 - 1 is V31() V32() integer ext-real set
[(f2 - 1),f2] is set
{(f2 - 1),f2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(f2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(f2 - 1),f2},{(f2 - 1)}} is non empty finite V39() set
p is set
H1 is set
j is set
j is set
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
s199 is Relation-like set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) \/ s199 is Relation-like set
p is set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
j is set
[p,j] is set
{p,j} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,j},{p}} is non empty finite V39() set
dom s199 is set
p is set
H1 is set
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
[(j - 1),j] is set
{(j - 1),j} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(j - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(j - 1),j},{(j - 1)}} is non empty finite V39() set
p is set
H1 is set
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
j is set
[p,j] is set
{p,j} is non empty finite set
{{p,j},{p}} is non empty finite V39() set
dom (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } is non empty set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . p is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
[(j - 1),j] is set
{(j - 1),j} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(j - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(j - 1),j},{(j - 1)}} is non empty finite V39() set
dom (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } is non empty set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
[(H2 - 1),H2] is set
{(H2 - 1),H2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H2 - 1),H2},{(H2 - 1)}} is non empty finite V39() set
p is set
H1 is set
[p,H1] is set
{p,H1} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,H1},{p}} is non empty finite V39() set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
[(j - 1),j] is set
{(j - 1),j} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(j - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(j - 1),j},{(j - 1)}} is non empty finite V39() set
j is set
[p,j] is set
{p,j} is non empty finite set
{{p,j},{p}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
[(H2 - 1),H2] is set
{(H2 - 1),H2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H2 - 1),H2},{(H2 - 1)}} is non empty finite V39() set
p is Relation-like Function-like set
dom p is set
dom (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } is non empty set
(dom (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } )) \/ (dom s199) is set
H1 is set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H1 is set
[H1,((id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H1)] is set
{H1,((id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H1)} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,((id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H1)},{H1}} is non empty finite V39() set
p . H1 is set
j is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
H1 is Relation-like Function-like set
rng H1 is set
(rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } )) \/ (rng H1) is set
rng p is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
[(H2 - 1),j] is set
{(H2 - 1),j} is non empty finite set
{(H2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H2 - 1),j},{(H2 - 1)}} is non empty finite V39() set
H1 is Relation-like Function-like set
rng H1 is set
rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
(rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } )) \/ (rng H1) is set
rng p is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng p is set
rng p is set
j is set
rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) }
H1 is Relation-like Function-like set
rng H1 is set
(rng (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } )) \/ (rng H1) is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is set
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
[(H2 - 1),H2] is set
{(H2 - 1),H2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H2 - 1),H2},{(H2 - 1)}} is non empty finite V39() set
j is set
j is set
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Sgm ((Seg (G + 1)) \ {s1})) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (Sgm ((Seg (G + 1)) \ {s1}))) is finite len (Sgm ((Seg (G + 1)) \ {s1})) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Sgm ((Seg (G + 1)) \ {s1})) ) } is set
dom (Sgm ((Seg (G + 1)) \ {s1})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (s299 + 1) is non empty finite s299 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= s299 + 1 ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
{H2} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (s299 + 1)) \ {H2} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (s299 + 1)) \ {H2}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (s299 + 1)) \ {H2})) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(rng p) |` (Sgm ((Seg (G + 1)) \ {s1})) is Relation-like NAT -defined NAT -valued rng p -valued NAT -valued Function-like finite FinSubsequence-like V191() V192() V193() V194() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
[(H1 - 1),H1] is set
{(H1 - 1),H1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H1 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H1 - 1),H1},{(H1 - 1)}} is non empty finite V39() set
s1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s1 + 1) - 1 is V31() V32() integer ext-real set
H2 is V31() V32() integer ext-real set
(G + 1) - 1 is V31() V32() integer ext-real set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Sgm ((Seg (G + 1)) \ {s1})) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (Sgm ((Seg (G + 1)) \ {s1}))) is finite len (Sgm ((Seg (G + 1)) \ {s1})) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (Sgm ((Seg (G + 1)) \ {s1})) ) } is set
dom (Sgm ((Seg (G + 1)) \ {s1})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (s299 + 1) is non empty finite s299 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s299 + 1 ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
{H2} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (s299 + 1)) \ {H2} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (s299 + 1)) \ {H2}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (s299 + 1)) \ {H2})) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(rng p) |` (Sgm ((Seg (G + 1)) \ {s1})) is Relation-like NAT -defined NAT -valued rng p -valued NAT -valued Function-like finite FinSubsequence-like V191() V192() V193() V194() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite set
(rng p) |` (Sgm ((Seg (G + 1)) \ {s1})) is Relation-like NAT -defined NAT -valued rng p -valued NAT -valued Function-like finite FinSubsequence-like V191() V192() V193() V194() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite set
(rng p) |` (Sgm ((Seg (G + 1)) \ {s1})) is Relation-like NAT -defined NAT -valued rng p -valued NAT -valued Function-like finite FinSubsequence-like V191() V192() V193() V194() Element of bool [:NAT,NAT:]
bool [:NAT,NAT:] is non empty non trivial non finite set
dom (Sgm ((Seg (G + 1)) \ {s1})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len (Sgm ((Seg (G + 1)) \ {s1})) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (Sgm ((Seg (G + 1)) \ {s1}))) is finite len (Sgm ((Seg (G + 1)) \ {s1})) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Sgm ((Seg (G + 1)) \ {s1})) ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (H2 + 1) is non empty finite H2 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= H2 + 1 ) } is set
{H1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (H2 + 1)) \ {H1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (H2 + 1)) \ {H1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (H2 + 1)) \ {H1})) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (H2 + 1) is non empty finite H2 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= H2 + 1 ) } is set
{H1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (H2 + 1)) \ {H1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (H2 + 1)) \ {H1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (H2 + 1)) \ {H1})) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - 1 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p " is Relation-like Function-like set
dom (p ") is set
rng (p ") is set
[:(dom (p ")),(rng (p ")):] is Relation-like set
bool [:(dom (p ")),(rng (p ")):] is non empty set
dom H1 is set
(dom (id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } )) \/ (dom H1) is set
j is set
H1 . j is set
[j,(H1 . j)] is set
{j,(H1 . j)} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,(H1 . j)},{j}} is non empty finite V39() set
p . j is set
j is set
H2 is set
p . j is set
p . H2 is set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . j is set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H2 is set
H1 . H2 is set
[H2,(p . H2)] is set
{H2,(p . H2)} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,(p . H2)},{H2}} is non empty finite V39() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
[(H1 - 1),H1] is set
{(H1 - 1),H1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H1 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H1 - 1),H1},{(H1 - 1)}} is non empty finite V39() set
H1 . j is set
[j,(p . j)] is set
{j,(p . j)} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,(p . j)},{j}} is non empty finite V39() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - 1 is V31() V32() integer ext-real set
[(s299 - 1),s299] is set
{(s299 - 1),s299} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(s299 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(s299 - 1),s299},{(s299 - 1)}} is non empty finite V39() set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) . H2 is set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 . H2 is set
[H2,(p . H2)] is set
{H2,(p . H2)} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,(p . H2)},{H2}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
[(H1 - 1),H1] is set
{(H1 - 1),H1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H1 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H1 - 1),H1},{(H1 - 1)}} is non empty finite V39() set
p ~ is Relation-like set
(Sgm ((Seg (G + 1)) \ {s1})) ~ is Relation-like NAT -defined NAT -valued V191() V192() V193() V194() Element of bool [:NAT,NAT:]
((Sgm ((Seg (G + 1)) \ {s1})) ~) | (dom (p ")) is Relation-like NAT -defined dom (p ") -defined NAT -defined NAT -valued RAT -valued V191() V192() V193() V194() Element of bool [:NAT,NAT:]
f1 is finite set
j is Relation-like dom (p ") -defined rng (p ") -valued Function-like Element of bool [:(dom (p ")),(rng (p ")):]
rng j is Element of bool (rng (p "))
bool (rng (p ")) is non empty set
dom j is Element of bool (dom (p "))
bool (dom (p ")) is non empty set
j is Relation-like REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) \/ H1 is Relation-like set
((id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) \/ H1) ~ is Relation-like set
(id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) ~ is Relation-like { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } -defined { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } -valued Function-like Element of bool [: { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } , { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } :]
H1 ~ is Relation-like set
((id { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } ) ~) \/ (H1 ~) is Relation-like set
H2 is V31() V32() ext-real Element of REAL
dom j is complex-membered ext-real-membered real-membered Element of bool REAL
O /\ (dom j) is complex-membered ext-real-membered real-membered Element of bool REAL
j . H2 is V31() V32() ext-real set
[H2,(j . H2)] is set
{H2,(j . H2)} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{H2} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{H2,(j . H2)},{H2}} is non empty finite V39() set
s299 is V31() V32() ext-real Element of REAL
j . s299 is V31() V32() ext-real set
[s299,(j . s299)] is set
{s299,(j . s299)} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{s299} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{s299,(j . s299)},{s299}} is non empty finite V39() set
[(j . s299),s299] is set
{(j . s299),s299} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{(j . s299)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{(j . s299),s299},{(j . s299)}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
[(H1 - 1),H1] is set
{(H1 - 1),H1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H1 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H1 - 1),H1},{(H1 - 1)}} is non empty finite V39() set
s1 - 1 is V31() V32() integer ext-real set
s299 is V31() V32() integer ext-real set
s299 + 1 is V31() V32() integer ext-real set
H2 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
[(j . H2),H2] is set
{(j . H2),H2} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{(j . H2)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{(j . H2),H2},{(j . H2)}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
[(H1 - 1),H1] is set
{(H1 - 1),H1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H1 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H1 - 1),H1},{(H1 - 1)}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
[(j . s299),s299] is set
{(j . s299),s299} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{(j . s299)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{(j . s299),s299},{(j . s299)}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
[(H2 - 1),H2] is set
{(H2 - 1),H2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(H2 - 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(H2 - 1),H2},{(H2 - 1)}} is non empty finite V39() set
j | O is Relation-like REAL -defined O -defined REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
(p ") .: O is set
H2 is set
j .: O is complex-membered ext-real-membered real-membered Element of bool REAL
Sgm ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not s1 <= b<sub>1</sub> ) } \/ { (b<sub>1</sub> - 1) where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in O & not b<sub>1</sub> <= s1 ) } ) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm (j .: O) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
((Sgm ((Seg (G + 1)) \ {s1})) ") | O is Relation-like Function-like set
(Sgm O) * (((Sgm ((Seg (G + 1)) \ {s1})) ") | O) is Relation-like NAT -defined Function-like finite set
O |` (Sgm O) is Relation-like NAT -defined NAT -valued O -valued NAT -valued Function-like finite FinSubsequence-like V191() V192() V193() V194() Element of bool [:NAT,NAT:]
(O |` (Sgm O)) * ((Sgm ((Seg (G + 1)) \ {s1})) ") is Relation-like NAT -defined Function-like finite set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s29 is set
s1 . (s29 + 1) is set
Del (s1,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s29} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s29}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {s29})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s1 ) } is set
(len s1) - 1 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
f1 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (i + 1) is non empty finite i + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= i + 1 ) } is set
s199 is set
Sgm s199 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s199) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
f2 is set
Sgm f2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm f2) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
s199 is set
{(s29 + 1)} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{(s29 + 1)} /\ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() set
s199 \/ {(s29 + 1)} is non empty set
(s199 \/ {(s29 + 1)}) \ {s29} is Element of bool (s199 \/ {(s29 + 1)})
bool (s199 \/ {(s29 + 1)}) is non empty set
s199 is set
f2 is set
Sgm f2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
rng (Sgm f2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p is finite set
Sgm p is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
dom (Sgm p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
card p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
Seg (card p) is finite card p -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= card p ) } is set
H1 is finite set
H1 \/ {(s29 + 1)} is non empty finite set
(H1 \/ {(s29 + 1)}) \ {s29} is finite Element of bool (H1 \/ {(s29 + 1)})
bool (H1 \/ {(s29 + 1)}) is non empty finite V39() set
card ((H1 \/ {(s29 + 1)}) \ {s29}) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
card (H1 \/ {(s29 + 1)}) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
card {s29} is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
(card (H1 \/ {(s29 + 1)})) - (card {s29}) is V31() V32() integer ext-real set
(card (H1 \/ {(s29 + 1)})) - 1 is V31() V32() integer ext-real set
1 + s29 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Sgm H1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
rng (Sgm H1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (Sgm H1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j is set
(Sgm H1) . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is set
(Sgm H1) . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
len (Sgm H1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (Sgm H1)) is finite len (Sgm H1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Sgm H1) ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - H2 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(H2 + 1) - H2 is V31() V32() integer ext-real set
1 + H2 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(Sgm H1) . (H2 + 1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 . H2 is set
s19 . (H2 + 1) is set
i is Element of (O,G)
j is Element of (O,G)
H is non empty unital Group-like associative (O) (O,G)
H9 is non empty unital Group-like associative (O) (O)
K is non empty unital Group-like associative (O) (O,G)
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . ((Sgm H1) . H2) is set
the carrier of H9 is non empty set
K9 is non empty unital Group-like associative (O) (O,H9) (O,H9)
the carrier of K9 is non empty set
(O,H9,K9) is non empty unital Group-like associative (O) (O)
(O,H9,K9) is set
(O,H9,K9) is Relation-like [:(O,H9,K9),(O,H9,K9):] -defined (O,H9,K9) -valued Function-like quasi_total Element of bool [:[:(O,H9,K9),(O,H9,K9):],(O,H9,K9):]
[:(O,H9,K9),(O,H9,K9):] is Relation-like set
[:[:(O,H9,K9),(O,H9,K9):],(O,H9,K9):] is Relation-like set
bool [:[:(O,H9,K9),(O,H9,K9):],(O,H9,K9):] is non empty set
(O,H9,K9) is Relation-like O -defined Funcs ((O,H9,K9),(O,H9,K9)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H9,K9),(O,H9,K9))):]
Funcs ((O,H9,K9),(O,H9,K9)) is functional non empty set
[:O,(Funcs ((O,H9,K9),(O,H9,K9))):] is Relation-like set
bool [:O,(Funcs ((O,H9,K9),(O,H9,K9))):] is non empty set
(O,(O,H9,K9),(O,H9,K9),(O,H9,K9)) is (O) (O)
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
card H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
(card H1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } is set
H2 is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29} is non empty set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is non empty set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } /\ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 - s29 is V31() V32() integer ext-real set
(s29 + 1) - s29 is V31() V32() integer ext-real set
H2 is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } /\ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)} is non empty set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)}) \ {s29} is Element of bool ((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)})
bool ((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)}) is non empty set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \ {s29} is Element of bool (( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } )
bool (( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is non empty set
{(s29 + 1)} \ {s29} is trivial finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool {(s29 + 1)}
bool {(s29 + 1)} is non empty finite V39() set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \ {s29}) \/ ({(s29 + 1)} \ {s29}) is set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \ {s29}) \/ {(s29 + 1)} is non empty set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \ {s29} is Element of bool ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29})
bool ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } \ {s29} is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) }
bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is non empty set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \ {s29}) \/ ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } \ {s29}) is set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \ {s29}) \/ ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } \ {s29})) \/ {(s29 + 1)} is non empty set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) \ {s29}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)} is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29} is Element of bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) }
bool { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } is non empty set
{s29} \ {s29} is trivial finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool {s29}
bool {s29} is non empty finite V39() set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ ({s29} \ {s29}) is set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ ({s29} \ {s29})) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ ({s29} \ {s29})) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)} is non empty set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ {} is set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ {}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is set
((( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \ {s29}) \/ {}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)} is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {} is set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is set
(( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) \/ {(s29 + 1)} is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {(s29 + 1)} is non empty set
( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {(s29 + 1)}) \/ { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is non empty set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Sgm ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {(s29 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm {(s29 + 1)} is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ (Sgm {(s29 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {s29})) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm {s29} is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ (Sgm {s29}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
<*s29*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,s29] is set
{1,s29} is non empty finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{1,s29},{1}} is non empty finite V39() set
{[1,s29]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(Sgm ( { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } \/ {(s29 + 1)})) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
<*(s29 + 1)*> is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() V194() increasing V196() V197() V198() Element of 1 -tuples_on NAT
1 -tuples_on NAT is FinSequenceSet of NAT
[1,(s29 + 1)] is set
{1,(s29 + 1)} is non empty finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{1,(s29 + 1)},{1}} is non empty finite V39() set
