:: MESFUNC9 semantic presentation

REAL is non empty V29() V73() V74() V75() V79() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() Element of bool REAL
bool REAL is non empty set
ExtREAL is non empty V74() set
[:NAT,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:NAT,ExtREAL:] is non empty set
bool ExtREAL is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() set
bool omega is non empty set
bool NAT is non empty set
COMPLEX is non empty V29() V73() V79() set
[:NAT,REAL:] is non empty Relation-like complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty set
bool (bool REAL) is non empty set
RAT is non empty V29() V73() V74() V75() V76() V79() set
bool RAT is non empty set
INT is non empty V29() V73() V74() V75() V76() V77() V79() set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
{{},1} is non empty V73() V74() V75() V76() V77() V78() set
[:COMPLEX,COMPLEX:] is non empty Relation-like complex-valued set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty Relation-like complex-valued set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty Relation-like complex-valued ext-real-valued real-valued set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty Relation-like complex-valued ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty Relation-like RAT -valued complex-valued ext-real-valued real-valued set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty Relation-like RAT -valued complex-valued ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is non empty set
0. is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() Element of ExtREAL
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() Element of NAT
+infty is non empty non real ext-real positive non negative Element of ExtREAL
-infty is non empty non real ext-real non positive negative Element of ExtREAL
{-infty} is non empty V74() set
{+infty} is non empty V74() set
- 1 is non empty V11() real ext-real non positive negative Element of REAL
+infty is non empty non real ext-real positive non negative set
-infty is non empty non real ext-real non positive negative set
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom S is Element of bool X
bool X is non empty set
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S + M) is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
rng M is V74() Element of bool ExtREAL
M " {+infty} is Element of bool X
rng S is V74() Element of bool ExtREAL
S " {+infty} is Element of bool X
M " {-infty} is Element of bool X
(S " {+infty}) /\ (M " {-infty}) is Element of bool X
S " {-infty} is Element of bool X
(S " {-infty}) /\ (M " {+infty}) is Element of bool X
((S " {+infty}) /\ (M " {-infty})) \/ ((S " {-infty}) /\ (M " {+infty})) is Element of bool X
((dom S) /\ (dom M)) \ {} is Element of bool X
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom S is Element of bool X
bool X is non empty set
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S - M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S - M) is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
rng S is V74() Element of bool ExtREAL
S " {+infty} is Element of bool X
rng M is V74() Element of bool ExtREAL
M " {-infty} is Element of bool X
M " {+infty} is Element of bool X
(M " {+infty}) /\ (S " {+infty}) is Element of bool X
S " {-infty} is Element of bool X
(M " {-infty}) /\ (S " {-infty}) is Element of bool X
((M " {+infty}) /\ (S " {+infty})) \/ ((M " {-infty}) /\ (S " {-infty})) is Element of bool X
((dom S) /\ (dom M)) \ {} is Element of bool X
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S + M) is Element of bool X
bool X is non empty set
dom S is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
E is set
(S + M) . E is ext-real Element of ExtREAL
S . E is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
(S . E) + (M . E) is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S + M) is Element of bool X
bool X is non empty set
dom S is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
E is set
(S + M) . E is ext-real Element of ExtREAL
S . E is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
(S . E) + (M . E) is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S - M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S - M) is Element of bool X
bool X is non empty set
dom S is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
E is set
(S - M) . E is ext-real Element of ExtREAL
S . E is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
(S . E) - (M . E) is ext-real Element of ExtREAL
K111((M . E)) is ext-real set
K110((S . E),K111((M . E))) is ext-real set