{[1,(s29 + 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*> is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is Relation-like NAT -defined NAT -valued Function-like non empty finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s29*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } )) + (len <*s29*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } )) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len <*(s29 + 1)*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } )) + (len <*(s29 + 1)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is set
dom ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
Seg (len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>)) is non empty finite len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len ((Sgm { b<sub>1</sub> where b<sub>2</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) ) } is set
dom ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) . s299 is set
(((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } )) . s299 is set
(Sgm H1) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } )) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm f2) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm f2) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
rng (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom <*s29*> is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } )) + H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } )) + H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*s29*> . H1 is set
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) . s299 is set
(((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } )) . s299 is set
(Sgm H1) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm f2) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom <*(s29 + 1)*> is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
<*(s29 + 1)*> . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*(s29 + 1)*>) ^ (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } )) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom <*s29*> is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
dom (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>)) + H1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len ((Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not s29 <= b<sub>1</sub> ) } ) ^ <*s29*>)) + H1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(Sgm f2) . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm f2) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
rng (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (Sgm { b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in H1 & not b<sub>1</sub> <= s29 + 1 ) } ) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s1 * (Sgm H1) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
dom (s1 * (Sgm H1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s1 * (Sgm f2) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
dom (s1 * (Sgm f2)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H2 is set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(s1 * (Sgm f2)) . H2 is set
(Sgm f2) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . ((Sgm f2) . H2) is set
s1 . ((Sgm H1) . H2) is set
(s1 * (Sgm H1)) . H2 is set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(s1 * (Sgm f2)) . H2 is set
(Sgm f2) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . ((Sgm f2) . H2) is set
s1 . ((Sgm H1) . H2) is set
(s1 * (Sgm H1)) . H2 is set
(Sgm H1) . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 is set
Sgm s199 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s199) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
s199 is set
Sgm s199 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s199) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
(Seg (i + 1)) \ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (i + 1)) \ {s29}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (i + 1)) \ {s29})) " is Relation-like Function-like set
(Sgm s199) * ((Sgm ((Seg (i + 1)) \ {s29})) ") is Relation-like NAT -defined Function-like finite set
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= i ) } is set
f2 is set
Sgm f2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
len (Del (s1,s29)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom (Sgm ((Seg (i + 1)) \ {s29})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng (Sgm ((Seg (i + 1)) \ {s29})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(rng (Sgm ((Seg (i + 1)) \ {s29}))):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(rng (Sgm ((Seg (i + 1)) \ {s29}))):] is non empty finite V39() set
H2 is Relation-like dom (Sgm ((Seg (i + 1)) \ {s29})) -defined rng (Sgm ((Seg (i + 1)) \ {s29})) -valued Function-like quasi_total finite V191() V192() V193() V194() Element of bool [:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(rng (Sgm ((Seg (i + 1)) \ {s29}))):]
rng H2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (rng (Sgm ((Seg (i + 1)) \ {s29})))
bool (rng (Sgm ((Seg (i + 1)) \ {s29}))) is non empty finite V39() set
s299 is set
rng (Sgm s199) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(Seg 2) \ {s29} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm p is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm p) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
s1 * H2 is Relation-like dom (Sgm ((Seg (i + 1)) \ {s29})) -defined (O,G) -valued Function-like finite Element of bool [:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(O,G):]
[:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(O,G):] is Relation-like set
bool [:(dom (Sgm ((Seg (i + 1)) \ {s29}))),(O,G):] is non empty set
(s1 * H2) * (Sgm p) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
H2 " is Relation-like Function-like set
(H2 ") * (s1 * H2) is Relation-like (O,G) -valued Function-like set
(Sgm s199) * ((H2 ") * (s1 * H2)) is Relation-like NAT -defined (O,G) -valued Function-like finite set
(H2 ") * H2 is Relation-like RAT -valued rng (Sgm ((Seg (i + 1)) \ {s29})) -valued Function-like V191() V192() V193() V194() set
((H2 ") * H2) * s1 is Relation-like (O,G) -valued Function-like set
(Sgm s199) * (((H2 ") * H2) * s1) is Relation-like NAT -defined (O,G) -valued Function-like finite set
id (rng H2) is Relation-like rng H2 -defined rng H2 -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:(rng H2),(rng H2):]
[:(rng H2),(rng H2):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(rng H2),(rng H2):] is non empty finite V39() set
s1 * (id (rng H2)) is Relation-like rng H2 -defined (O,G) -valued Function-like finite Element of bool [:(rng H2),(O,G):]
[:(rng H2),(O,G):] is Relation-like set
bool [:(rng H2),(O,G):] is non empty set
(s1 * (id (rng H2))) * (Sgm s199) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
(id (rng H2)) * (Sgm s199) is Relation-like NAT -defined RAT -valued INT -valued rng H2 -valued Function-like finite V191() V192() V193() V194() Element of bool [:NAT,(rng H2):]
[:NAT,(rng H2):] is Relation-like RAT -valued INT -valued V191() V192() V193() V194() set
bool [:NAT,(rng H2):] is non empty set
s1 * ((id (rng H2)) * (Sgm s199)) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
f2 is set
Sgm f2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm f2) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s1 ) } is set
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s1) + s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 + (len s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + (len s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s2 ) } is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( b<sub>1</sub> in dom s1 & s1 . b<sub>1</sub> = s2 . b<sub>1</sub> ) } is set
s1 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s2 . 1 is set
p is set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . f1 is set
s2 . f1 is set
p is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
sup p is V31() V32() ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s199 is set
s2 . s199 is set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s199 is set
s2 . s199 is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s199 is set
s2 . s199 is set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 - 1 is V31() V32() integer ext-real set
(s199 + 1) - 1 is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s199 is set
s2 . s199 is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (s199 + 1) is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . p is set
s2 . p is set
s199 is Element of (O,G)
f2 is Element of (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . j is set
s2 . j is set
p is non empty unital Group-like associative (O) (O,G)
the carrier of p is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element set
H1 is non empty unital Group-like associative (O) (O,G)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . j is set
s2 . j is set
j is non empty unital Group-like associative (O) (O,p) (O,p)
the carrier of j is non empty set
the carrier of (O,G) is non empty set
(O,p,j) is non empty unital Group-like associative (O) (O)
(O,p,j) is set
(O,p,j) is Relation-like [:(O,p,j),(O,p,j):] -defined (O,p,j) -valued Function-like quasi_total Element of bool [:[:(O,p,j),(O,p,j):],(O,p,j):]
[:(O,p,j),(O,p,j):] is Relation-like set
[:[:(O,p,j),(O,p,j):],(O,p,j):] is Relation-like set
bool [:[:(O,p,j),(O,p,j):],(O,p,j):] is non empty set
(O,p,j) is Relation-like O -defined Funcs ((O,p,j),(O,p,j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,j),(O,p,j))):]
Funcs ((O,p,j),(O,p,j)) is functional non empty set
[:O,(Funcs ((O,p,j),(O,p,j))):] is Relation-like set
bool [:O,(Funcs ((O,p,j),(O,p,j))):] is non empty set
(O,(O,p,j),(O,p,j),(O,p,j)) is (O) (O)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . j is set
s2 . j is set
1 + s199 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . (s199 + 1) is set
s2 . (s199 + 1) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s199 is set
s2 . s199 is set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s29 is set
s2 . s29 is set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s29 + 1) is set
s2 . (s29 + 1) is set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s29 is set
s2 . s29 is set
s1 . (s29 + 1) is set
s2 . (s29 + 1) is set
p is set
Sgm p is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm p) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
(Sgm p) . (s29 + 1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (Sgm p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng (Sgm p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 . s199 is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 . p is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . p is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . s29 is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . p is set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . (s29 + f1) is set
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 + (f1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real positive non negative set
s19 . (s29 + (f1 + 1)) is set
s199 is Element of (O,G)
s199 is non empty unital Group-like associative (O) (O,G)
s29 + (f1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s29 + f1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is non empty unital Group-like associative (O) (O,G)
s19 . (s29 + (f1 + 1)) is set
p is non empty unital Group-like associative (O) (O,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s19) is finite len s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s19 ) } is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s199 is non empty unital Group-like associative (O) (O,G)
1 + (len s19) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len s19) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s29 + f1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s19) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s199 is non empty unital Group-like associative (O) (O,G)
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 + 0 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . (s29 + 0) is set
i is non empty unital Group-like associative (O) (O,G)
p - s29 is V31() V32() integer ext-real set
s29 - s29 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (O,G,s2) is finite set
len (O,G,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s2)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom (O,G,s2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is set
(O,G,s2) . s29 is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg (len (O,G,s2)) is finite len (O,G,s2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (O,G,s2) ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + p is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s2 ) } is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 . p is set
s2 . (p + 1) is set
f1 is non empty unital Group-like associative (O) (O) (O,G)
i is non empty unital Group-like associative (O) (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,f1) (O,f1)
(O,f1,s199) is non empty unital Group-like associative (O) (O)
(O,f1,s199) is set
(O,f1,s199) is Relation-like [:(O,f1,s199),(O,f1,s199):] -defined (O,f1,s199) -valued Function-like quasi_total Element of bool [:[:(O,f1,s199),(O,f1,s199):],(O,f1,s199):]
[:(O,f1,s199),(O,f1,s199):] is Relation-like set
[:[:(O,f1,s199),(O,f1,s199):],(O,f1,s199):] is Relation-like set
bool [:[:(O,f1,s199),(O,f1,s199):],(O,f1,s199):] is non empty set
(O,f1,s199) is Relation-like O -defined Funcs ((O,f1,s199),(O,f1,s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f1,s199),(O,f1,s199))):]
Funcs ((O,f1,s199),(O,f1,s199)) is functional non empty set
[:O,(Funcs ((O,f1,s199),(O,f1,s199))):] is Relation-like set
bool [:O,(Funcs ((O,f1,s199),(O,f1,s199))):] is non empty set
(O,(O,f1,s199),(O,f1,s199),(O,f1,s199)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . s2 is set
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s2 is set
s1 . (s2 + 1) is set
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
rng (O,G,s1) is finite set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s29 is non empty unital Group-like associative (O) (O) (O)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . 2 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 . 1 is set
(O,(O,G)) is non empty unital Group-like associative (O) (O) (O,(O,G)) (O,(O,G))
f1 is non empty unital Group-like associative (O) (O,(O,G))
i is non empty unital Group-like associative (O) (O,(O,G)) (O,(O,G))
(O,(O,G),i) is non empty unital Group-like associative (O) (O)
(O,(O,G),i) is set
(O,(O,G),i) is Relation-like [:(O,(O,G),i),(O,(O,G),i):] -defined (O,(O,G),i) -valued Function-like quasi_total Element of bool [:[:(O,(O,G),i),(O,(O,G),i):],(O,(O,G),i):]
[:(O,(O,G),i),(O,(O,G),i):] is Relation-like set
[:[:(O,(O,G),i),(O,(O,G),i):],(O,(O,G),i):] is Relation-like set
bool [:[:(O,(O,G),i),(O,(O,G),i):],(O,(O,G),i):] is non empty set
(O,(O,G),i) is Relation-like O -defined Funcs ((O,(O,G),i),(O,(O,G),i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G),i),(O,(O,G),i))):]
Funcs ((O,(O,G),i),(O,(O,G),i)) is functional non empty set
[:O,(Funcs ((O,(O,G),i),(O,(O,G),i))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G),i),(O,(O,G),i))):] is non empty set
(O,(O,(O,G),i),(O,(O,G),i),(O,(O,G),i)) is (O) (O)
s199 is non empty unital Group-like associative (O) (O,G)
s199 is non empty unital Group-like associative (O) (O,s199) (O,s199)
(O,s199,s199) is non empty unital Group-like associative (O) (O)
(O,s199,s199) is set
(O,s199,s199) is Relation-like [:(O,s199,s199),(O,s199,s199):] -defined (O,s199,s199) -valued Function-like quasi_total Element of bool [:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):]
[:(O,s199,s199),(O,s199,s199):] is Relation-like set
[:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):] is Relation-like set
bool [:[:(O,s199,s199),(O,s199,s199):],(O,s199,s199):] is non empty set
(O,s199,s199) is Relation-like O -defined Funcs ((O,s199,s199),(O,s199,s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,s199),(O,s199,s199))):]
Funcs ((O,s199,s199),(O,s199,s199)) is functional non empty set
[:O,(Funcs ((O,s199,s199),(O,s199,s199))):] is Relation-like set
bool [:O,(Funcs ((O,s199,s199),(O,s199,s199))):] is non empty set
(O,(O,s199,s199),(O,s199,s199),(O,s199,s199)) is (O) (O)
rng (O,G,s1) is finite set
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (O,G,s1) ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s2 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (O,G,s1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s1 ) } is set
f1 is non empty unital Group-like associative (O) (O) (O,G)
p is non empty unital Group-like associative (O) (O) (O,G)
s29 is non empty unital Group-like associative (O) (O) (O)
i is non empty unital Group-like associative (O) (O,f1) (O,f1)
(O,f1,i) is non empty unital Group-like associative (O) (O)
(O,f1,i) is set
(O,f1,i) is Relation-like [:(O,f1,i),(O,f1,i):] -defined (O,f1,i) -valued Function-like quasi_total Element of bool [:[:(O,f1,i),(O,f1,i):],(O,f1,i):]
[:(O,f1,i),(O,f1,i):] is Relation-like set
[:[:(O,f1,i),(O,f1,i):],(O,f1,i):] is Relation-like set
bool [:[:(O,f1,i),(O,f1,i):],(O,f1,i):] is non empty set
(O,f1,i) is Relation-like O -defined Funcs ((O,f1,i),(O,f1,i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f1,i),(O,f1,i))):]
Funcs ((O,f1,i),(O,f1,i)) is functional non empty set
[:O,(Funcs ((O,f1,i),(O,f1,i))):] is Relation-like set
bool [:O,(Funcs ((O,f1,i),(O,f1,i))):] is non empty set
(O,(O,f1,i),(O,f1,i),(O,f1,i)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s19 is set
s1 . (s19 + 1) is set
Del (s1,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s19} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {s19})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
Del ((O,G,s1),s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom (O,G,s1)) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (O,G,s1)) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (O,G,s1)) \ {s19})) * (O,G,s1) is Relation-like NAT -defined Function-like finite set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Del (s1,s19)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s1 ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (O,G,s1) ) } is set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Del ((O,G,s1),1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(dom (O,G,s1)) \ {1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (O,G,s1)) \ {1}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (O,G,s1)) \ {1})) * (O,G,s1) is Relation-like NAT -defined Function-like finite set
len (Del ((O,G,s1),1)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Del ((O,G,s1),s19)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s19 + 1) - 1 is V31() V32() integer ext-real set
(len s1) - 1 is V31() V32() integer ext-real set
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (O,G,s1) ) } is set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len (Del ((O,G,s1),s19)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom (Del ((O,G,s1),s19)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len (Del ((O,G,s1),s19))) is finite len (Del ((O,G,s1),s19)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Del ((O,G,s1),s19)) ) } is set
f2 is non empty unital Group-like associative (O) (O,G)
s2 . i is set
1 + i is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s1)) - (len (Del ((O,G,s1),s19))) is V31() V32() integer ext-real set
((len (O,G,s1)) - (len (Del ((O,G,s1),s19)))) + (len (Del ((O,G,s1),s19))) is V31() V32() integer ext-real set
0 + (len (Del ((O,G,s1),s19))) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is non empty unital Group-like associative (O) (O,f2) (O,f2)
s2 . (i + 1) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (H1 + 1) is set
s1 . H1 is set
(O,G,s1) . i is set
(O,f2,p) is non empty unital Group-like associative (O) (O)
(O,f2,p) is set
(O,f2,p) is Relation-like [:(O,f2,p),(O,f2,p):] -defined (O,f2,p) -valued Function-like quasi_total Element of bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):]
[:(O,f2,p),(O,f2,p):] is Relation-like set
[:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is Relation-like set
bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is non empty set
(O,f2,p) is Relation-like O -defined Funcs ((O,f2,p),(O,f2,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f2,p),(O,f2,p))):]
Funcs ((O,f2,p),(O,f2,p)) is functional non empty set
[:O,(Funcs ((O,f2,p),(O,f2,p))):] is Relation-like set
bool [:O,(Funcs ((O,f2,p),(O,f2,p))):] is non empty set
(O,(O,f2,p),(O,f2,p),(O,f2,p)) is (O) (O)
(Del ((O,G,s1),s19)) . i is set
s1 . i is set
s1 . (i + 1) is set
(O,G,s1) . i is set
(O,f2,p) is non empty unital Group-like associative (O) (O)
(O,f2,p) is set
(O,f2,p) is Relation-like [:(O,f2,p),(O,f2,p):] -defined (O,f2,p) -valued Function-like quasi_total Element of bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):]
[:(O,f2,p),(O,f2,p):] is Relation-like set
[:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is Relation-like set
bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is non empty set
(O,f2,p) is Relation-like O -defined Funcs ((O,f2,p),(O,f2,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f2,p),(O,f2,p))):]
Funcs ((O,f2,p),(O,f2,p)) is functional non empty set
[:O,(Funcs ((O,f2,p),(O,f2,p))):] is Relation-like set
bool [:O,(Funcs ((O,f2,p),(O,f2,p))):] is non empty set
(O,(O,f2,p),(O,f2,p),(O,f2,p)) is (O) (O)
(Del ((O,G,s1),s19)) . i is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
1 + s19 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (H1 + 1) is set
s1 . H1 is set
(O,G,s1) . s199 is set
(O,f2,p) is non empty unital Group-like associative (O) (O)
(O,f2,p) is set
(O,f2,p) is Relation-like [:(O,f2,p),(O,f2,p):] -defined (O,f2,p) -valued Function-like quasi_total Element of bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):]
[:(O,f2,p),(O,f2,p):] is Relation-like set
[:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is Relation-like set
bool [:[:(O,f2,p),(O,f2,p):],(O,f2,p):] is non empty set
(O,f2,p) is Relation-like O -defined Funcs ((O,f2,p),(O,f2,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f2,p),(O,f2,p))):]
Funcs ((O,f2,p),(O,f2,p)) is functional non empty set
[:O,(Funcs ((O,f2,p),(O,f2,p))):] is Relation-like set
bool [:O,(Funcs ((O,f2,p),(O,f2,p))):] is non empty set
(O,(O,f2,p),(O,f2,p),(O,f2,p)) is (O) (O)
(Del ((O,G,s1),s19)) . i is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . s19 is set
Del (s1,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s19} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {s19})) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
Del (s2,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom s2) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s2) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s2) \ {s19})) * s2 is Relation-like NAT -defined Function-like finite set
s1 . s19 is set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s19 + 1) is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
O is set
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom G),(dom G):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom G),(dom G):] is non empty finite V39() set
s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Del (G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s2} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom G) \ {s2} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom G) \ {s2}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom G) \ {s2})) * G is Relation-like NAT -defined Function-like finite set
dom (Del (G,s2)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (Del (G,s2))),(dom (Del (G,s2))):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):] is non empty finite V39() set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Del (s1,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{s19} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {s19})) * s1 is Relation-like NAT -defined Function-like finite set
len G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 is Relation-like dom G -defined dom G -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom G),(dom G):]
s29 " is Relation-like dom G -defined dom G -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom G),(dom G):]
(s29 ") . s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
rng (s29 ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
bool (dom G) is non empty finite V39() set
Seg (len G) is finite len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len G ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len (Del (s1,s19)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (Del (G,s2)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
the Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):] is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
dom {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool NAT
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty trivial proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered with_non-empty_elements ext-real non positive non negative V191() V192() V193() V194() increasing V196() V197() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool RAT
bool RAT is non empty non trivial non finite set
[:(dom {}),(rng {}):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom {}),(rng {}):] is non empty finite V39() set
[:{},{}:] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:{},{}:] is non empty finite V39() set
f1 is Relation-like non-empty empty-yielding dom {} -defined rng {} -valued Function-like one-to-one constant functional empty total quasi_total epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool [:(dom {}),(rng {}):]
i is Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional empty total quasi_total epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() Element of bool [:{},{}:]
s199 is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
s199 " is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(s199 ") . p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 is non empty unital Group-like associative (O) (O)
(Del (G,s2)) . p is set
f2 is non empty unital Group-like associative (O) (O)
(Del (s1,s19)) . H1 is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(dom s1) \ {s19} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= i ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(dom G) \ {s2} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {s19}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm ((dom G) \ {s2}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom G) \ {s2})) " is Relation-like Function-like set
s29 * ((Sgm ((dom G) \ {s2})) ") is Relation-like dom G -defined Function-like finite set
(Sgm ((dom s1) \ {s19})) * (s29 * ((Sgm ((dom G) \ {s2})) ")) is Relation-like NAT -defined Function-like finite set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len s1 ) } is set
rng (Sgm ((dom G) \ {s2})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng (Sgm ((dom s1) \ {s19})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 . s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 * (s29 ") is Relation-like dom G -defined RAT -valued dom G -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom G),(dom G):]
(s29 * (s29 ")) . s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative set
id (dom G) is Relation-like dom G -defined dom G -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:(dom G),(dom G):]
(id (dom G)) . s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative set
((Sgm ((dom s1) \ {s19})) * (s29 * ((Sgm ((dom G) \ {s2})) "))) " is Relation-like Function-like set
(Sgm ((dom s1) \ {s19})) " is Relation-like Function-like set
(s29 ") * ((Sgm ((dom s1) \ {s19})) ") is Relation-like dom G -defined Function-like finite set
(Sgm ((dom G) \ {s2})) * ((s29 ") * ((Sgm ((dom s1) \ {s19})) ")) is Relation-like NAT -defined Function-like finite set
((dom G) \ {s2}) \/ {s2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom G) \/ {s2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
card (((dom G) \ {s2}) \/ {s2}) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
card ((dom G) \ {s2}) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
card {s2} is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
(card ((dom G) \ {s2})) + (card {s2}) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(card ((dom G) \ {s2})) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
card (Seg (len G)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
((dom s1) \ {s19}) \/ {s19} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s1) \/ {s19} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
dom ((Sgm ((dom G) \ {s2})) ") is set
rng s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
s299 is set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
s29 . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (s29 * ((Sgm ((dom G) \ {s2})) ")) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
s299 is set
s29 . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
rng ((Sgm ((dom s1) \ {s19})) * (s29 * ((Sgm ((dom G) \ {s2})) "))) is finite set
rng (s29 * ((Sgm ((dom G) \ {s2})) ")) is finite set
rng ((Sgm ((dom G) \ {s2})) ") is set
dom (Sgm ((dom G) \ {s2})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg p is finite p -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= p ) } is set
card (((dom s1) \ {s19}) \/ {s19}) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
card ((dom s1) \ {s19}) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
card {s19} is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of omega
(card ((dom s1) \ {s19})) + (card {s19}) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(card ((dom s1) \ {s19})) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
card (Seg (len s1)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of omega
dom (Sgm ((dom s1) \ {s19})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom ((Sgm ((dom s1) \ {s19})) * (s29 * ((Sgm ((dom G) \ {s2})) "))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s299 is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like total quasi_total finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
H1 is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
j is Relation-like Function-like set
H is Relation-like Function-like set
H ~ is Relation-like set
K is Relation-like Function-like set
H9 is Relation-like Function-like set
H9 ~ is Relation-like set
s299 is Relation-like Function-like set
i is Relation-like Function-like set
K9 is Relation-like Function-like set
i9 is Relation-like Function-like set
i9 " is Relation-like Function-like set
H2 is Relation-like set
H2 ~ is Relation-like set
i9 is Relation-like set
JH is Relation-like set
i9 * JH is Relation-like set
(i9 * JH) ~ is Relation-like set
JK is Relation-like set
JK ~ is Relation-like set
((i9 * JH) ~) * (JK ~) is Relation-like set
JH ~ is Relation-like set
i9 ~ is Relation-like set
(JH ~) * (i9 ~) is Relation-like set
((JH ~) * (i9 ~)) * (JK ~) is Relation-like set
JH is Relation-like set
JH ~ is Relation-like set
(JH ~) ~ is Relation-like set
k19 is Relation-like set
((JH ~) ~) * k19 is Relation-like set
JK is Relation-like set
(((JH ~) ~) * k19) * JK is Relation-like set
j9 is Relation-like Function-like set
s199 is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
dom (Sgm ((dom s1) \ {s19})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (Sgm ((dom s1) \ {s19}))),(rng (Sgm ((dom s1) \ {s19}))):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (Sgm ((dom s1) \ {s19}))),(rng (Sgm ((dom s1) \ {s19}))):] is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(Sgm ((dom s1) \ {s19})) . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 " is Relation-like dom (Del (G,s2)) -defined dom (Del (G,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),(dom (Del (G,s2))):]
(Sgm ((dom s1) \ {s19})) * (s199 ") is Relation-like dom (Del (G,s2)) -defined NAT -valued RAT -valued Function-like finite V191() V192() V193() V194() Element of bool [:(dom (Del (G,s2))),NAT:]
[:(dom (Del (G,s2))),NAT:] is Relation-like RAT -valued INT -valued V191() V192() V193() V194() set
bool [:(dom (Del (G,s2))),NAT:] is non empty set
(s29 ") * (Sgm ((dom G) \ {s2})) is Relation-like NAT -defined RAT -valued dom G -valued Function-like finite V191() V192() V193() V194() Element of bool [:NAT,(dom G):]
[:NAT,(dom G):] is Relation-like RAT -valued INT -valued V191() V192() V193() V194() set
bool [:NAT,(dom G):] is non empty set
((s29 ") * (Sgm ((dom G) \ {s2}))) * ((Sgm ((dom s1) \ {s19})) ") is Relation-like NAT -defined Function-like finite set
(((s29 ") * (Sgm ((dom G) \ {s2}))) * ((Sgm ((dom s1) \ {s19})) ")) * (Sgm ((dom s1) \ {s19})) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V191() V192() V193() V194() set
((Sgm ((dom s1) \ {s19})) ") * (Sgm ((dom s1) \ {s19})) is Relation-like NAT -valued RAT -valued Function-like V191() V192() V193() V194() set
((s29 ") * (Sgm ((dom G) \ {s2}))) * (((Sgm ((dom s1) \ {s19})) ") * (Sgm ((dom s1) \ {s19}))) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V191() V192() V193() V194() set
id ((dom s1) \ {s19}) is Relation-like (dom s1) \ {s19} -defined (dom s1) \ {s19} -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:((dom s1) \ {s19}),((dom s1) \ {s19}):]
[:((dom s1) \ {s19}),((dom s1) \ {s19}):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:((dom s1) \ {s19}),((dom s1) \ {s19}):] is non empty finite V39() set
(id ((dom s1) \ {s19})) * ((s29 ") * (Sgm ((dom G) \ {s2}))) is Relation-like NAT -defined RAT -valued INT -valued (dom s1) \ {s19} -valued Function-like finite V191() V192() V193() V194() Element of bool [:NAT,((dom s1) \ {s19}):]
[:NAT,((dom s1) \ {s19}):] is Relation-like RAT -valued INT -valued V191() V192() V193() V194() set
bool [:NAT,((dom s1) \ {s19}):] is non empty set
(id ((dom s1) \ {s19})) * (s29 ") is Relation-like dom G -defined RAT -valued INT -valued (dom s1) \ {s19} -valued Function-like one-to-one finite V191() V192() V193() V194() Element of bool [:(dom G),((dom s1) \ {s19}):]
[:(dom G),((dom s1) \ {s19}):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom G),((dom s1) \ {s19}):] is non empty finite V39() set
((id ((dom s1) \ {s19})) * (s29 ")) * (Sgm ((dom G) \ {s2})) is Relation-like NAT -defined RAT -valued INT -valued (dom s1) \ {s19} -valued Function-like finite V191() V192() V193() V194() Element of bool [:NAT,((dom s1) \ {s19}):]
((dom s1) \ {s19}) |` (s29 ") is Relation-like dom G -defined dom G -valued (dom s1) \ {s19} -valued dom G -valued Function-like finite V191() V192() V193() V194() Element of bool [:(dom G),(dom G):]
(((dom s1) \ {s19}) |` (s29 ")) * (Sgm ((dom G) \ {s2})) is Relation-like NAT -defined RAT -valued dom G -valued Function-like finite V191() V192() V193() V194() Element of bool [:NAT,(dom G):]
(s199 ") . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (s199 ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom (Del (G,s2)))
bool (dom (Del (G,s2))) is non empty finite V39() set
rng (s199 ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom (Del (G,s2)))
Seg (len (Del (s1,s19))) is finite len (Del (s1,s19)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len (Del (s1,s19)) ) } is set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom (Del (s1,s19)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(Sgm ((dom G) \ {s2})) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (Sgm ((dom G) \ {s2})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
f2 is non empty unital Group-like associative (O) (O)
(Del (G,s2)) . H1 is set
p is non empty unital Group-like associative (O) (O)
(Del (s1,s19)) . j is set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
G . H1 is set
rng s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
dom (s29 ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom G)
(s29 ") . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . H2 is set
((Sgm ((dom s1) \ {s19})) * (s199 ")) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative set
(((dom s1) \ {s19}) |` (s29 ")) . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is non empty unital Group-like associative (O) (O)
(O,s1) is non empty set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s19 is Relation-like NAT -defined (O,s1) -valued Function-like finite FinSequence-like FinSubsequence-like (O,s1) FinSequence of (O,s1)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg s29 is finite s29 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s29 ) } is set
[:(Seg s29),(Seg s29):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(Seg s29),(Seg s29):] is non empty finite V39() set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is non empty unital Group-like associative (O) (O)
(O,s1) is non empty set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s2)),(dom (O,G,s2)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s2)),(dom (O,G,s2)):] is non empty finite V39() set
s19 is Relation-like NAT -defined (O,s1) -valued Function-like finite FinSequence-like FinSubsequence-like (O,s1) FinSequence of (O,s1)
(O,s1,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
the Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):] is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom f1),(dom f1):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom f1),(dom f1):] is non empty finite V39() set
s199 is Relation-like dom f1 -defined dom f1 -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom f1),(dom f1):]
s199 " is Relation-like dom f1 -defined dom f1 -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom f1),(dom f1):]
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(s199 ") . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is non empty unital Group-like associative (O) (O)
f1 . H1 is set
p is non empty unital Group-like associative (O) (O)
i . j is set
len (O,G,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (O,s1,s19) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
(len s2) - 1 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
[:(Seg i),(Seg i):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(Seg i),(Seg i):] is non empty finite V39() set
s199 is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
len (O,G,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s2)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s199 " is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
len (O,s1,s19) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,s1,s19)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom f2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom f2),(dom f2):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom f2),(dom f2):] is non empty finite V39() set
s199 is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
H1 is Relation-like dom f2 -defined dom f2 -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom f2),(dom f2):]
H1 " is Relation-like dom f2 -defined dom f2 -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom f2),(dom f2):]
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . s299 is set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . H1 is set
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (s299 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 . (H1 + 1) is set
len f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len f2) is finite len f2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len f2 ) } is set
(len f2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + s299 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(H1 ") . s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
1 + (len f2) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len f2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H is Element of (O,G)
K is Element of (O,G)
H9 is non empty unital Group-like associative (O) (O,G)
K9 is non empty unital Group-like associative (O) (O,G)
j is non empty unital Group-like associative (O) (O)
f2 . s299 is set
H2 is non empty unital Group-like associative (O) (O)
p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
p . H1 is set
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (Seg i)
bool (Seg i) is non empty finite V39() set
rng s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (Seg i)
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + H1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len s19) is finite len s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s19 ) } is set
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + (len p) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len p) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j9 is Element of (O,s1)
JK is Element of (O,s1)
JK is non empty unital Group-like associative (O) (O,s1)
JH is non empty unital Group-like associative (O) (O,s1)
i9 is non empty unital Group-like associative (O) (O,H9) (O,H9)
(O,H9,i9) is non empty unital Group-like associative (O) (O)
(O,H9,i9) is set
(O,H9,i9) is Relation-like [:(O,H9,i9),(O,H9,i9):] -defined (O,H9,i9) -valued Function-like quasi_total Element of bool [:[:(O,H9,i9),(O,H9,i9):],(O,H9,i9):]
[:(O,H9,i9),(O,H9,i9):] is Relation-like set
[:[:(O,H9,i9),(O,H9,i9):],(O,H9,i9):] is Relation-like set
bool [:[:(O,H9,i9),(O,H9,i9):],(O,H9,i9):] is non empty set
(O,H9,i9) is Relation-like O -defined Funcs ((O,H9,i9),(O,H9,i9)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H9,i9),(O,H9,i9))):]
Funcs ((O,H9,i9),(O,H9,i9)) is functional non empty set
[:O,(Funcs ((O,H9,i9),(O,H9,i9))):] is Relation-like set
bool [:O,(Funcs ((O,H9,i9),(O,H9,i9))):] is non empty set
(O,(O,H9,i9),(O,H9,i9),(O,H9,i9)) is (O) (O)
JH is non empty unital Group-like associative (O) (O,JK) (O,JK)
(O,JK,JH) is non empty unital Group-like associative (O) (O)
(O,JK,JH) is set
(O,JK,JH) is Relation-like [:(O,JK,JH),(O,JK,JH):] -defined (O,JK,JH) -valued Function-like quasi_total Element of bool [:[:(O,JK,JH),(O,JK,JH):],(O,JK,JH):]
[:(O,JK,JH),(O,JK,JH):] is Relation-like set
[:[:(O,JK,JH),(O,JK,JH):],(O,JK,JH):] is Relation-like set
bool [:[:(O,JK,JH),(O,JK,JH):],(O,JK,JH):] is non empty set
(O,JK,JH) is Relation-like O -defined Funcs ((O,JK,JH),(O,JK,JH)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,JK,JH),(O,JK,JH))):]
Funcs ((O,JK,JH),(O,JK,JH)) is functional non empty set
[:O,(Funcs ((O,JK,JH),(O,JK,JH))):] is Relation-like set
bool [:O,(Funcs ((O,JK,JH),(O,JK,JH))):] is non empty set
(O,(O,JK,JH),(O,JK,JH),(O,JK,JH)) is (O) (O)
Seg (len (O,s1,s19)) is finite len (O,s1,s19) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,s1,s19) ) } is set
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
f1 is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
len (O,G,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (O,s1,s19) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
[:(Seg i),(Seg i):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(Seg i),(Seg i):] is non empty finite V39() set
the Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):] is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f2 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,s1)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
s199 . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . j is set
s19 . H2 is set
H1 is non empty unital Group-like associative (O) (O,f2) (O,f2)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (j + 1) is set
j is non empty unital Group-like associative (O) (O,p) (O,p)
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 . (H2 + 1) is set
(O,f2,H1) is non empty unital Group-like associative (O) (O)
(O,f2,H1) is set
(O,f2,H1) is Relation-like [:(O,f2,H1),(O,f2,H1):] -defined (O,f2,H1) -valued Function-like quasi_total Element of bool [:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):]
[:(O,f2,H1),(O,f2,H1):] is Relation-like set
[:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):] is Relation-like set
bool [:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):] is non empty set
(O,f2,H1) is Relation-like O -defined Funcs ((O,f2,H1),(O,f2,H1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f2,H1),(O,f2,H1))):]
Funcs ((O,f2,H1),(O,f2,H1)) is functional non empty set
[:O,(Funcs ((O,f2,H1),(O,f2,H1))):] is Relation-like set
bool [:O,(Funcs ((O,f2,H1),(O,f2,H1))):] is non empty set
(O,(O,f2,H1),(O,f2,H1),(O,f2,H1)) is (O) (O)
(O,p,j) is non empty unital Group-like associative (O) (O)
(O,p,j) is set
(O,p,j) is Relation-like [:(O,p,j),(O,p,j):] -defined (O,p,j) -valued Function-like quasi_total Element of bool [:[:(O,p,j),(O,p,j):],(O,p,j):]
[:(O,p,j),(O,p,j):] is Relation-like set
[:[:(O,p,j),(O,p,j):],(O,p,j):] is Relation-like set
bool [:[:(O,p,j),(O,p,j):],(O,p,j):] is non empty set
(O,p,j) is Relation-like O -defined Funcs ((O,p,j),(O,p,j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,j),(O,p,j))):]
Funcs ((O,p,j),(O,p,j)) is functional non empty set
[:O,(Funcs ((O,p,j),(O,p,j))):] is Relation-like set
bool [:O,(Funcs ((O,p,j),(O,p,j))):] is non empty set
(O,(O,p,j),(O,p,j),(O,p,j)) is (O) (O)
(len (O,s1,s19)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (O,G,s2)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg i is finite i -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
[:(Seg i),(Seg i):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(Seg i),(Seg i):] is non empty finite V39() set
f1 " is Relation-like dom (O,G,s2) -defined dom (O,G,s2) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s2)),(dom (O,G,s2)):]
s199 is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
f2 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,s1)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 is Relation-like Seg i -defined Seg i -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(Seg i),(Seg i):]
s199 . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . j is set
s19 . H2 is set
H1 is non empty unital Group-like associative (O) (O,f2) (O,f2)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (j + 1) is set
j is non empty unital Group-like associative (O) (O,p) (O,p)
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 . (H2 + 1) is set
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (Seg i)
bool (Seg i) is non empty finite V39() set
rng s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (Seg i)
dom (O,s1,s19) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(O,s1,s19) . H2 is set
(O,p,j) is non empty unital Group-like associative (O) (O)
(O,p,j) is set
(O,p,j) is Relation-like [:(O,p,j),(O,p,j):] -defined (O,p,j) -valued Function-like quasi_total Element of bool [:[:(O,p,j),(O,p,j):],(O,p,j):]
[:(O,p,j),(O,p,j):] is Relation-like set
[:[:(O,p,j),(O,p,j):],(O,p,j):] is Relation-like set
bool [:[:(O,p,j),(O,p,j):],(O,p,j):] is non empty set
(O,p,j) is Relation-like O -defined Funcs ((O,p,j),(O,p,j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,j),(O,p,j))):]
Funcs ((O,p,j),(O,p,j)) is functional non empty set
[:O,(Funcs ((O,p,j),(O,p,j))):] is Relation-like set
bool [:O,(Funcs ((O,p,j),(O,p,j))):] is non empty set
(O,(O,p,j),(O,p,j),(O,p,j)) is (O) (O)
(O,G,s2) . j is set