X is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S - M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S - M) is Element of bool X
bool X is non empty set
dom S is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
E is set
(S - M) . E is ext-real Element of ExtREAL
S . E is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
(S . E) - (M . E) is ext-real Element of ExtREAL
K111((M . E)) is ext-real set
K110((S . E),K111((M . E))) is ext-real set
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim X is ext-real Element of ExtREAL
S is V11() real ext-real set
M is V11() real ext-real set
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
X . S is ext-real Element of ExtREAL
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim X is ext-real Element of ExtREAL
S is ext-real set
rng X is non empty V55() V74() Element of bool ExtREAL
M is ext-real set
dom X is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
E is set
X . E is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
X . F is ext-real Element of ExtREAL
superior_realsequence X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
{ (X . b₁) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : 0 <= b₁ } is set
M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
M . 0 is ext-real Element of ExtREAL
E is non empty V74() Element of bool ExtREAL
sup E is ext-real set
F is set
dom X is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
L is set
X . L is ext-real Element of ExtREAL
inf M is ext-real Element of ExtREAL
rng M is non empty V55() V74() Element of bool ExtREAL
inf (rng M) is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . F is ext-real Element of ExtREAL
dom M is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
F is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
X . L is ext-real Element of ExtREAL
(superior_realsequence X) . 0 is ext-real Element of ExtREAL
lim_sup X is ext-real Element of ExtREAL
inf (superior_realsequence X) is ext-real Element of ExtREAL
rng (superior_realsequence X) is non empty V55() V74() Element of bool ExtREAL
inf (rng (superior_realsequence X)) is ext-real Element of ExtREAL
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim X is ext-real Element of ExtREAL
S is ext-real set
rng X is non empty V55() V74() Element of bool ExtREAL
M is ext-real set
dom X is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
E is set
X . E is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
X . F is ext-real Element of ExtREAL
inferior_realsequence X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
{ (X . b₁) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : 0 <= b₁ } is set
M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
M . 0 is ext-real Element of ExtREAL
E is non empty V74() Element of bool ExtREAL
inf E is ext-real set
F is set
dom X is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
L is set
X . L is ext-real Element of ExtREAL
sup M is ext-real Element of ExtREAL
rng M is non empty V55() V74() Element of bool ExtREAL
sup (rng M) is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . F is ext-real Element of ExtREAL
dom M is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
F is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
X . L is ext-real Element of ExtREAL
(inferior_realsequence X) . 0 is ext-real Element of ExtREAL
lim_inf X is ext-real Element of ExtREAL
sup (inferior_realsequence X) is ext-real Element of ExtREAL
rng (inferior_realsequence X) is non empty V55() V74() Element of bool ExtREAL
sup (rng (inferior_realsequence X)) is ext-real Element of ExtREAL
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim X is ext-real Element of ExtREAL
S is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim S is ext-real Element of ExtREAL
(lim X) + (lim S) is ext-real Element of ExtREAL
M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim M is ext-real Element of ExtREAL
dom M is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
E is set
M . E is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . F is ext-real Element of ExtREAL
M . F is ext-real Element of ExtREAL
X . F is ext-real Element of ExtREAL
(X . F) + (S . F) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . E is ext-real Element of ExtREAL
E is V11() real ext-real set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . L is ext-real Element of ExtREAL
X . L is ext-real Element of ExtREAL
S . L is ext-real Element of ExtREAL
R_EAL E is ext-real Element of ExtREAL
(R_EAL E) + 0. is ext-real Element of ExtREAL
(X . L) + (S . L) is ext-real Element of ExtREAL