(O,f2,H1) is non empty unital Group-like associative (O) (O)
(O,f2,H1) is set
(O,f2,H1) is Relation-like [:(O,f2,H1),(O,f2,H1):] -defined (O,f2,H1) -valued Function-like quasi_total Element of bool [:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):]
[:(O,f2,H1),(O,f2,H1):] is Relation-like set
[:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):] is Relation-like set
bool [:[:(O,f2,H1),(O,f2,H1):],(O,f2,H1):] is non empty set
(O,f2,H1) is Relation-like O -defined Funcs ((O,f2,H1),(O,f2,H1)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,f2,H1),(O,f2,H1))):]
Funcs ((O,f2,H1),(O,f2,H1)) is functional non empty set
[:O,(Funcs ((O,f2,H1),(O,f2,H1))):] is Relation-like set
bool [:O,(Funcs ((O,f2,H1),(O,f2,H1))):] is non empty set
(O,(O,f2,H1),(O,f2,H1),(O,f2,H1)) is (O) (O)
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s29 is non empty unital Group-like associative (O) (O,G)
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s19 is set
p is non empty unital Group-like associative (O) (O,s29) (O,s29)
s1 . (s19 + 1) is set
(O,s29,p) is non empty unital Group-like associative (O) (O)
(O,s29,p) is set
(O,s29,p) is Relation-like [:(O,s29,p),(O,s29,p):] -defined (O,s29,p) -valued Function-like quasi_total Element of bool [:[:(O,s29,p),(O,s29,p):],(O,s29,p):]
[:(O,s29,p),(O,s29,p):] is Relation-like set
[:[:(O,s29,p),(O,s29,p):],(O,s29,p):] is Relation-like set
bool [:[:(O,s29,p),(O,s29,p):],(O,s29,p):] is non empty set
(O,s29,p) is Relation-like O -defined Funcs ((O,s29,p),(O,s29,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s29,p),(O,s29,p))):]
Funcs ((O,s29,p),(O,s29,p)) is functional non empty set
[:O,(Funcs ((O,s29,p),(O,s29,p))):] is Relation-like set
bool [:O,(Funcs ((O,s29,p),(O,s29,p))):] is non empty set
(O,(O,s29,p),(O,s29,p),(O,s29,p)) is (O) (O)
s29 is non empty unital Group-like associative (O) (O,G)
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s19 is set
p is non empty unital Group-like associative (O) (O,s29) (O,s29)
s1 . (s19 + 1) is set
(O,s29,p) is non empty unital Group-like associative (O) (O)
(O,s29,p) is set
(O,s29,p) is Relation-like [:(O,s29,p),(O,s29,p):] -defined (O,s29,p) -valued Function-like quasi_total Element of bool [:[:(O,s29,p),(O,s29,p):],(O,s29,p):]
[:(O,s29,p),(O,s29,p):] is Relation-like set
[:[:(O,s29,p),(O,s29,p):],(O,s29,p):] is Relation-like set
bool [:[:(O,s29,p),(O,s29,p):],(O,s29,p):] is non empty set
(O,s29,p) is Relation-like O -defined Funcs ((O,s29,p),(O,s29,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s29,p),(O,s29,p))):]
Funcs ((O,s29,p),(O,s29,p)) is functional non empty set
[:O,(Funcs ((O,s29,p),(O,s29,p))):] is Relation-like set
bool [:O,(Funcs ((O,s29,p),(O,s29,p))):] is non empty set
(O,(O,s29,p),(O,s29,p),(O,s29,p)) is (O) (O)
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i is non empty unital Group-like associative (O) (O,G)
s1 . f1 is set
s199 is non empty unital Group-like associative (O) (O,i) (O,i)
s1 . (f1 + 1) is set
(O,i,s199) is non empty unital Group-like associative (O) (O)
(O,i,s199) is set
(O,i,s199) is Relation-like [:(O,i,s199),(O,i,s199):] -defined (O,i,s199) -valued Function-like quasi_total Element of bool [:[:(O,i,s199),(O,i,s199):],(O,i,s199):]
[:(O,i,s199),(O,i,s199):] is Relation-like set
[:[:(O,i,s199),(O,i,s199):],(O,i,s199):] is Relation-like set
bool [:[:(O,i,s199),(O,i,s199):],(O,i,s199):] is non empty set
(O,i,s199) is Relation-like O -defined Funcs ((O,i,s199),(O,i,s199)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,i,s199),(O,i,s199))):]
Funcs ((O,i,s199),(O,i,s199)) is functional non empty set
[:O,(Funcs ((O,i,s199),(O,i,s199))):] is Relation-like set
bool [:O,(Funcs ((O,i,s199),(O,i,s199))):] is non empty set
(O,(O,i,s199),(O,i,s199),(O,i,s199)) is (O) (O)
(O,G,s1) . s19 is set
(s19 + 1) - 1 is V31() V32() integer ext-real set
(len s1) - 1 is V31() V32() integer ext-real set
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,s1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,s1) ) } is set
f1 is non empty unital Group-like associative (O) (O)
G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
G - 1 is V31() V32() integer ext-real set
O + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O - 1 is V31() V32() integer ext-real set
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
G . O is set
<*(G . O)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(G . O)] is set
{1,(G . O)} is non empty finite set
{{1,(G . O)},{1}} is non empty finite V39() set
{[1,(G . O)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len G) - O is V31() V32() integer ext-real set
s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O + s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 ^ s29 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f1 is set
<*f1*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,f1] is set
{1,f1} is non empty finite set
{{1,f1},{1}} is non empty finite V39() set
{[1,f1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
p ^ <*f1*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 . O is set
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O - 1 is V31() V32() integer ext-real set
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del (G,O) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{O} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom G) \ {O} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom G) \ {O}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom G) \ {O})) * G is Relation-like NAT -defined Function-like finite set
s1 ^ s2 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s19 is set
<*s19*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,s19] is set
{1,s19} is non empty finite set
{{1,s19},{1}} is non empty finite V39() set
{[1,s19]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s1 ^ <*s19*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(s1 ^ <*s19*>) ^ s2 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len (Del (G,O)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (Del (G,O))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (s1 ^ <*s19*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (s1 ^ <*s19*>)) + (len s2) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s1) + 1) + (len s2) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O + (len s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) + (len s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (s1 ^ s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len <*s19*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
<*s19*> ^ s2 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
s1 ^ (<*s19*> ^ s2) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len (<*s19*> ^ s2) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O - 1) + (len (<*s19*> ^ s2)) is V31() V32() integer ext-real set
(len G) - (O - 1) is V31() V32() integer ext-real set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(s1 ^ s2) . s29 is set
s1 . s29 is set
G . p is set
(Del (G,O)) . s29 is set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s29 + 1) - O is V31() V32() integer ext-real set
((s29 + 1) - O) + 1 is V31() V32() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s29 + 1) - (O - 1) is V31() V32() integer ext-real set
(s1 ^ s2) . s29 is set
s29 - (len s1) is V31() V32() integer ext-real set
s2 . (s29 - (len s1)) is set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 - 1 is V31() V32() integer ext-real set
s2 . (f1 - 1) is set
(<*s19*> ^ s2) . f1 is set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s1 ^ (<*s19*> ^ s2)) . (p + 1) is set
(Del (G,O)) . s29 is set
(s1 ^ s2) . s29 is set
(Del (G,O)) . s29 is set
(s1 ^ s2) . s29 is set
(Del (G,O)) . s29 is set
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
G is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
G ^ s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Del ((G ^ s1),O) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (G ^ s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{O} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom (G ^ s1)) \ {O} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (G ^ s1)) \ {O}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (G ^ s1)) \ {O})) * (G ^ s1) is Relation-like NAT -defined Function-like finite set
Del (G,O) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom G) \ {O} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom G) \ {O}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom G) \ {O})) * G is Relation-like NAT -defined Function-like finite set
(Del (G,O)) ^ s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(len G) + O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Del ((G ^ s1),((len G) + O)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
{((len G) + O)} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom (G ^ s1)) \ {((len G) + O)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (G ^ s1)) \ {((len G) + O)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (G ^ s1)) \ {((len G) + O)})) * (G ^ s1) is Relation-like NAT -defined Function-like finite set
Del (s1,O) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom s1) \ {O} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s1) \ {O}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s1) \ {O})) * s1 is Relation-like NAT -defined Function-like finite set
G ^ (Del (s1,O)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (G ^ s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len G) + (len s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (G ^ s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O - 1 is V31() V32() integer ext-real set
0 + (len G) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O - 1) + (len G) is V31() V32() integer ext-real set
dom (G ^ s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(G ^ s1) . ((len G) + O) is set
<*((G ^ s1) . ((len G) + O))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((G ^ s1) . ((len G) + O))] is set
{1,((G ^ s1) . ((len G) + O))} is non empty finite set
{{1,((G ^ s1) . ((len G) + O))},{1}} is non empty finite V39() set
{[1,((G ^ s1) . ((len G) + O))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
((len G) + O) - 1 is V31() V32() integer ext-real set
(len (G ^ s1)) - ((len G) + O) is V31() V32() integer ext-real set
p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
p ^ <*((G ^ s1) . ((len G) + O))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(p ^ <*((G ^ s1) . ((len G) + O))*>) ^ f1 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p ^ f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
<*((G ^ s1) . ((len G) + O))*> ^ f1 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
p ^ (<*((G ^ s1) . ((len G) + O))*> ^ f1) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
G ^ i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i ^ <*((G ^ s1) . ((len G) + O))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(i ^ <*((G ^ s1) . ((len G) + O))*>) ^ f1 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
G ^ ((i ^ <*((G ^ s1) . ((len G) + O))*>) ^ f1) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
G ^ (i ^ <*((G ^ s1) . ((len G) + O))*>) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(G ^ (i ^ <*((G ^ s1) . ((len G) + O))*>)) ^ f1 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
(len G) + (O - 1) is V31() V32() integer ext-real set
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len G) + (len i) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i ^ f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(G ^ s1) . O is set
<*((G ^ s1) . O)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,((G ^ s1) . O)] is set
{1,((G ^ s1) . O)} is non empty finite set
{{1,((G ^ s1) . O)},{1}} is non empty finite V39() set
{[1,((G ^ s1) . O)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (G ^ s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
dom (G ^ s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O - 1 is V31() V32() integer ext-real set
(len (G ^ s1)) - O is V31() V32() integer ext-real set
f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f1 ^ <*((G ^ s1) . O)*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(f1 ^ <*((G ^ s1) . O)*>) ^ i is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (f1 ^ <*((G ^ s1) . O)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O - 1) + 1 is V31() V32() integer ext-real set
s199 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(f1 ^ <*((G ^ s1) . O)*>) ^ s199 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
s199 ^ s1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(f1 ^ <*((G ^ s1) . O)*>) ^ (s199 ^ s1) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
f1 ^ s199 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f1 ^ i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
O is non empty set
G is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s1 is Element of O
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Ins (G,s2,s1) is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
G | s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
Seg s2 is finite s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s2 ) } is set
G | (Seg s2) is Relation-like NAT -defined Seg s2 -defined NAT -defined O -valued Function-like finite FinSubsequence-like set
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(G | s2) ^ <*s1*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
G /^ s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
((G | s2) ^ <*s1*>) ^ (G /^ s2) is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
Del ((Ins (G,s2,s1)),(s2 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (Ins (G,s2,s1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{(s2 + 1)} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom (Ins (G,s2,s1))) \ {(s2 + 1)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (Ins (G,s2,s1))) \ {(s2 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (Ins (G,s2,s1))) \ {(s2 + 1)})) * (Ins (G,s2,s1)) is Relation-like NAT -defined O -valued Function-like finite set
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on O
1 -tuples_on O is FinSequenceSet of O
(G | s2) ^ <*s1*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
len G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len G) is finite len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
len (G | s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len ((G | s2) ^ <*s1*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len <*s1*> is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len (G | s2)) + (len <*s1*>) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Del (((G | s2) ^ <*s1*>),(s2 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom ((G | s2) ^ <*s1*>) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
(dom ((G | s2) ^ <*s1*>)) \ {(s2 + 1)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom ((G | s2) ^ <*s1*>)) \ {(s2 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom ((G | s2) ^ <*s1*>)) \ {(s2 + 1)})) * ((G | s2) ^ <*s1*>) is Relation-like NAT -defined O -valued Function-like finite set
(Del (((G | s2) ^ <*s1*>),(s2 + 1))) ^ (G /^ s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(G | s2) ^ (G /^ s2) is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . s2 is set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . s2 is set
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,s1) ) } is set
(len (O,G,s1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
1 + (len (O,G,s1)) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len (O,G,s1)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . s2 is set
s1 . (s2 + 1) is set
s19 is Element of (O,G)
s29 is Element of (O,G)
p is non empty unital Group-like associative (O) (O) (O,G)
f1 is non empty unital Group-like associative (O) (O) (O,G)
i is non empty unital Group-like associative (O) (O) (O,p) (O,p)
(O,p,i) is non empty unital Group-like associative (O) (O)
(O,p,i) is set
(O,p,i) is Relation-like [:(O,p,i),(O,p,i):] -defined (O,p,i) -valued Function-like quasi_total Element of bool [:[:(O,p,i),(O,p,i):],(O,p,i):]
[:(O,p,i),(O,p,i):] is Relation-like set
[:[:(O,p,i),(O,p,i):],(O,p,i):] is Relation-like set
bool [:[:(O,p,i),(O,p,i):],(O,p,i):] is non empty set
(O,p,i) is Relation-like O -defined Funcs ((O,p,i),(O,p,i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,p,i),(O,p,i))):]
Funcs ((O,p,i),(O,p,i)) is functional non empty set
[:O,(Funcs ((O,p,i),(O,p,i))):] is Relation-like set
bool [:O,(Funcs ((O,p,i),(O,p,i))):] is non empty set
(O,(O,p,i),(O,p,i),(O,p,i)) is (O) (O)
(O,(O,p,i)) is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
the carrier of (O,p,i) is non empty set
the multF of (O,p,i) is Relation-like [: the carrier of (O,p,i), the carrier of (O,p,i):] -defined the carrier of (O,p,i) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):]
[: the carrier of (O,p,i), the carrier of (O,p,i):] is Relation-like non empty set
[:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):] is Relation-like non empty set
bool [:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):] is non empty set
the of (O,p,i) is Relation-like O -defined Funcs ( the carrier of (O,p,i), the carrier of (O,p,i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):]
Funcs ( the carrier of (O,p,i), the carrier of (O,p,i)) is functional non empty set
[:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):] is non empty set
(O, the carrier of (O,p,i), the multF of (O,p,i), the of (O,p,i)) is (O) (O)
(O,(O,p,i)) is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
s199 is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
s199 is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
(O,p,i) is Relation-like the carrier of p -defined the carrier of (O,p,i) -valued Function-like non empty total quasi_total unity-preserving multiplicative (O,p,(O,p,i)) Element of bool [: the carrier of p, the carrier of (O,p,i):]
the carrier of p is non empty set
[: the carrier of p, the carrier of (O,p,i):] is Relation-like non empty set
bool [: the carrier of p, the carrier of (O,p,i):] is non empty set
(O,p,(O,p,i),(O,p,i)) is non empty unital Group-like associative (O) (O) (O,p) (O,p)
the carrier of s199 is non empty set
(O,p,i) " the carrier of s199 is Element of bool the carrier of p
bool the carrier of p is non empty set
s199 is non empty unital Group-like associative (O) (O) (O,p)
the carrier of s199 is non empty set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is Element of (O,G)
Ins (s1,f2,p) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
s1 | f2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
Seg f2 is finite f2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= f2 ) } is set
s1 | (Seg f2) is Relation-like NAT -defined Seg f2 -defined NAT -defined (O,G) -valued Function-like finite FinSubsequence-like set
<*p*> is Relation-like NAT -defined (O,G) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (O,G)
[1,p] is set
{1,p} is non empty finite set
{{1,p},{1}} is non empty finite V39() set
{[1,p]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(s1 | f2) ^ <*p*> is Relation-like NAT -defined (O,G) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
s1 /^ f2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
((s1 | f2) ^ <*p*>) ^ (s1 /^ f2) is Relation-like NAT -defined (O,G) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
len (Ins (s1,f2,p)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom (Ins (s1,f2,p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len (Ins (s1,f2,p))) is finite len (Ins (s1,f2,p)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (Ins (s1,f2,p)) ) } is set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(j + 1) - 1 is V31() V32() integer ext-real set
f2 - 1 is V31() V32() integer ext-real set
j is non empty unital Group-like associative (O) (O,G)
(Ins (s1,f2,p)) . j is set
H2 is non empty unital Group-like associative (O) (O,G)
(Ins (s1,f2,p)) . (j + 1) is set
len (s1 | f2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (s1 | f2)) is finite len (s1 | f2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (s1 | f2) ) } is set
dom (s1 | f2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
0 + f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
- 1 is V31() V32() integer ext-real non positive set
(- 1) + f2 is V31() V32() integer ext-real set
(f2 - 1) + 1 is V31() V32() integer ext-real set
(Ins (s1,f2,p)) /. (j + 1) is Element of (O,G)
s1 /. (j + 1) is Element of (O,G)
s1 . (j + 1) is set
(Ins (s1,f2,p)) /. j is Element of (O,G)
s1 /. j is Element of (O,G)
s1 . j is set
len (s1 | f2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (s1 | f2)) is finite len (s1 | f2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (s1 | f2) ) } is set
dom (s1 | f2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(Ins (s1,f2,p)) /. j is Element of (O,G)
s1 /. j is Element of (O,G)
s1 . j is set
(Ins (s1,f2,p)) /. (f2 + 1) is Element of (O,G)
(Ins (s1,f2,p)) /. (f2 + 1) is Element of (O,G)
(f2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(Ins (s1,f2,p)) /. ((f2 + 1) + 1) is Element of (O,G)
s1 /. (f2 + 1) is Element of (O,G)
j - 1 is V31() V32() integer ext-real set
(0 + 1) - 1 is V31() V32() integer ext-real set
(len (Ins (s1,f2,p))) - 1 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(f2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((f2 + 1) + 1) - 1 is V31() V32() integer ext-real set
1 + H1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(H1 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(Ins (s1,f2,p)) /. ((H1 + 1) + 1) is Element of (O,G)
s1 /. (H1 + 1) is Element of (O,G)
s1 . (H1 + 1) is set
(Ins (s1,f2,p)) /. (H1 + 1) is Element of (O,G)
s1 /. H1 is Element of (O,G)
s1 . H1 is set
(Ins (s1,f2,p)) . (len (Ins (s1,f2,p))) is set
(Ins (s1,f2,p)) /. (len (Ins (s1,f2,p))) is Element of (O,G)
(Ins (s1,f2,p)) /. ((len s1) + 1) is Element of (O,G)
s1 /. (len s1) is Element of (O,G)
s1 . (len s1) is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(Ins (s1,f2,p)) . 1 is set
(Ins (s1,f2,p)) /. 1 is Element of (O,G)
s1 /. 1 is Element of (O,G)
s1 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom j is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
len j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len j) is finite len j -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len j ) } is set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(j + 1) - 1 is V31() V32() integer ext-real set
H2 is non empty unital Group-like associative (O) (O,G)
j . j is set
s299 is non empty unital Group-like associative (O) (O,H2) (O,H2)
j . (j + 1) is set
len (s1 | f2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (s1 | f2)) is finite len (s1 | f2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (s1 | f2) ) } is set
dom (s1 | f2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
0 + f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
- 1 is V31() V32() integer ext-real non positive set
(- 1) + f2 is V31() V32() integer ext-real set
(f2 - 1) + 1 is V31() V32() integer ext-real set
j /. (j + 1) is Element of (O,G)
s1 /. (j + 1) is Element of (O,G)
s1 . (j + 1) is set
j /. j is Element of (O,G)
s1 /. j is Element of (O,G)
s1 . j is set
(O,H2,s299) is non empty unital Group-like associative (O) (O)
(O,H2,s299) is set
(O,H2,s299) is Relation-like [:(O,H2,s299),(O,H2,s299):] -defined (O,H2,s299) -valued Function-like quasi_total Element of bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):]
[:(O,H2,s299),(O,H2,s299):] is Relation-like set
[:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is Relation-like set
bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is non empty set
(O,H2,s299) is Relation-like O -defined Funcs ((O,H2,s299),(O,H2,s299)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):]
Funcs ((O,H2,s299),(O,H2,s299)) is functional non empty set
[:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is Relation-like set
bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is non empty set
(O,(O,H2,s299),(O,H2,s299),(O,H2,s299)) is (O) (O)
len (s1 | f2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (s1 | f2)) is finite len (s1 | f2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (s1 | f2) ) } is set
dom (s1 | f2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j /. (f2 + 1) is Element of (O,G)
j /. j is Element of (O,G)
s1 /. j is Element of (O,G)
s1 . j is set
(O,H2,s299) is non empty unital Group-like associative (O) (O)
(O,H2,s299) is set
(O,H2,s299) is Relation-like [:(O,H2,s299),(O,H2,s299):] -defined (O,H2,s299) -valued Function-like quasi_total Element of bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):]
[:(O,H2,s299),(O,H2,s299):] is Relation-like set
[:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is Relation-like set
bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is non empty set
(O,H2,s299) is Relation-like O -defined Funcs ((O,H2,s299),(O,H2,s299)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):]
Funcs ((O,H2,s299),(O,H2,s299)) is functional non empty set
[:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is Relation-like set
bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is non empty set
(O,(O,H2,s299),(O,H2,s299),(O,H2,s299)) is (O) (O)
H1 is non empty unital Group-like associative (O) (O,p) (O,p)
j /. (f2 + 1) is Element of (O,G)
(f2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j /. ((f2 + 1) + 1) is Element of (O,G)
s1 /. (f2 + 1) is Element of (O,G)
(O,H2,s299) is non empty unital Group-like associative (O) (O)
(O,H2,s299) is set
(O,H2,s299) is Relation-like [:(O,H2,s299),(O,H2,s299):] -defined (O,H2,s299) -valued Function-like quasi_total Element of bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):]
[:(O,H2,s299),(O,H2,s299):] is Relation-like set
[:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is Relation-like set
bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is non empty set
(O,H2,s299) is Relation-like O -defined Funcs ((O,H2,s299),(O,H2,s299)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):]
Funcs ((O,H2,s299),(O,H2,s299)) is functional non empty set
[:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is Relation-like set
bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is non empty set
(O,(O,H2,s299),(O,H2,s299),(O,H2,s299)) is (O) (O)
the carrier of i is non empty set
j - 1 is V31() V32() integer ext-real set
(0 + 1) - 1 is V31() V32() integer ext-real set
(len j) - 1 is V31() V32() integer ext-real set
(f2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((f2 + 1) + 1) - 1 is V31() V32() integer ext-real set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + H2 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(H2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j /. ((H2 + 1) + 1) is Element of (O,G)
s1 /. (H2 + 1) is Element of (O,G)
s1 . (H2 + 1) is set
j /. (H2 + 1) is Element of (O,G)
s1 /. H2 is Element of (O,G)
s1 . H2 is set
(O,H2,s299) is non empty unital Group-like associative (O) (O)
(O,H2,s299) is set
(O,H2,s299) is Relation-like [:(O,H2,s299),(O,H2,s299):] -defined (O,H2,s299) -valued Function-like quasi_total Element of bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):]
[:(O,H2,s299),(O,H2,s299):] is Relation-like set
[:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is Relation-like set
bool [:[:(O,H2,s299),(O,H2,s299):],(O,H2,s299):] is non empty set
(O,H2,s299) is Relation-like O -defined Funcs ((O,H2,s299),(O,H2,s299)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):]
Funcs ((O,H2,s299),(O,H2,s299)) is functional non empty set
[:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is Relation-like set
bool [:O,(Funcs ((O,H2,s299),(O,H2,s299))):] is non empty set
(O,(O,H2,s299),(O,H2,s299),(O,H2,s299)) is (O) (O)
{(f2 + 1)} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom j) \ {(f2 + 1)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Del (j,(f2 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom j is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom j) \ {(f2 + 1)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom j) \ {(f2 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom j) \ {(f2 + 1)})) * j is Relation-like NAT -defined (O,G) -valued Function-like finite set
(O,(O,p,i)) is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
the carrier of (O,p,i) is non empty set
the multF of (O,p,i) is Relation-like [: the carrier of (O,p,i), the carrier of (O,p,i):] -defined the carrier of (O,p,i) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):]
[: the carrier of (O,p,i), the carrier of (O,p,i):] is Relation-like non empty set
[:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):] is Relation-like non empty set
bool [:[: the carrier of (O,p,i), the carrier of (O,p,i):], the carrier of (O,p,i):] is non empty set
the of (O,p,i) is Relation-like O -defined Funcs ( the carrier of (O,p,i), the carrier of (O,p,i)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):]
Funcs ( the carrier of (O,p,i), the carrier of (O,p,i)) is functional non empty set
[:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,p,i), the carrier of (O,p,i))):] is non empty set
(O, the carrier of (O,p,i), the multF of (O,p,i), the of (O,p,i)) is (O) (O)
(O,(O,p,i)) is non empty unital Group-like associative (O) (O) (O,(O,p,i)) (O,(O,p,i))
s19 is non empty unital Group-like associative (O) (O,G)
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s2 is set
s29 is non empty unital Group-like associative (O) (O,s19) (O,s19)
s1 . (s2 + 1) is set
(O,s19,s29) is non empty unital Group-like associative (O) (O)
(O,s19,s29) is set
(O,s19,s29) is Relation-like [:(O,s19,s29),(O,s19,s29):] -defined (O,s19,s29) -valued Function-like quasi_total Element of bool [:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):]
[:(O,s19,s29),(O,s19,s29):] is Relation-like set
[:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):] is Relation-like set
bool [:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):] is non empty set
(O,s19,s29) is Relation-like O -defined Funcs ((O,s19,s29),(O,s19,s29)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s19,s29),(O,s19,s29))):]
Funcs ((O,s19,s29),(O,s19,s29)) is functional non empty set
[:O,(Funcs ((O,s19,s29),(O,s19,s29))):] is Relation-like set
bool [:O,(Funcs ((O,s19,s29),(O,s19,s29))):] is non empty set
(O,(O,s19,s29),(O,s19,s29),(O,s19,s29)) is (O) (O)
s19 is non empty unital Group-like associative (O) (O,G)
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . s2 is set
s29 is non empty unital Group-like associative (O) (O,s19) (O,s19)
s1 . (s2 + 1) is set
(O,s19,s29) is non empty unital Group-like associative (O) (O)
(O,s19,s29) is set
(O,s19,s29) is Relation-like [:(O,s19,s29),(O,s19,s29):] -defined (O,s19,s29) -valued Function-like quasi_total Element of bool [:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):]
[:(O,s19,s29),(O,s19,s29):] is Relation-like set
[:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):] is Relation-like set
bool [:[:(O,s19,s29),(O,s19,s29):],(O,s19,s29):] is non empty set
(O,s19,s29) is Relation-like O -defined Funcs ((O,s19,s29),(O,s19,s29)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s19,s29),(O,s19,s29))):]
Funcs ((O,s19,s29),(O,s19,s29)) is functional non empty set
[:O,(Funcs ((O,s19,s29),(O,s19,s29))):] is Relation-like set
bool [:O,(Funcs ((O,s19,s29),(O,s19,s29))):] is non empty set
(O,(O,s19,s29),(O,s19,s29),(O,s19,s29)) is (O) (O)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
(s2 + 1) - 1 is V31() V32() integer ext-real set
(O,G,s1) . s2 is set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . s19 is set
s2 . s19 is set
s1 . (s19 + 1) is set
s2 . (s19 + 1) is set
s2 . s29 is set
p is Element of (O,G)
f1 is Element of (O,G)
i is Element of (O,G)
s199 is non empty unital Group-like associative (O) (O) (O,G)
s199 is non empty unital Group-like associative (O) (O) (O,G)
f2 is non empty unital Group-like associative (O) (O) (O,G)
H1 is non empty unital Group-like associative (O) (O) (O,s199) (O,s199)
p is non empty unital Group-like associative (O) (O) (O,s199) (O,s199)
(O,s199,p) is non empty unital Group-like associative (O) (O)
(O,s199,p) is set
(O,s199,p) is Relation-like [:(O,s199,p),(O,s199,p):] -defined (O,s199,p) -valued Function-like quasi_total Element of bool [:[:(O,s199,p),(O,s199,p):],(O,s199,p):]
[:(O,s199,p),(O,s199,p):] is Relation-like set
[:[:(O,s199,p),(O,s199,p):],(O,s199,p):] is Relation-like set
bool [:[:(O,s199,p),(O,s199,p):],(O,s199,p):] is non empty set
(O,s199,p) is Relation-like O -defined Funcs ((O,s199,p),(O,s199,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,s199,p),(O,s199,p))):]
Funcs ((O,s199,p),(O,s199,p)) is functional non empty set
[:O,(Funcs ((O,s199,p),(O,s199,p))):] is Relation-like set
bool [:O,(Funcs ((O,s199,p),(O,s199,p))):] is non empty set
(O,(O,s199,p),(O,s199,p),(O,s199,p)) is (O) (O)
j is non empty unital Group-like associative (O) (O,H1) (O,H1)
(O,H1,j) is non empty unital Group-like associative (O) (O)
(O,H1,j) is set
(O,H1,j) is Relation-like [:(O,H1,j),(O,H1,j):] -defined (O,H1,j) -valued Function-like quasi_total Element of bool [:[:(O,H1,j),(O,H1,j):],(O,H1,j):]
[:(O,H1,j),(O,H1,j):] is Relation-like set
[:[:(O,H1,j),(O,H1,j):],(O,H1,j):] is Relation-like set
bool [:[:(O,H1,j),(O,H1,j):],(O,H1,j):] is non empty set
(O,H1,j) is Relation-like O -defined Funcs ((O,H1,j),(O,H1,j)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H1,j),(O,H1,j))):]
Funcs ((O,H1,j),(O,H1,j)) is functional non empty set
[:O,(Funcs ((O,H1,j),(O,H1,j))):] is Relation-like set
bool [:O,(Funcs ((O,H1,j),(O,H1,j))):] is non empty set
(O,(O,H1,j),(O,H1,j),(O,H1,j)) is (O) (O)
j is non empty unital Group-like associative (O) (O) (O,(O,s199,p)) (O,(O,s199,p))
(O,(O,s199,p)) is non empty unital Group-like associative (O) (O) (O,(O,s199,p)) (O,(O,s199,p))
the carrier of (O,s199,p) is non empty set
the multF of (O,s199,p) is Relation-like [: the carrier of (O,s199,p), the carrier of (O,s199,p):] -defined the carrier of (O,s199,p) -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of (O,s199,p), the carrier of (O,s199,p):], the carrier of (O,s199,p):]
[: the carrier of (O,s199,p), the carrier of (O,s199,p):] is Relation-like non empty set
[:[: the carrier of (O,s199,p), the carrier of (O,s199,p):], the carrier of (O,s199,p):] is Relation-like non empty set
bool [:[: the carrier of (O,s199,p), the carrier of (O,s199,p):], the carrier of (O,s199,p):] is non empty set
the of (O,s199,p) is Relation-like O -defined Funcs ( the carrier of (O,s199,p), the carrier of (O,s199,p)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of (O,s199,p), the carrier of (O,s199,p))):]
Funcs ( the carrier of (O,s199,p), the carrier of (O,s199,p)) is functional non empty set
[:O,(Funcs ( the carrier of (O,s199,p), the carrier of (O,s199,p))):] is Relation-like set
bool [:O,(Funcs ( the carrier of (O,s199,p), the carrier of (O,s199,p))):] is non empty set
(O, the carrier of (O,s199,p), the multF of (O,s199,p), the of (O,s199,p)) is (O) (O)
the carrier of H1 is non empty set
union (O,s199,p) is set
H2 is non empty unital Group-like associative (O) (O) (O,H1) (O,H1)
(O,H1) is non empty unital Group-like associative (O) (O) (O,H1) (O,H1)
the multF of H1 is Relation-like [: the carrier of H1, the carrier of H1:] -defined the carrier of H1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:]
[: the carrier of H1, the carrier of H1:] is Relation-like non empty set
[:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is Relation-like non empty set
bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is non empty set
the of H1 is Relation-like O -defined Funcs ( the carrier of H1, the carrier of H1) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of H1, the carrier of H1)):]
Funcs ( the carrier of H1, the carrier of H1) is functional non empty set
[:O,(Funcs ( the carrier of H1, the carrier of H1)):] is Relation-like set
bool [:O,(Funcs ( the carrier of H1, the carrier of H1)):] is non empty set
(O, the carrier of H1, the multF of H1, the of H1) is (O) (O)
(O,H1,H2) is non empty unital Group-like associative (O) (O)
(O,H1,H2) is set
(O,H1,H2) is Relation-like [:(O,H1,H2),(O,H1,H2):] -defined (O,H1,H2) -valued Function-like quasi_total Element of bool [:[:(O,H1,H2),(O,H1,H2):],(O,H1,H2):]
[:(O,H1,H2),(O,H1,H2):] is Relation-like set
[:[:(O,H1,H2),(O,H1,H2):],(O,H1,H2):] is Relation-like set
bool [:[:(O,H1,H2),(O,H1,H2):],(O,H1,H2):] is non empty set
(O,H1,H2) is Relation-like O -defined Funcs ((O,H1,H2),(O,H1,H2)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,H1,H2),(O,H1,H2))):]
Funcs ((O,H1,H2),(O,H1,H2)) is functional non empty set
[:O,(Funcs ((O,H1,H2),(O,H1,H2))):] is Relation-like set
bool [:O,(Funcs ((O,H1,H2),(O,H1,H2))):] is non empty set
(O,(O,H1,H2),(O,H1,H2),(O,H1,H2)) is (O) (O)
the carrier of p is non empty set
the multF of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is Relation-like non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is Relation-like non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the of p is Relation-like O -defined Funcs ( the carrier of p, the carrier of p) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of p, the carrier of p)):]
Funcs ( the carrier of p, the carrier of p) is functional non empty set
[:O,(Funcs ( the carrier of p, the carrier of p)):] is Relation-like set
bool [:O,(Funcs ( the carrier of p, the carrier of p)):] is non empty set
(O, the carrier of p, the multF of p, the of p) is (O) (O)
(O,(O,s199,p)) is non empty unital Group-like associative (O) (O) (O,(O,s199,p)) (O,(O,s199,p))
H2 is non empty unital Group-like associative (O) (O) (O,H1) (O,H1)
(O,H1,H2) is set
union (O,H1,H2) is set
1_ (O,s199,p) is non being_of_order_0 Element of the carrier of (O,s199,p)
{(1_ (O,s199,p))} is non empty trivial finite 1 -element set
union {(1_ (O,s199,p))} is set
(O,s199,p) is Element of bool the carrier of s199
the carrier of s199 is non empty set
bool the carrier of s199 is non empty set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
(s19 + 1) - 1 is V31() V32() integer ext-real set
(O,G,s1) . s19 is set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . s2 is set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . s19 is set
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 . s2 is set
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s2 + 1) is set
0 + s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s2 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s1) - 1) + 1 is V31() V32() integer ext-real set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is non empty unital Group-like associative (O) (O,G)
s2 is non empty unital Group-like associative (O) (O,G)
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s19) - 1 is V31() V32() integer ext-real set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 . s29 is set
s29 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 . (s29 + 1) is set
((len s19) - 1) + 1 is V31() V32() integer ext-real set
0 + s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s29 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (len s19) is finite len s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s19 ) } is set
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
id (dom (O,G,s1)) is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
[:(dom (O,G,s1)),(dom (O,G,s1)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s1)),(dom (O,G,s1)):] is non empty finite V39() set
s29 is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like total quasi_total finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
rng s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom (O,G,s1))
bool (dom (O,G,s1)) is non empty finite V39() set
p is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
f1 " is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
(f1 ") . s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i is non empty unital Group-like associative (O) (O)
(O,G,s1) . s199 is set
s199 is non empty unital Group-like associative (O) (O)
(O,G,s1) . f2 is set
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . 1 is set
<*(s1 . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(s1 . 1)] is set
{1,(s1 . 1)} is non empty finite set
{{1,(s1 . 1)},{1}} is non empty finite V39() set
{[1,(s1 . 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
<*1*> is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like V191() V192() V193() V194() increasing V196() V197() V198() Element of 1 -tuples_on NAT
1 -tuples_on NAT is FinSequenceSet of NAT
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng s2 is finite Element of bool (O,G)
bool (O,G) is non empty set
s19 is non empty set
[:s19,(O,G):] is Relation-like non empty set
bool [:s19,(O,G):] is non empty set
p is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
rng <*1*> is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
Sgm p is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s199 is Relation-like NAT -defined s19 -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of s19
s199 is Relation-like s19 -defined (O,G) -valued Function-like non empty total quasi_total Element of bool [:s19,(O,G):]
s199 * s199 is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s199 . 1 is set
<*(s199 . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(s199 . 1)] is set
{1,(s199 . 1)} is non empty finite set
{{1,(s199 . 1)},{1}} is non empty finite V39() set
{[1,(s199 . 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s2 * (Sgm p) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
<*(O,G)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(O,G)] is set
{1,(O,G)} is non empty finite set
{{1,(O,G)},{1}} is non empty finite V39() set
{[1,(O,G)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s19 is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
Sgm s19 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm s19) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm s29 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm s29) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s19 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
Sgm s19 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm s19) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
<*(s1 . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(s1 . 1)] is set
{1,(s1 . 1)} is non empty finite set
{{1,(s1 . 1)},{1}} is non empty finite V39() set
{[1,(s1 . 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
s2 . 1 is set
<*(s2 . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(s2 . 1)] is set
{1,(s2 . 1)} is non empty finite set
{{1,(s2 . 1)},{1}} is non empty finite V39() set
{[1,(s2 . 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s1)),(dom (O,G,s1)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s1)),(dom (O,G,s1)):] is non empty finite V39() set
p is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
len (O,G,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len (O,G,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p . f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s1) . (p . f1) is set
(O,G,s2) . f1 is set
dom (O,G,s2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng (O,G,s2) is finite set
Seg (len (O,G,s1)) is finite len (O,G,s1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,s1) ) } is set
Seg (len (O,G,s2)) is finite len (O,G,s2) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,s2) ) } is set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1) . f2 is set
rng (O,G,s1) is finite set
p " is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
(p ") . f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p is non empty unital Group-like associative (O) (O) (O)
H1 is non empty unital Group-like associative (O) (O) (O)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s1) - 1 is V31() V32() integer ext-real set
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) - 1 is V31() V32() integer ext-real set
((len s1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((len s1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg s29 is finite s29 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s29 ) } is set
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 - 1 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 div s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s19 div s199) * s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 mod s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((s19 div s199) * s199) + (s19 mod s199) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s19 div s199) - 1 is V31() V32() integer ext-real set
((s19 div s199) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((s19 div s199) - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
(s19 div s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s1) - 1) * s199 is V31() V32() integer ext-real set
((s19 div s199) * s199) / s199 is V31() V32() ext-real non negative set
(((len s1) - 1) * s199) / s199 is V31() V32() ext-real set
(s19 div s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((s19 div s199) + 1) - 1 is V31() V32() integer ext-real set
(((s19 div s199) + 1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((((s19 div s199) + 1) - 1) * ((len s2) - 1)) + (s19 mod s199) is V31() V32() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s19 + 1) - 1 is V31() V32() integer ext-real set
s29 - 1 is V31() V32() integer ext-real set
s199 * (s19 div s199) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + (s199 * (s19 div s199)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s19 mod s199) + (s199 * (s19 div s199)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i * s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((len s1) - 1) * s199 is V31() V32() integer ext-real set
(((len s1) - 1) * s199) / s199 is V31() V32() ext-real set
((s19 div s199) * s199) / s199 is V31() V32() ext-real non negative set
(s19 mod s199) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 - 1 is V31() V32() integer ext-real set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s19 - 1 is V31() V32() integer ext-real set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len f1) - 1 is V31() V32() integer ext-real set
(s1 - 1) * ((len f1) - 1) is V31() V32() integer ext-real set
((s1 - 1) * ((len f1) - 1)) + s2 is V31() V32() integer ext-real set
(s19 - 1) * ((len f1) - 1) is V31() V32() integer ext-real set
((s19 - 1) * ((len f1) - 1)) + s29 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 / f2 is V31() V32() ext-real non negative set
s2 mod f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p * f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(p * f2) + s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((p * f2) + s2) mod f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 * f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(H1 * f2) + s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H1 * f2) + s29) mod f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 mod f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p * (f2 / f2) is V31() V32() ext-real non negative set
(H1 * f2) / f2 is V31() V32() ext-real non negative set