E is V11() real ext-real set
F is V11() real ext-real set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . E0 is ext-real Element of ExtREAL
S . E0 is ext-real Element of ExtREAL
X . E0 is ext-real Element of ExtREAL
R_EAL F is ext-real Element of ExtREAL
(R_EAL F) + 0. is ext-real Element of ExtREAL
(X . E0) + (S . E0) is ext-real Element of ExtREAL
F is V11() real ext-real set
E + F is V11() real ext-real set
R_EAL (E + F) is ext-real Element of ExtREAL
E1 is V11() real ext-real set
R_EAL F is ext-real Element of ExtREAL
L is V11() real ext-real Element of REAL
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 / 2 is V11() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
max (I,H) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . G is ext-real Element of ExtREAL
(M . G) - (R_EAL (E + F)) is ext-real Element of ExtREAL
K111((R_EAL (E + F))) is ext-real set
K110((M . G),K111((R_EAL (E + F)))) is ext-real set
|.((M . G) - (R_EAL (E + F))).| is ext-real Element of ExtREAL
S . G is ext-real Element of ExtREAL
(S . G) - (lim S) is ext-real Element of ExtREAL
K111((lim S)) is ext-real set
K110((S . G),K111((lim S))) is ext-real set
|.((S . G) - (lim S)).| is ext-real Element of ExtREAL
(S . G) - (R_EAL F) is ext-real Element of ExtREAL
K111((R_EAL F)) is ext-real set
K110((S . G),K111((R_EAL F))) is ext-real set
X . G is ext-real Element of ExtREAL
(X . G) - (lim X) is ext-real Element of ExtREAL
K111((lim X)) is ext-real set
K110((X . G),K111((lim X))) is ext-real set
|.((X . G) - (lim X)).| is ext-real Element of ExtREAL
R_EAL E is ext-real Element of ExtREAL
(X . G) - (R_EAL E) is ext-real Element of ExtREAL
K111((R_EAL E)) is ext-real set
K110((X . G),K111((R_EAL E))) is ext-real set
G is V11() real ext-real Element of REAL
G is V11() real ext-real Element of REAL
|.((S . G) - (R_EAL F)).| is ext-real Element of ExtREAL
|.G.| is V11() real ext-real Element of REAL
|.((X . G) - (R_EAL E)).| is ext-real Element of ExtREAL
|.G.| is V11() real ext-real Element of REAL
|.((S . G) - (R_EAL F)).| + |.((X . G) - (R_EAL E)).| is ext-real Element of ExtREAL
|.G.| + |.G.| is V11() real ext-real Element of REAL
E0 is V11() real ext-real Element of REAL
(R_EAL E) + (R_EAL F) is ext-real Element of ExtREAL
(M . G) - (R_EAL F) is ext-real Element of ExtREAL
K110((M . G),K111((R_EAL F))) is ext-real set
((M . G) - (R_EAL F)) - (R_EAL E) is ext-real Element of ExtREAL
K110(((M . G) - (R_EAL F)),K111((R_EAL E))) is ext-real set
(X . G) + (S . G) is ext-real Element of ExtREAL
((X . G) + (S . G)) - (R_EAL F) is ext-real Element of ExtREAL
K110(((X . G) + (S . G)),K111((R_EAL F))) is ext-real set
(((X . G) + (S . G)) - (R_EAL F)) - (R_EAL E) is ext-real Element of ExtREAL
K110((((X . G) + (S . G)) - (R_EAL F)),K111((R_EAL E))) is ext-real set
(X . G) + ((S . G) - (R_EAL F)) is ext-real Element of ExtREAL
((X . G) + ((S . G) - (R_EAL F))) - (R_EAL E) is ext-real Element of ExtREAL
K110(((X . G) + ((S . G) - (R_EAL F))),K111((R_EAL E))) is ext-real set
((S . G) - (R_EAL F)) + ((X . G) - (R_EAL E)) is ext-real Element of ExtREAL
(E1 / 2) + (E1 / 2) is V11() real ext-real Element of REAL
E0 is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
E0 + L is V11() real ext-real Element of REAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is Element of X
M # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
S # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
F is set
L is V11() real ext-real set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(S # E) . E1 is ext-real Element of ExtREAL
(M # E) . E1 is ext-real Element of ExtREAL
M . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
(M . E1) . E is ext-real Element of ExtREAL
(M . E1) | F is Relation-like X -defined F -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((M . E1) | F) . E is ext-real Element of ExtREAL
S . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . E1) . E is ext-real Element of ExtREAL
(S # E) . E1 is ext-real Element of ExtREAL
L is V11() real ext-real set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(S # E) . E1 is ext-real Element of ExtREAL
(M # E) . E1 is ext-real Element of ExtREAL
M . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
(M . E1) . E is ext-real Element of ExtREAL
(M . E1) | F is Relation-like X -defined F -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((M . E1) | F) . E is ext-real Element of ExtREAL
S . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . E1) . E is ext-real Element of ExtREAL
(S # E) . E1 is ext-real Element of ExtREAL