p * 1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 * 1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is V31() V32() integer ext-real set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 - 1 is V31() V32() integer ext-real set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s29) - 1 is V31() V32() integer ext-real set
p is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len p) - 1 is V31() V32() integer ext-real set
(s2 - 1) * ((len p) - 1) is V31() V32() integer ext-real set
((s2 - 1) * ((len p) - 1)) + s19 is V31() V32() integer ext-real set
((len s29) - 1) * ((len p) - 1) is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
((len s29) - 1) - 1 is V31() V32() integer ext-real set
(((len s29) - 1) - 1) * ((len p) - 1) is V31() V32() integer ext-real set
1 * ((len p) - 1) is V31() V32() integer ext-real set
(((len s29) - 1) * ((len p) - 1)) - (1 * ((len p) - 1)) is V31() V32() integer ext-real set
((((len s29) - 1) * ((len p) - 1)) - (1 * ((len p) - 1))) + ((len p) - 1) is V31() V32() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s2) - 1 is V31() V32() integer ext-real set
(len s1) - 1 is V31() V32() integer ext-real set
((len s1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((len s1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
1 - 1 is V31() V32() integer ext-real set
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 - 1 is V31() V32() integer ext-real set
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 / s19 is V31() V32() ext-real non negative set
(len s1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg p is finite p -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
1 mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 * s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s29 * s19) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((s29 * s19) + 1) mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 * s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(f2 * s19) + s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((f2 * s19) + s199) mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 * (s19 / s19) is V31() V32() ext-real non negative set
(f2 * s19) / s19 is V31() V32() ext-real non negative set
s29 * 1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f2 * 1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
- 1 is V31() V32() integer ext-real non positive set
(- 1) + i is V31() V32() integer ext-real set
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (i + 1) is set
s1 . i is set
s2 . s199 is set
s199 is non empty unital Group-like associative (O) (O,G)
f2 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
(O,G,f2,p) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s199,(O,G,f2,p)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s199) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s199 is non empty set
(O,G,(O,G,f2,p)) is Element of bool the carrier of G
the carrier of (O,G,f2,p) is non empty set
(O,G,s199) \/ (O,G,(O,G,f2,p)) is Element of bool the carrier of G
(O,G,((O,G,s199) \/ (O,G,(O,G,f2,p)))) is non empty unital Group-like associative (O) (O) (O,G)
H1 is non empty unital Group-like associative (O) (O) (O,G)
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((j - 1) * ((len s2) - 1)) + j is V31() V32() integer ext-real set
H2 is non empty unital Group-like associative (O) (O,G)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (j + 1) is set
s299 is non empty unital Group-like associative (O) (O,G)
s1 . j is set
H1 is non empty unital Group-like associative (O) (O,G)
s2 . j is set
(O,G,s299,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2,(O,G,s299,H1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2) is Element of bool the carrier of G
the carrier of H2 is non empty set
(O,G,(O,G,s299,H1)) is Element of bool the carrier of G
the carrier of (O,G,s299,H1) is non empty set
(O,G,H2) \/ (O,G,(O,G,s299,H1)) is Element of bool the carrier of G
(O,G,((O,G,H2) \/ (O,G,(O,G,s299,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((H2 - 1) * ((len s2) - 1)) + s299 is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((j - 1) * ((len s2) - 1)) + j is V31() V32() integer ext-real set
H2 is non empty unital Group-like associative (O) (O,G)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (j + 1) is set
s299 is non empty unital Group-like associative (O) (O,G)
s1 . j is set
H1 is non empty unital Group-like associative (O) (O,G)
s2 . j is set
(O,G,s299,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2,(O,G,s299,H1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2) is Element of bool the carrier of G
the carrier of H2 is non empty set
(O,G,(O,G,s299,H1)) is Element of bool the carrier of G
the carrier of (O,G,s299,H1) is non empty set
(O,G,H2) \/ (O,G,(O,G,s299,H1)) is Element of bool the carrier of G
(O,G,((O,G,H2) \/ (O,G,(O,G,s299,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
s199 is non empty unital Group-like associative (O) (O,G)
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (i + 1) is set
f2 is non empty unital Group-like associative (O) (O,G)
s1 . i is set
p is non empty unital Group-like associative (O) (O,G)
s2 . s199 is set
(O,G,f2,p) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s199,(O,G,f2,p)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s199) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s199 is non empty set
(O,G,(O,G,f2,p)) is Element of bool the carrier of G
the carrier of (O,G,f2,p) is non empty set
(O,G,s199) \/ (O,G,(O,G,f2,p)) is Element of bool the carrier of G
(O,G,((O,G,s199) \/ (O,G,(O,G,f2,p)))) is non empty unital Group-like associative (O) (O) (O,G)
f1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f1 . i is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((s199 - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
f2 is non empty unital Group-like associative (O) (O,G)
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s199 + 1) is set
p is non empty unital Group-like associative (O) (O,G)
s1 . s199 is set
H1 is non empty unital Group-like associative (O) (O,G)
s2 . s199 is set
(O,G,p,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,f2,(O,G,p,H1)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,f2) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of f2 is non empty set
(O,G,(O,G,p,H1)) is Element of bool the carrier of G
the carrier of (O,G,p,H1) is non empty set
(O,G,f2) \/ (O,G,(O,G,p,H1)) is Element of bool the carrier of G
(O,G,((O,G,f2) \/ (O,G,(O,G,p,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((s199 - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((s199 - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s199 + 1) is set
s1 . s199 is set
s2 . s199 is set
f2 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
H1 is non empty unital Group-like associative (O) (O,G)
(O,G,p,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,f2,(O,G,p,H1)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,f2) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of f2 is non empty set
(O,G,(O,G,p,H1)) is Element of bool the carrier of G
the carrier of (O,G,p,H1) is non empty set
(O,G,f2) \/ (O,G,(O,G,p,H1)) is Element of bool the carrier of G
(O,G,((O,G,f2) \/ (O,G,(O,G,p,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
i is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
i . p is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((s199 - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
f2 is non empty unital Group-like associative (O) (O,G)
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s199 + 1) is set
p is non empty unital Group-like associative (O) (O,G)
s1 . s199 is set
H1 is non empty unital Group-like associative (O) (O,G)
s2 . s199 is set
(O,G,p,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,f2,(O,G,p,H1)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,f2) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of f2 is non empty set
(O,G,(O,G,p,H1)) is Element of bool the carrier of G
the carrier of (O,G,p,H1) is non empty set
(O,G,f2) \/ (O,G,(O,G,p,H1)) is Element of bool the carrier of G
(O,G,((O,G,f2) \/ (O,G,(O,G,p,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i . (len i) is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s199 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len s199) - 1 is V31() V32() integer ext-real set
s199 . s199 is set
s199 . (s199 + 1) is set
f2 is non empty unital Group-like associative (O) (O)
0 + s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s199 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s199) - 1) + 1 is V31() V32() integer ext-real set
Seg (len s199) is finite len s199 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s199 ) } is set
dom s199 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
dom i is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
i . s199 is set
i . (s199 + 1) is set
f2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
f2 - 1 is V31() V32() integer ext-real set
(f2 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((f2 - 1) * ((len s2) - 1)) + p is V31() V32() integer ext-real set
f2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (f2 + 1) is set
s1 . f2 is set
s2 . p is set
H1 is non empty unital Group-like associative (O) (O) (O,G)
j is non empty unital Group-like associative (O) (O) (O,G)
j is non empty unital Group-like associative (O) (O) (O,G)
(O,G,j,j) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,(O,G,j,j)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,H1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H1 is non empty set
(O,G,(O,G,j,j)) is Element of bool the carrier of G
the carrier of (O,G,j,j) is non empty set
(O,G,H1) \/ (O,G,(O,G,j,j)) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,(O,G,j,j)))) is non empty unital Group-like associative (O) (O) (O,G)
H2 is non empty unital Group-like associative (O) (O,G)
s299 is non empty unital Group-like associative (O) (O,G)
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s2 . H1 is set
0 + p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + p is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is non empty unital Group-like associative (O) (O) (O,G)
((f2 - 1) * ((len s2) - 1)) + H1 is V31() V32() integer ext-real set
(O,G,j,H2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,(O,G,j,H2)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,j,H2)) is Element of bool the carrier of G
the carrier of (O,G,j,H2) is non empty set
(O,G,H1) \/ (O,G,(O,G,j,H2)) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,(O,G,j,H2)))) is non empty unital Group-like associative (O) (O) (O,G)
0 + (f2 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + (f2 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
the carrier of H2 is non empty set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(f2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((len s1) - 1) + 1 is V31() V32() integer ext-real set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s1 . ((f2 + 1) + 1) is set
s2 . 1 is set
s299 is non empty unital Group-like associative (O) (O) (O,G)
i is non empty unital Group-like associative (O) (O) (O,G)
the carrier of i is non empty set
H is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
the carrier of (O,G) is non empty set
the carrier of i /\ the carrier of (O,G) is set
j is non empty unital Group-like associative (O) (O) (O,G)
(f2 + 1) - 1 is V31() V32() integer ext-real set
((f2 + 1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((f2 + 1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
(O,G,i,j) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s299,(O,G,i,j)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s299) is Element of bool the carrier of G
the carrier of s299 is non empty set
(O,G,(O,G,i,j)) is Element of bool the carrier of G
the carrier of (O,G,i,j) is non empty set
(O,G,s299) \/ (O,G,(O,G,i,j)) is Element of bool the carrier of G
(O,G,((O,G,s299) \/ (O,G,(O,G,i,j)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s299,i) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,i) is Element of bool the carrier of G
(O,G,s299) \/ (O,G,i) is Element of bool the carrier of G
(O,G,((O,G,s299) \/ (O,G,i))) is non empty unital Group-like associative (O) (O) (O,G)
H is non empty strict unital Group-like associative Subgroup of H2
the carrier of s299 is non empty set
the multF of s299 is Relation-like [: the carrier of s299, the carrier of s299:] -defined the carrier of s299 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of s299, the carrier of s299:], the carrier of s299:]
[: the carrier of s299, the carrier of s299:] is Relation-like non empty set
[:[: the carrier of s299, the carrier of s299:], the carrier of s299:] is Relation-like non empty set
bool [:[: the carrier of s299, the carrier of s299:], the carrier of s299:] is non empty set
multMagma(# the carrier of s299, the multF of s299 #) is non empty strict multMagma
the carrier of j is non empty set
K is Element of the carrier of H2
the multF of H1 is Relation-like [: the carrier of H1, the carrier of H1:] -defined the carrier of H1 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:]
[: the carrier of H1, the carrier of H1:] is Relation-like non empty set
[:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is Relation-like non empty set
bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is non empty set
multMagma(# the carrier of H1, the multF of H1 #) is non empty strict multMagma
i9 is set
K * H is Element of bool the carrier of H2
bool the carrier of H2 is non empty set
carr H is Element of bool the carrier of H2
the carrier of H is non empty set
K * (carr H) is Element of bool the carrier of H2
K350( the carrier of H2,K) is non empty trivial finite 1 -element Element of bool the carrier of H2
K350( the carrier of H2,K) * (carr H) is Element of bool the carrier of H2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H2 : ( b₁ in K350( the carrier of H2,K) & b₂ in carr H ) } is set
j9 is Element of the carrier of H2
K * j9 is Element of the carrier of H2
the multF of H2 is Relation-like [: the carrier of H2, the carrier of H2:] -defined the carrier of H2 -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of H2, the carrier of H2:], the carrier of H2:]
[: the carrier of H2, the carrier of H2:] is Relation-like non empty set
[:[: the carrier of H2, the carrier of H2:], the carrier of H2:] is Relation-like non empty set
bool [:[: the carrier of H2, the carrier of H2:], the carrier of H2:] is non empty set
the multF of H2 . (K,j9) is Element of the carrier of H2
H9 is Element of the carrier of j
JK is Element of the carrier of j
H9 * JK is Element of the carrier of j
the multF of j is Relation-like [: the carrier of j, the carrier of j:] -defined the carrier of j -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of j, the carrier of j:], the carrier of j:]
[: the carrier of j, the carrier of j:] is Relation-like non empty set
[:[: the carrier of j, the carrier of j:], the carrier of j:] is Relation-like non empty set
bool [:[: the carrier of j, the carrier of j:], the carrier of j:] is non empty set
the multF of j . (H9,JK) is Element of the carrier of j
K9 is non empty unital Group-like associative normal Subgroup of j
H9 * K9 is Element of bool the carrier of j
bool the carrier of j is non empty set
carr K9 is Element of bool the carrier of j
the carrier of K9 is non empty set
H9 * (carr K9) is Element of bool the carrier of j
K350( the carrier of j,H9) is non empty trivial finite 1 -element Element of bool the carrier of j
K350( the carrier of j,H9) * (carr K9) is Element of bool the carrier of j
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of j : ( b₁ in K350( the carrier of j,H9) & b₂ in carr K9 ) } is set
K9 * H9 is Element of bool the carrier of j
(carr K9) * H9 is Element of bool the carrier of j
(carr K9) * K350( the carrier of j,H9) is Element of bool the carrier of j
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of j : ( b₁ in carr K9 & b₂ in K350( the carrier of j,H9) ) } is set
JH is Element of the carrier of j
JH * H9 is Element of the carrier of j
the multF of j . (JH,H9) is Element of the carrier of j
JH is Element of the carrier of H2
JH * K is Element of the carrier of H2
the multF of H2 . (JH,K) is Element of the carrier of H2
H * K is Element of bool the carrier of H2
(carr H) * K is Element of bool the carrier of H2
(carr H) * K350( the carrier of H2,K) is Element of bool the carrier of H2
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H2 : ( b₁ in carr H & b₂ in K350( the carrier of H2,K) ) } is set
(O,H2) is non empty unital Group-like associative (O) (O) (O,H2) (O,H2)
0 * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((s199 - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
s199 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (s199 + 1) is set
s1 . s199 is set
s2 . s199 is set
1 mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p * s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(p * s19) + s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((p * s19) + s199) mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 mod s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is non empty unital Group-like associative (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
(p * s19) / s19 is V31() V32() ext-real non negative set
0 / s19 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
p * 1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j is non empty unital Group-like associative (O) (O,G)
i . 1 is set
H1 is non empty unital Group-like associative (O) (O,G)
(O,G,j,j) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,(O,G,j,j)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H1 is non empty set
(O,G,(O,G,j,j)) is Element of bool the carrier of G
the carrier of (O,G,j,j) is non empty set
(O,G,H1) \/ (O,G,(O,G,j,j)) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,(O,G,j,j)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
(O,G,H1) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
H2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H1 - 1 is V31() V32() integer ext-real set
(H1 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((H1 - 1) * ((len s2) - 1)) + H2 is V31() V32() integer ext-real set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (H1 + 1) is set
s1 . H1 is set
s2 . H2 is set
H2 . s299 is set
s299 is non empty unital Group-like associative (O) (O,G)
i is non empty unital Group-like associative (O) (O,G)
j is non empty unital Group-like associative (O) (O,G)
(O,G,i,j) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s299,(O,G,i,j)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s299) is Element of bool the carrier of G
the carrier of s299 is non empty set
(O,G,(O,G,i,j)) is Element of bool the carrier of G
the carrier of (O,G,i,j) is non empty set
(O,G,s299) \/ (O,G,(O,G,i,j)) is Element of bool the carrier of G
(O,G,((O,G,s299) \/ (O,G,(O,G,i,j)))) is non empty unital Group-like associative (O) (O) (O,G)
H is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
K is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
K - 1 is V31() V32() integer ext-real set
(K - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
H9 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((K - 1) * ((len s2) - 1)) + H9 is V31() V32() integer ext-real set
s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 - 1 is V31() V32() integer ext-real set
(K - 1) * s19 is V31() V32() integer ext-real set
(s29 - 1) * s19 is V31() V32() integer ext-real set
s29 * s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + (s29 * s19) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + (s29 * s19) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 * s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(s29 * s19) - (1 * s19) is V31() V32() integer ext-real set
((s29 * s19) - (1 * s19)) + s19 is V31() V32() integer ext-real set
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
dom s19 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg (len s19) is finite len s19 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s19 ) } is set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (i + 1) is set
s1 . i is set
s2 . s199 is set
s19 . f1 is set
s199 is non empty unital Group-like associative (O) (O,G)
f2 is non empty unital Group-like associative (O) (O,G)
p is non empty unital Group-like associative (O) (O,G)
(O,G,f2,p) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s199,(O,G,f2,p)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,s199) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of s199 is non empty set
(O,G,(O,G,f2,p)) is Element of bool the carrier of G
the carrier of (O,G,f2,p) is non empty set
(O,G,s199) \/ (O,G,(O,G,f2,p)) is Element of bool the carrier of G
(O,G,((O,G,s199) \/ (O,G,(O,G,f2,p)))) is non empty unital Group-like associative (O) (O) (O,G)
s29 . f1 is set
s19 . f1 is set
s29 . f1 is set
Seg (len s29) is finite len s29 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s29 ) } is set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1,s2) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
rng s1 is finite Element of bool (O,G)
bool (O,G) is non empty set
(len s1) - 1 is V31() V32() integer ext-real set
(len s2) - 1 is V31() V32() integer ext-real set
((len s1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((len s1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
{ [b₁,(((b₁ - 1) * ((len s2) - 1)) + 1)] where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
s199 is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((s199 - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[s199,(((s199 - 1) * ((len s2) - 1)) + 1)] is set
{s199,(((s199 - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{s199} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{s199,(((s199 - 1) * ((len s2) - 1)) + 1)},{s199}} is non empty finite V39() set
H1 is set
j is set
j is set
H2 is set
[j,H2] is set
{j,H2} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,H2},{j}} is non empty finite V39() set
s199 is set
s199 is Relation-like set
rng s199 is set
f2 is set
[f2,s199] is set
{f2,s199} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,s199},{f2}} is non empty finite V39() set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p - 1 is V31() V32() integer ext-real set
(p - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((p - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[p,(((p - 1) * ((len s2) - 1)) + 1)] is set
{p,(((p - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{p} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{p,(((p - 1) * ((len s2) - 1)) + 1)},{p}} is non empty finite V39() set
s199 is set
f2 is set
[s199,f2] is set
{s199,f2} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,f2},{s199}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
(H1 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((H1 - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[H1,(((H1 - 1) * ((len s2) - 1)) + 1)] is set
{H1,(((H1 - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{H1} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{H1,(((H1 - 1) * ((len s2) - 1)) + 1)},{H1}} is non empty finite V39() set
p is set
[s199,p] is set
{s199,p} is non empty finite set
{{s199,p},{s199}} is non empty finite V39() set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[j,(((j - 1) * ((len s2) - 1)) + 1)] is set
{j,(((j - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{j} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{j,(((j - 1) * ((len s2) - 1)) + 1)},{j}} is non empty finite V39() set