lim (M # E) is ext-real Element of ExtREAL
L is V11() real ext-real set
R_EAL L is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(S # E) . H is ext-real Element of ExtREAL
((S # E) . H) - (R_EAL L) is ext-real Element of ExtREAL
K111((R_EAL L)) is ext-real set
K110(((S # E) . H),K111((R_EAL L))) is ext-real set
|.(((S # E) . H) - (R_EAL L)).| is ext-real Element of ExtREAL
(M # E) . H is ext-real Element of ExtREAL
M . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
(M . H) . E is ext-real Element of ExtREAL
(M . H) | F is Relation-like X -defined F -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((M . H) | F) . E is ext-real Element of ExtREAL
S . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . H) . E is ext-real Element of ExtREAL
((M # E) . H) - (lim (M # E)) is ext-real Element of ExtREAL
K111((lim (M # E))) is ext-real set
K110(((M # E) . H),K111((lim (M # E)))) is ext-real set
|.(((M # E) . H) - (lim (M # E))).| is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom F is Element of bool X
eq_dom (F,+infty) is Element of bool X
E /\ (eq_dom (F,+infty)) is Element of bool X
M . (E /\ (eq_dom (F,+infty))) is ext-real Element of ExtREAL
Integral (M,F) is ext-real Element of ExtREAL
max+ F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ F)) is ext-real Element of ExtREAL
max- F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- F)) is ext-real Element of ExtREAL
(integral+ (M,(max+ F))) - (integral+ (M,(max- F))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- F)))) is ext-real set
K110((integral+ (M,(max+ F))),K111((integral+ (M,(max- F))))) is ext-real set
F | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F | E) is Element of bool X
(dom F) /\ E is Element of bool X
L is Element of S
F | L is Relation-like X -defined L -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(F | L)) is ext-real Element of ExtREAL
integral+ (M,(F | E)) is ext-real Element of ExtREAL
Integral (M,(F | E)) is ext-real Element of ExtREAL
max+ (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F | E))) is ext-real Element of ExtREAL
max- (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F | E)))) - (integral+ (M,(max- (F | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F | E))))) is ext-real set
K110((integral+ (M,(max+ (F | E)))),K111((integral+ (M,(max- (F | E)))))) is ext-real set
(dom F) /\ L is Element of bool X
chi (L,X) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(chi (L,X)) | L is Relation-like X -defined L -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E1 is Relation-like Function-like set
dom E1 is set
H is set
rng E1 is set
I is set
E1 . I is set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I (#) ((chi (L,X)) | L) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
H is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . I) is Element of bool X
dom (chi (L,X)) is Element of bool X
dom ((chi (L,X)) | L) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I (#) ((chi (L,X)) | L) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is set
(H . I) . H is ext-real Element of ExtREAL
G is Element of X
(chi (L,X)) . G is ext-real Element of ExtREAL
1. is ext-real Element of ExtREAL
((chi (L,X)) | L) . G is ext-real Element of ExtREAL
R_EAL I is ext-real Element of ExtREAL
(R_EAL I) * 1. is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . I) is Element of bool X
I is set
(H . I) . I is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I . H is ext-real Element of ExtREAL
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(H . H)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(H . G)) is ext-real Element of ExtREAL
dom (F | L) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
(H . H) . G is ext-real Element of ExtREAL
(H . G) . G is ext-real Element of ExtREAL
dom (H . H) is Element of bool X
dom (H . G) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . H) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is set
(H . H) . G is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I . H is ext-real Element of ExtREAL
R_EAL H is ext-real Element of ExtREAL
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . H) is Element of bool X
M . (dom (H . H)) is ext-real Element of ExtREAL
(R_EAL H) * (M . (dom (H . H))) is ext-real Element of ExtREAL
G is set
(H . H) . G is ext-real Element of ExtREAL
integral' (M,(H . H)) is ext-real Element of ExtREAL
E0 is Element of S
M . E0 is ext-real Element of ExtREAL
M . L is ext-real Element of ExtREAL
rng I is non empty V55() V74() Element of bool ExtREAL
H is V11() real ext-real set