s199 is set
dom s199 is set
f2 is set
[s199,f2] is set
{s199,f2} is non empty finite set
{s199} is non empty trivial finite 1 -element set
{{s199,f2},{s199}} is non empty finite V39() set
p is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p - 1 is V31() V32() integer ext-real set
(p - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((p - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[p,(((p - 1) * ((len s2) - 1)) + 1)] is set
{p,(((p - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{p} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{p,(((p - 1) * ((len s2) - 1)) + 1)},{p}} is non empty finite V39() set
s199 is Relation-like Function-like set
rng s199 is set
dom s199 is set
[:(dom s199),(rng s199):] is Relation-like set
bool [:(dom s199),(rng s199):] is non empty set
f2 is Relation-like dom s199 -defined rng s199 -valued Function-like Element of bool [:(dom s199),(rng s199):]
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
p is Relation-like REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
p .: (Seg (len s1)) is finite complex-membered ext-real-membered real-membered V210() V211() V212() Element of bool REAL
[:(dom s1),(rng s1):] is Relation-like finite set
bool [:(dom s1),(rng s1):] is non empty finite V39() set
id (dom s1) is Relation-like dom s1 -defined dom s1 -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:(dom s1),(dom s1):]
[:(dom s1),(dom s1):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom s1),(dom s1):] is non empty finite V39() set
s299 is Relation-like dom s1 -defined rng s1 -valued finite Element of bool [:(dom s1),(rng s1):]
(id (dom s1)) * s299 is Relation-like dom s1 -defined rng s1 -valued finite Element of bool [:(dom s1),(rng s1):]
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 - 1 is V31() V32() integer ext-real set
(len s1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
len (O,G,s1,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (O,G,s1,s2) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Seg H1 is finite H1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= H1 ) } is set
H2 is set
H2 is finite complex-membered ext-real-membered real-membered V210() V211() V212() Element of bool REAL
s299 is set
[s299,H2] is set
{s299,H2} is non empty finite set
{s299} is non empty trivial finite 1 -element set
{{s299,H2},{s299}} is non empty finite V39() set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[i,(((i - 1) * ((len s2) - 1)) + 1)] is set
{i,(((i - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{i} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{i,(((i - 1) * ((len s2) - 1)) + 1)},{i}} is non empty finite V39() set
1 - 1 is V31() V32() integer ext-real set
j is V31() V32() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 is set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - 1 is V31() V32() integer ext-real set
(s299 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((s299 - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[H2,(((s299 - 1) * ((len s2) - 1)) + 1)] is set
{H2,(((s299 - 1) * ((len s2) - 1)) + 1)} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,(((s299 - 1) * ((len s2) - 1)) + 1)},{H2}} is non empty finite V39() set
dom p is complex-membered ext-real-membered real-membered Element of bool REAL
H2 is set
s299 is set
[H2,s299] is set
{H2,s299} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,s299},{H2}} is non empty finite V39() set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[i,(((i - 1) * ((len s2) - 1)) + 1)] is set
{i,(((i - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{i} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{i,(((i - 1) * ((len s2) - 1)) + 1)},{i}} is non empty finite V39() set
rng p is complex-membered ext-real-membered real-membered Element of bool REAL
(O,G,s1,s2) * p is Relation-like REAL -defined (O,G) -valued Function-like Element of bool [:REAL,(O,G):]
[:REAL,(O,G):] is Relation-like non empty non trivial non finite set
bool [:REAL,(O,G):] is non empty non trivial non finite set
dom ((O,G,s1,s2) * p) is complex-membered ext-real-membered real-membered Element of bool REAL
H2 is set
p . H2 is V31() V32() ext-real set
[H2,(p . H2)] is set
{H2,(p . H2)} is non empty finite set
{H2} is non empty trivial finite 1 -element set
{{H2,(p . H2)},{H2}} is non empty finite V39() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - 1 is V31() V32() integer ext-real set
(s299 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((s299 - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[s299,(((s299 - 1) * ((len s2) - 1)) + 1)] is set
{s299,(((s299 - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{s299} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{s299,(((s299 - 1) * ((len s2) - 1)) + 1)},{s299}} is non empty finite V39() set
((O,G,s1,s2) * p) . H2 is set
(O,G,s1,s2) . (p . H2) is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1,s2) . j is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
s1 . (len s1) is set
s1 . H2 is set
s299 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(s299 + 1) - 1 is V31() V32() integer ext-real set
s2 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
s1 . (s299 + 1) is set
s1 . s299 is set
K is non empty unital Group-like associative (O) (O) (O,G)
the carrier of K is non empty set
K9 is set
the carrier of (O,G) is non empty set
H9 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of H9 is non empty set
the carrier of K /\ the carrier of H9 is set
0 + s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
1 + s299 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H is non empty unital Group-like associative (O) (O) (O,G)
((O,G,s1,s2) * p) . H2 is set
(O,G,s1,s2) . (p . H2) is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1,s2) . j is set
(O,G,K,H9) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H,(O,G,K,H9)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H is non empty set
(O,G,(O,G,K,H9)) is Element of bool the carrier of G
the carrier of (O,G,K,H9) is non empty set
(O,G,H) \/ (O,G,(O,G,K,H9)) is Element of bool the carrier of G
(O,G,((O,G,H) \/ (O,G,(O,G,K,H9)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H,K) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,K) is Element of bool the carrier of G
(O,G,H) \/ (O,G,K) is Element of bool the carrier of G
(O,G,((O,G,H) \/ (O,G,K))) is non empty unital Group-like associative (O) (O) (O,G)
s1 . H2 is set
H2 is V31() V32() ext-real Element of REAL
(Seg (len s1)) /\ (dom p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool REAL
p . H2 is V31() V32() ext-real set
[H2,(p . H2)] is set
{H2,(p . H2)} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{H2} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{H2,(p . H2)},{H2}} is non empty finite V39() set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[i,(((i - 1) * ((len s2) - 1)) + 1)] is set
{i,(((i - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{i} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{i,(((i - 1) * ((len s2) - 1)) + 1)},{i}} is non empty finite V39() set
s299 is V31() V32() ext-real Element of REAL
p . s299 is V31() V32() ext-real set
[s299,(p . s299)] is set
{s299,(p . s299)} is non empty finite complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{s299} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered V208() V209() V210() V211() V212() set
{{s299,(p . s299)},{s299}} is non empty finite V39() set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[j,(((j - 1) * ((len s2) - 1)) + 1)] is set
{j,(((j - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{j} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{j,(((j - 1) * ((len s2) - 1)) + 1)},{j}} is non empty finite V39() set
p | (Seg (len s1)) is Relation-like REAL -defined Seg (len s1) -defined REAL -defined REAL -valued Function-like finite V191() V192() V193() Element of bool [:REAL,REAL:]
H2 is set
s299 is set
[s299,H2] is set
{s299,H2} is non empty finite set
{s299} is non empty trivial finite 1 -element set
{{s299,H2},{s299}} is non empty finite V39() set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
[i,(((i - 1) * ((len s2) - 1)) + 1)] is set
{i,(((i - 1) * ((len s2) - 1)) + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{i} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{i,(((i - 1) * ((len s2) - 1)) + 1)},{i}} is non empty finite V39() set
j is V31() V32() integer ext-real set
Sgm H2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(O,G,s1,s2) * (Sgm H2) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
Sgm (Seg (len s1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
p * (Sgm (Seg (len s1))) is Relation-like NAT -defined REAL -valued Function-like finite V191() V192() V193() Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial non finite V191() V192() V193() set
bool [:NAT,REAL:] is non empty non trivial non finite set
(O,G,s1,s2) * (p * (Sgm (Seg (len s1)))) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
((O,G,s1,s2) * p) * (Sgm (Seg (len s1))) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
s1 * (Sgm (Seg (len s1))) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
idseq (len s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(idseq (len s1)) * s1 is Relation-like NAT -defined (O,G) -valued Function-like finite set
id (Seg (len s1)) is Relation-like Seg (len s1) -defined Seg (len s1) -valued RAT -valued INT -valued Function-like one-to-one total quasi_total finite V191() V192() V193() V194() increasing V197() Element of bool [:(Seg (len s1)),(Seg (len s1)):]
[:(Seg (len s1)),(Seg (len s1)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(Seg (len s1)),(Seg (len s1)):] is non empty finite V39() set
s1 * (id (Seg (len s1))) is Relation-like Seg (len s1) -defined (O,G) -valued Function-like finite Element of bool [:(Seg (len s1)),(O,G):]
[:(Seg (len s1)),(O,G):] is Relation-like set
bool [:(Seg (len s1)),(O,G):] is non empty set
s1 * (id (dom s1)) is Relation-like dom s1 -defined (O,G) -valued Function-like finite Element of bool [:(dom s1),(O,G):]
[:(dom s1),(O,G):] is Relation-like set
bool [:(dom s1),(O,G):] is non empty set
s19 is finite complex-membered ext-real-membered real-membered V210() V211() V212() Element of bool REAL
Sgm s19 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(O,G,s1,s2) * (Sgm s19) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1,s2) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s2,s1) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(len s1) - 1 is V31() V32() integer ext-real set
1 - 1 is V31() V32() integer ext-real set
(len s2) - 1 is V31() V32() integer ext-real set
len (O,G,s1,s2) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((len s1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((len s1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
len (O,G,s2,s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 * ((len s2) - 1) is V31() V32() integer ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
{ [(((b₂ - 1) * ((len s1) - 1)) + b₁),(((b₁ - 1) * ((len s2) - 1)) + b₂)] where b₁, b₂ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= (len s1) - 1 & 1 <= b₂ & b₂ <= (len s2) - 1 ) } is set
f1 is set
s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s199 - 1 is V31() V32() integer ext-real set
(s199 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((s199 - 1) * ((len s1) - 1)) + i is V31() V32() integer ext-real set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i - 1) * ((len s2) - 1)) + s199 is V31() V32() integer ext-real set
[(((s199 - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + s199)] is set
{(((s199 - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + s199)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((s199 - 1) * ((len s1) - 1)) + i)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((s199 - 1) * ((len s1) - 1)) + i),(((i - 1) * ((len s2) - 1)) + s199)},{(((s199 - 1) * ((len s1) - 1)) + i)}} is non empty finite V39() set
p is set
H1 is set
j is set
j is set
[j,j] is set
{j,j} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,j},{j}} is non empty finite V39() set
(O,G,(O,G,s1,s2)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,(O,G,s1,s2)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(O,G,(O,G,s2,s1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f2 is set
p is set
[f2,p] is set
{f2,p} is non empty finite set
{f2} is non empty trivial finite 1 -element set
{{f2,p},{f2}} is non empty finite V39() set
f1 is Relation-like set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((j - 1) * ((len s1) - 1)) + j is V31() V32() integer ext-real set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + j is V31() V32() integer ext-real set
[(((j - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + j)] is set
{(((j - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + j)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((j - 1) * ((len s1) - 1)) + j)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((j - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + j)},{(((j - 1) * ((len s1) - 1)) + j)}} is non empty finite V39() set
H1 is set
[f2,H1] is set
{f2,H1} is non empty finite set
{{f2,H1},{f2}} is non empty finite V39() set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s299 - 1 is V31() V32() integer ext-real set
(s299 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((s299 - 1) * ((len s1) - 1)) + H2 is V31() V32() integer ext-real set
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((H2 - 1) * ((len s2) - 1)) + s299 is V31() V32() integer ext-real set
[(((s299 - 1) * ((len s1) - 1)) + H2),(((H2 - 1) * ((len s2) - 1)) + s299)] is set
{(((s299 - 1) * ((len s1) - 1)) + H2),(((H2 - 1) * ((len s2) - 1)) + s299)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((s299 - 1) * ((len s1) - 1)) + H2)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((s299 - 1) * ((len s1) - 1)) + H2),(((H2 - 1) * ((len s2) - 1)) + s299)},{(((s299 - 1) * ((len s1) - 1)) + H2)}} is non empty finite V39() set
len (O,G,(O,G,s1,s2)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,(O,G,s1,s2))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j is set
Seg (len (O,G,(O,G,s1,s2))) is finite len (O,G,(O,G,s1,s2)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,(O,G,s1,s2)) ) } is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
0 + (((len s1) - 1) * ((len s2) - 1)) is V31() V32() integer ext-real set
1 + (((len s1) - 1) * ((len s2) - 1)) is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (H1 + 1) is non empty finite H1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= H1 + 1 ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((H2 - 1) * ((len s2) - 1)) + s299 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
(H1 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H1 - 1) * ((len s1) - 1)) + H2 is V31() V32() integer ext-real set
i is set
[i,j] is set
{i,j} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,j},{i}} is non empty finite V39() set
f2 is Relation-like Function-like set
rng f2 is set
j is set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (H1 + 1) is non empty finite H1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= H1 + 1 ) } is set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((H2 - 1) * ((len s1) - 1)) + s299 is V31() V32() integer ext-real set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H2 - 1) * ((len s2) - 1)) + H1 is V31() V32() integer ext-real set
i is set
[j,i] is set
{j,i} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,i},{j}} is non empty finite V39() set
dom f2 is set
p is set
j is set
[j,p] is set
{j,p} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,p},{j}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H2 - 1) * ((len s1) - 1)) + j is V31() V32() integer ext-real set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + H2 is V31() V32() integer ext-real set
[(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)] is set
{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((H2 - 1) * ((len s1) - 1)) + j)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)},{(((H2 - 1) * ((len s1) - 1)) + j)}} is non empty finite V39() set
s299 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is set
j is set
[p,j] is set
{p,j} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,j},{p}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H2 - 1) * ((len s1) - 1)) + j is V31() V32() integer ext-real set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + H2 is V31() V32() integer ext-real set
[(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)] is set
{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((H2 - 1) * ((len s1) - 1)) + j)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)},{(((H2 - 1) * ((len s1) - 1)) + j)}} is non empty finite V39() set
s299 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
[:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):] is non empty finite V39() set
p is Relation-like dom (O,G,(O,G,s1,s2)) -defined dom (O,G,(O,G,s1,s2)) -valued Function-like total quasi_total finite V191() V192() V193() V194() Element of bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):]
H1 is set
j is set
p . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
p . j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
[H1,(p . H1)] is set
{H1,(p . H1)} is non empty finite set
{H1} is non empty trivial finite 1 -element set
{{H1,(p . H1)},{H1}} is non empty finite V39() set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H2 - 1 is V31() V32() integer ext-real set
(H2 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H2 - 1) * ((len s1) - 1)) + j is V31() V32() integer ext-real set
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s2) - 1)) + H2 is V31() V32() integer ext-real set
[(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)] is set
{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((H2 - 1) * ((len s1) - 1)) + j)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((H2 - 1) * ((len s1) - 1)) + j),(((j - 1) * ((len s2) - 1)) + H2)},{(((H2 - 1) * ((len s1) - 1)) + j)}} is non empty finite V39() set
[j,(p . j)] is set
{j,(p . j)} is non empty finite set
{j} is non empty trivial finite 1 -element set
{{j,(p . j)},{j}} is non empty finite V39() set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
H1 - 1 is V31() V32() integer ext-real set
(H1 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((H1 - 1) * ((len s1) - 1)) + s299 is V31() V32() integer ext-real set
s299 - 1 is V31() V32() integer ext-real set
(s299 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((s299 - 1) * ((len s2) - 1)) + H1 is V31() V32() integer ext-real set
[(((H1 - 1) * ((len s1) - 1)) + s299),(((s299 - 1) * ((len s2) - 1)) + H1)] is set
{(((H1 - 1) * ((len s1) - 1)) + s299),(((s299 - 1) * ((len s2) - 1)) + H1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((H1 - 1) * ((len s1) - 1)) + s299)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((H1 - 1) * ((len s1) - 1)) + s299),(((s299 - 1) * ((len s2) - 1)) + H1)},{(((H1 - 1) * ((len s1) - 1)) + s299)}} is non empty finite V39() set
H1 is Relation-like dom (O,G,(O,G,s1,s2)) -defined dom (O,G,(O,G,s1,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):]
len (O,G,(O,G,s2,s1)) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(len (O,G,(O,G,s2,s1))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
2 - 1 is V31() V32() integer ext-real set
H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is Relation-like dom (O,G,(O,G,s1,s2)) -defined dom (O,G,(O,G,s1,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):]
j " is Relation-like dom (O,G,(O,G,s1,s2)) -defined dom (O,G,(O,G,s1,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):]
(j ") . H1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
H2 is non empty unital Group-like associative (O) (O)
(O,G,(O,G,s1,s2)) . H1 is set
s299 is non empty unital Group-like associative (O) (O)
(O,G,(O,G,s2,s1)) . H2 is set
s299 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg s299 is finite s299 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s299 ) } is set
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
i - 1 is V31() V32() integer ext-real set
(i - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
((i - 1) * ((len s2) - 1)) + j is V31() V32() integer ext-real set
s1 . i is set
s2 . j is set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . (i + 1) is set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . (j + 1) is set
rng (j ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom (O,G,(O,G,s1,s2)))
bool (dom (O,G,(O,G,s1,s2))) is non empty finite V39() set
j . H2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
[H2,H1] is set
{H2,H1} is non empty finite V39() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{H2} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
{{H2,H1},{H2}} is non empty finite V39() set
j9 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
j9 - 1 is V31() V32() integer ext-real set
(j9 - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
i9 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((j9 - 1) * ((len s1) - 1)) + i9 is V31() V32() integer ext-real set
i9 - 1 is V31() V32() integer ext-real set
(i9 - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
((i9 - 1) * ((len s2) - 1)) + j9 is V31() V32() integer ext-real set
[(((j9 - 1) * ((len s1) - 1)) + i9),(((i9 - 1) * ((len s2) - 1)) + j9)] is set
{(((j9 - 1) * ((len s1) - 1)) + i9),(((i9 - 1) * ((len s2) - 1)) + j9)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{(((j9 - 1) * ((len s1) - 1)) + i9)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered V208() V209() V210() V211() V212() set
{{(((j9 - 1) * ((len s1) - 1)) + i9),(((i9 - 1) * ((len s2) - 1)) + j9)},{(((j9 - 1) * ((len s1) - 1)) + i9)}} is non empty finite V39() set
K9 is non empty unital Group-like associative (O) (O) (O,G)
K is non empty unital Group-like associative (O) (O) (O,G)
H9 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,K,H9) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,K9,(O,G,K,H9)) is non empty unital Group-like associative (O) (O) (O,G)
the carrier of G is non empty set
(O,G,K9) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of K9 is non empty set
(O,G,(O,G,K,H9)) is Element of bool the carrier of G
the carrier of (O,G,K,H9) is non empty set
(O,G,K9) \/ (O,G,(O,G,K,H9)) is Element of bool the carrier of G
(O,G,((O,G,K9) \/ (O,G,(O,G,K,H9)))) is non empty unital Group-like associative (O) (O) (O,G)
0 + (j + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + (j + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . 1 is set
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,K9,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
(O,G,K9) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,K9) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
(j + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s2 . ((j + 1) + 1) is set
((len s2) - 1) + 1 is V31() V32() integer ext-real set
Seg (len s2) is finite len s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s2 ) } is set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
H1 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of H1 is non empty set
H2 is set
the carrier of (O,G) is non empty set
the carrier of H1 /\ the carrier of (O,G) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(j + 1) - 1 is V31() V32() integer ext-real set
((j + 1) - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
(((j + 1) - 1) * ((len s1) - 1)) + 1 is V31() V32() integer ext-real set
(O,G,s2,s1) . (H2 + 1) is set
H2 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2,(O,G,H1,H1)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2) is Element of bool the carrier of G
the carrier of H2 is non empty set
(O,G,(O,G,H1,H1)) is Element of bool the carrier of G
the carrier of (O,G,H1,H1) is non empty set
(O,G,H2) \/ (O,G,(O,G,H1,H1)) is Element of bool the carrier of G
(O,G,((O,G,H2) \/ (O,G,(O,G,H1,H1)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2,H1) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1) is Element of bool the carrier of G
(O,G,H2) \/ (O,G,H1) is Element of bool the carrier of G
(O,G,((O,G,H2) \/ (O,G,H1))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,(O,G),(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
(O,G,(O,G)) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,(O,G)) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s2,s1) . (H2 + 1) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s2,s1) . (H2 + 1) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s2,s1) . (H2 + 1) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s1) - 1)) + i is V31() V32() integer ext-real set
(((j - 1) * ((len s1) - 1)) + i) + 1 is V31() V32() integer ext-real set
((j - 1) * ((len s1) - 1)) + (i + 1) is V31() V32() integer ext-real set
(O,G,s2,s1) . (H2 + 1) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s2,s1) . (H2 + 1) is set
H2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s2,s1) . (H2 + 1) is set
dom (O,G,(O,G,s2,s1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H is non empty unital Group-like associative (O) (O) (O,G)
(O,G,K,H) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,K9,(O,G,K,H)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,K,H)) is Element of bool the carrier of G
the carrier of (O,G,K,H) is non empty set
(O,G,K9) \/ (O,G,(O,G,K,H)) is Element of bool the carrier of G
(O,G,((O,G,K9) \/ (O,G,(O,G,K,H)))) is non empty unital Group-like associative (O) (O) (O,G)
j - 1 is V31() V32() integer ext-real set
(j - 1) * ((len s1) - 1) is V31() V32() integer ext-real set
((j - 1) * ((len s1) - 1)) + i is V31() V32() integer ext-real set
(O,G,s2,s1) . H2 is set
JK is non empty unital Group-like associative (O) (O,(O,G,K9,(O,G,K,H))) (O,(O,G,K9,(O,G,K,H)))
(O,(O,G,K9,(O,G,K,H)),JK) is non empty unital Group-like associative (O) (O)
(O,(O,G,K9,(O,G,K,H)),JK) is set
(O,(O,G,K9,(O,G,K,H)),JK) is Relation-like [:(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK):] -defined (O,(O,G,K9,(O,G,K,H)),JK) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK):],(O,(O,G,K9,(O,G,K,H)),JK):]
[:(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK):] is Relation-like set
[:[:(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK):],(O,(O,G,K9,(O,G,K,H)),JK):] is Relation-like set
bool [:[:(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK):],(O,(O,G,K9,(O,G,K,H)),JK):] is non empty set
(O,(O,G,K9,(O,G,K,H)),JK) is Relation-like O -defined Funcs ((O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK))):]
Funcs ((O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK)) is functional non empty set
[:O,(Funcs ((O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK))):] is non empty set
(O,(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK),(O,(O,G,K9,(O,G,K,H)),JK)) is (O) (O)
(O,G,H,K9) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H9,(O,G,H,K9)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H9) is Element of bool the carrier of G
the carrier of H9 is non empty set
(O,G,(O,G,H,K9)) is Element of bool the carrier of G
the carrier of (O,G,H,K9) is non empty set
(O,G,H9) \/ (O,G,(O,G,H,K9)) is Element of bool the carrier of G
(O,G,((O,G,H9) \/ (O,G,(O,G,H,K9)))) is non empty unital Group-like associative (O) (O) (O,G)
s2 . 1 is set
0 + (i + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
1 + (i + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(i + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 . ((i + 1) + 1) is set
((len s1) - 1) + 1 is V31() V32() integer ext-real set
Seg (len s1) is finite len s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len s1 ) } is set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
H3 is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is Relation-like non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is Relation-like non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the of G is Relation-like O -defined Funcs ( the carrier of G, the carrier of G) -valued Function-like total quasi_total Element of bool [:O,(Funcs ( the carrier of G, the carrier of G)):]
Funcs ( the carrier of G, the carrier of G) is functional non empty set
[:O,(Funcs ( the carrier of G, the carrier of G)):] is Relation-like set
bool [:O,(Funcs ( the carrier of G, the carrier of G)):] is non empty set
(O, the carrier of G, the multF of G, the of G) is (O) (O)
H2 is non empty unital Group-like associative (O) (O) (O,G)
the carrier of H2 is non empty set
x is set
the carrier of (O,G) is non empty set
the carrier of H2 /\ the carrier of (O,G) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(i + 1) - 1 is V31() V32() integer ext-real set
((i + 1) - 1) * ((len s2) - 1) is V31() V32() integer ext-real set
(((i + 1) - 1) * ((len s2) - 1)) + 1 is V31() V32() integer ext-real set
(O,G,s1,s2) . (H1 + 1) is set
H1 is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2,H3) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,(O,G,H2,H3)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1) is Element of bool the carrier of G
the carrier of H1 is non empty set
(O,G,(O,G,H2,H3)) is Element of bool the carrier of G
the carrier of (O,G,H2,H3) is non empty set
(O,G,H1) \/ (O,G,(O,G,H2,H3)) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,(O,G,H2,H3)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H1,H2) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H2) is Element of bool the carrier of G
(O,G,H1) \/ (O,G,H2) is Element of bool the carrier of G
(O,G,((O,G,H1) \/ (O,G,H2))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,H,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H9,(O,G,H,(O,G))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,H,(O,G))) is Element of bool the carrier of G
the carrier of (O,G,H,(O,G)) is non empty set
(O,G,H9) \/ (O,G,(O,G,H,(O,G))) is Element of bool the carrier of G
(O,G,((O,G,H9) \/ (O,G,(O,G,H,(O,G))))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H9,(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
(O,G,H9) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,H9) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G) is non empty unital Group-like associative (O) (O) (O,G) (O,G)
(O,G,(O,G),(O,G)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G)) is Element of bool the carrier of G
the carrier of (O,G) is non empty set
(O,G,(O,G)) \/ (O,G,(O,G)) is Element of bool the carrier of G
(O,G,((O,G,(O,G)) \/ (O,G,(O,G)))) is non empty unital Group-like associative (O) (O) (O,G)
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s1,s2) . (H1 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s1,s2) . (H1 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s1,s2) . (H1 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
((i - 1) * ((len s2) - 1)) + (j + 1) is V31() V32() integer ext-real set
(O,G,s1,s2) . (H1 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s1,s2) . (H1 + 1) is set
H1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s1,s2) . (H1 + 1) is set
(O,G,H,K) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,H9,(O,G,H,K)) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,(O,G,H,K)) is Element of bool the carrier of G
the carrier of (O,G,H,K) is non empty set
(O,G,H9) \/ (O,G,(O,G,H,K)) is Element of bool the carrier of G
(O,G,((O,G,H9) \/ (O,G,(O,G,H,K)))) is non empty unital Group-like associative (O) (O) (O,G)
(O,G,s1,s2) . H1 is set
JH is non empty unital Group-like associative (O) (O,(O,G,H9,(O,G,H,K))) (O,(O,G,H9,(O,G,H,K)))
(O,(O,G,H9,(O,G,H,K)),JH) is non empty unital Group-like associative (O) (O)
(O,(O,G,H9,(O,G,H,K)),JH) is set
(O,(O,G,H9,(O,G,H,K)),JH) is Relation-like [:(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH):] -defined (O,(O,G,H9,(O,G,H,K)),JH) -valued Function-like quasi_total Element of bool [:[:(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH):],(O,(O,G,H9,(O,G,H,K)),JH):]
[:(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH):] is Relation-like set
[:[:(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH):],(O,(O,G,H9,(O,G,H,K)),JH):] is Relation-like set
bool [:[:(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH):],(O,(O,G,H9,(O,G,H,K)),JH):] is non empty set
(O,(O,G,H9,(O,G,H,K)),JH) is Relation-like O -defined Funcs ((O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH)) -valued Function-like total quasi_total Element of bool [:O,(Funcs ((O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH))):]
Funcs ((O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH)) is functional non empty set
[:O,(Funcs ((O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH))):] is Relation-like set
bool [:O,(Funcs ((O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH))):] is non empty set
(O,(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH),(O,(O,G,H9,(O,G,H,K)),JH)) is (O) (O)
p is Relation-like dom (O,G,(O,G,s1,s2)) -defined dom (O,G,(O,G,s1,s2)) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,(O,G,s1,s2))),(dom (O,G,(O,G,s1,s2))):]
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s2,s1) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s1,s2) is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
O is set
G is non empty unital Group-like associative (O) (O)
(O,G) is non empty set
s1 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s2 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s29 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
dom s2 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm s29 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm s29) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s19 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
Sgm s19 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s2 * (Sgm s19) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s1)),(dom (O,G,s1)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s1)),(dom (O,G,s1)):] is non empty finite V39() set
(O,G,s2) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
s29 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm s29 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s29) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
[:NAT,(O,G):] is Relation-like non empty non trivial non finite set
bool [:NAT,(O,G):] is non empty non trivial non finite set
s19 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V31() V32() integer finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V191() V192() V193() V194() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V207() V210() V211() V212() V213() set
Sgm s19 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm s19) is Relation-like NAT -defined (O,G) -valued Function-like finite Element of bool [:NAT,(O,G):]
s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s1) + s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(len s1) + (s19 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
p is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
((len s1) + s19) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
(O,G,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(s19 + 1) + (len s1) is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
0 + (len s1) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (O,G,s29) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,s29) . i is set
Del (s29,i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{i} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom s29) \ {i} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom s29) \ {i}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom s29) \ {i})) * s29 is Relation-like NAT -defined (O,G) -valued Function-like finite set
dom s29 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s29 . i is set
s29 . (i + 1) is set
s199 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
s199 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s199) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Del ((O,G,s29),i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s29) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom (O,G,s29)) \ {i} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (O,G,s29)) \ {i}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (O,G,s29)) \ {i})) * (O,G,s29) is Relation-like NAT -defined Function-like finite set
(O,G,p) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
[:(dom (O,G,s29)),(dom (O,G,s29)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s29)),(dom (O,G,s29)):] is non empty finite V39() set
p is Relation-like dom (O,G,s29) -defined dom (O,G,s29) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s29)),(dom (O,G,s29)):]
rng (O,G,s29) is finite set
p " is Relation-like dom (O,G,s29) -defined dom (O,G,s29) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s29)),(dom (O,G,s29)):]
(p ") . i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
j is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(O,G,p) . j is set
Del (p,j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{j} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom p) \ {j} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom p) \ {j}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom p) \ {j})) * p is Relation-like NAT -defined (O,G) -valued Function-like finite set
rng (p ") is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool (dom (O,G,s29))
bool (dom (O,G,s29)) is non empty finite V39() set
len (O,G,s29) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
Seg (len (O,G,s29)) is finite len (O,G,s29) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= len (O,G,s29) ) } is set
len (O,G,p) is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
dom (O,G,p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
rng (O,G,p) is finite set
H1 is non empty unital Group-like associative (O) (O) (O)
H2 is non empty unital Group-like associative (O) (O) (O)
s299 is non empty unital Group-like associative (O) (O)
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
p . j is set
p . (j + 1) is set
s299 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (O,G)
s299 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s199 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s299) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Del ((O,G,p),j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
(dom (O,G,p)) \ {j} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (O,G,p)) \ {j}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (O,G,p)) \ {j})) * (O,G,p) is Relation-like NAT -defined Function-like finite set
dom (O,G,s199) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s199)),(dom (O,G,s199)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s199)),(dom (O,G,s199)):] is non empty finite V39() set
i is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,i) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,i)),(dom (O,G,i)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,i)),(dom (O,G,i)):] is non empty finite V39() set
j is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len i is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,j) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
H is Relation-like dom (O,G,i) -defined dom (O,G,i) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,i)),(dom (O,G,i)):]
K is Relation-like dom (O,G,s199) -defined dom (O,G,s199) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s199)),(dom (O,G,s199)):]
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s29) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s29)),(dom (O,G,s29)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s29)),(dom (O,G,s29)):] is non empty finite V39() set
p is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s29 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,p) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f1 is Relation-like dom (O,G,s29) -defined dom (O,G,s29) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s29)),(dom (O,G,s29)):]
(len s1) + 0 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s19) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s19)),(dom (O,G,s19)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s19)),(dom (O,G,s19)):] is non empty finite V39() set
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
(O,G,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
p is Relation-like dom (O,G,s19) -defined dom (O,G,s19) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s19)),(dom (O,G,s19)):]
s19 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
s29 is Relation-like NAT -defined (O,G) -valued Function-like finite FinSequence-like FinSubsequence-like (O,G) FinSequence of (O,G)
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
(O,G,s19) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (O,G,s19) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
[:(dom (O,G,s19)),(dom (O,G,s19)):] is Relation-like RAT -valued INT -valued finite V191() V192() V193() V194() set
bool [:(dom (O,G,s19)),(dom (O,G,s19)):] is non empty finite V39() set
(O,G,s29) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
p is Relation-like dom (O,G,s19) -defined dom (O,G,s19) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s19)),(dom (O,G,s19)):]
f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
(len s1) + f1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
p is Relation-like dom (O,G,s1) -defined dom (O,G,s1) -valued Function-like one-to-one total quasi_total onto bijective finite V191() V192() V193() V194() Element of bool [:(dom (O,G,s1)),(dom (O,G,s1)):]
O is Relation-like set
rng O is set
G is Relation-like set
(rng O) |` G is Relation-like set
O ~ is Relation-like set
G ~ is Relation-like set
dom (O ~) is set
(G ~) | (dom (O ~)) is Relation-like set
s1 is Relation-like set
s1 ~ is Relation-like set
s2 is Relation-like set
s2 ~ is Relation-like set
dom (s1 ~) is set
(s2 ~) | (dom (s1 ~)) is Relation-like set
rng s1 is set
(rng s1) |` s2 is Relation-like set
G is Relation-like set
s1 is Relation-like set
O is set
s1 | O is Relation-like set
G * (s1 | O) is Relation-like set
O |` G is Relation-like set
(O |` G) * s1 is Relation-like set
G is set
O is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Seg O is finite O -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= O ) } is set
s1 is Relation-like REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
dom s1 is complex-membered ext-real-membered real-membered Element of bool REAL
s1 | G is Relation-like REAL -defined G -defined REAL -defined REAL -valued Function-like V191() V192() V193() Element of bool [:REAL,REAL:]
s1 .: G is complex-membered ext-real-membered real-membered Element of bool REAL
Sgm (s1 .: G) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
Sgm G is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
s1 * (Sgm G) is Relation-like NAT -defined REAL -valued Function-like finite V191() V192() V193() Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial non finite V191() V192() V193() set
bool [:NAT,REAL:] is non empty non trivial non finite set
O is set
s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
s1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
Seg (s1 + 1) is non empty finite s1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s1 + 1 ) } is set
G is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
Sgm O is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
{G} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(Seg (s1 + 1)) \ {G} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((Seg (s1 + 1)) \ {G}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((Seg (s1 + 1)) \ {G})) " is Relation-like Function-like set
(Sgm O) * ((Sgm ((Seg (s1 + 1)) \ {G})) ") is Relation-like NAT -defined Function-like finite set
Seg s1 is finite s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s1 ) } is set
O is non empty set
s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal ext-real non negative set
G is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
dom G is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() Element of NAT
s1 is Element of O
Ins (G,s2,s1) is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
G | s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
Seg s2 is finite s2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT : ( 1 <= b₁ & b₁ <= s2 ) } is set
G | (Seg s2) is Relation-like NAT -defined Seg s2 -defined NAT -defined O -valued Function-like finite FinSubsequence-like set
<*s1*> is Relation-like NAT -defined O -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like FinSequence of O
[1,s1] is set
{1,s1} is non empty finite set
{{1,s1},{1}} is non empty finite V39() set
{[1,s1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(G | s2) ^ <*s1*> is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
G /^ s2 is Relation-like NAT -defined O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of O
((G | s2) ^ <*s1*>) ^ (G /^ s2) is Relation-like NAT -defined O -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of O
Del ((Ins (G,s2,s1)),(s2 + 1)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (Ins (G,s2,s1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
{(s2 + 1)} is non empty trivial finite V39() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V208() V209() V210() V211() V212() set
(dom (Ins (G,s2,s1))) \ {(s2 + 1)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
Sgm ((dom (Ins (G,s2,s1))) \ {(s2 + 1)}) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() V194() FinSequence of NAT
(Sgm ((dom (Ins (G,s2,s1))) \ {(s2 + 1)})) * (Ins (G,s2,s1)) is Relation-like NAT -defined O -valued Function-like finite set
O is non empty unital Group-like associative multMagma
the carrier of O is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[: the carrier of O, the carrier of G:] is Relation-like non empty set
bool [: the carrier of O, the carrier of G:] is non empty set
s1 is Relation-like NAT -defined the carrier of O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of O
dom s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of bool NAT
s2 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
s29 is Relation-like the carrier of O -defined the carrier of G -valued Function-like non empty total quasi_total unity-preserving multiplicative Element of bool [: the carrier of O, the carrier of G:]
len s1 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s19 is Relation-like NAT -defined INT -valued Function-like finite FinSequence-like FinSubsequence-like V191() V192() V193() FinSequence of INT
len s19 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
len s2 is epsilon-transitive epsilon-connected ordinal natural V31() V32() integer finite cardinal V46() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V210() V211() V212() Element of NAT
s1 |^ s19 is Relation-like NAT -defined the carrier of O -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of O
Product (s1 |^ s19) is Element of the carrier of O
s29 . (Product (s1 |^ s19)) is Element of the carrier of G
s2 |^ s19 is Relation-like NAT -defined the carrier of G -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (s2 |^ s19) is Element of the carrier of G