I . 1 is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
H . 1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . 1) is Element of bool X
M . (dom (H . 1)) is ext-real Element of ExtREAL
(R_EAL 1) * (M . (dom (H . 1))) is ext-real Element of ExtREAL
(R_EAL 1) * (M . L) is ext-real Element of ExtREAL
dom I is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
G is V11() real ext-real Element of REAL
G is V11() real ext-real Element of REAL
G / G is V11() real ext-real Element of REAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I . G is ext-real Element of ExtREAL
R_EAL G is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
M . (dom (H . G)) is ext-real Element of ExtREAL
(R_EAL G) * (M . (dom (H . G))) is ext-real Element of ExtREAL
(R_EAL G) * (M . L) is ext-real Element of ExtREAL
G * G is V11() real ext-real Element of REAL
(G / G) * G is V11() real ext-real Element of REAL
G / G is V11() real ext-real Element of REAL
G / (G / G) is V11() real ext-real Element of REAL
dom I is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I . G is ext-real Element of ExtREAL
I . H is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
R_EAL G is ext-real Element of ExtREAL
(R_EAL G) * (M . L) is ext-real Element of ExtREAL
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . H) is Element of bool X
R_EAL H is ext-real Element of ExtREAL
(R_EAL H) * (M . L) is ext-real Element of ExtREAL
lim I is ext-real Element of ExtREAL
sup I is ext-real Element of ExtREAL
sup (rng I) is ext-real Element of ExtREAL
H is Element of X
H # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (H # H) is ext-real Element of ExtREAL
(F | L) . H is ext-real Element of ExtREAL
F . H is ext-real Element of ExtREAL
rng (H # H) is non empty V55() V74() Element of bool ExtREAL
G is V11() real ext-real set
G is V11() real ext-real Element of REAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
(H . G) . H is ext-real Element of ExtREAL
(H # H) . G is ext-real Element of ExtREAL
dom (H # H) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(H # H) . G is ext-real Element of ExtREAL
(H # H) . G is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
(H . G) . H is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
(H . G) . H is ext-real Element of ExtREAL
sup (rng (H # H)) is ext-real set
sup (H # H) is ext-real Element of ExtREAL
sup (rng (H # H)) is ext-real Element of ExtREAL
sup (rng I) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
chi (E,X) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
Integral (M,(chi (E,X))) is ext-real Element of ExtREAL
max+ (chi (E,X)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (chi (E,X)))) is ext-real Element of ExtREAL
max- (chi (E,X)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (chi (E,X)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (chi (E,X))))) - (integral+ (M,(max- (chi (E,X))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (chi (E,X)))))) is ext-real set
K110((integral+ (M,(max+ (chi (E,X))))),K111((integral+ (M,(max- (chi (E,X))))))) is ext-real set
M . E is ext-real Element of ExtREAL
(chi (E,X)) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((chi (E,X)) | E)) is ext-real Element of ExtREAL
max+ ((chi (E,X)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((chi (E,X)) | E))) is ext-real Element of ExtREAL
max- ((chi (E,X)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((chi (E,X)) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((chi (E,X)) | E)))) - (integral+ (M,(max- ((chi (E,X)) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((chi (E,X)) | E))))) is ext-real set
K110((integral+ (M,(max+ ((chi (E,X)) | E)))),K111((integral+ (M,(max- ((chi (E,X)) | E)))))) is ext-real set
F is Element of S
F \ E is Element of S
E0 is Element of X
dom (max- (chi (E,X))) is Element of bool X
(chi (E,X)) . E0 is ext-real Element of ExtREAL
- ((chi (E,X)) . E0) is ext-real Element of ExtREAL
max ((- ((chi (E,X)) . E0)),0.) is ext-real set
(max- (chi (E,X))) . E0 is ext-real Element of ExtREAL
(chi (E,X)) . E0 is ext-real Element of ExtREAL
- ((chi (E,X)) . E0) is ext-real Element of ExtREAL
max ((- ((chi (E,X)) . E0)),0.) is ext-real set
(max- (chi (E,X))) . E0 is ext-real Element of ExtREAL
dom (chi (E,X)) is Element of bool X
dom (max+ (chi (E,X))) is Element of bool X
F /\ (F \ E) is Element of S
(max+ (chi (E,X))) | (F \ E) is Relation-like X -defined F \ E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((max+ (chi (E,X))) | (F \ E)) is Element of bool X
E0 is Element of X
(chi (E,X)) . E0 is ext-real Element of ExtREAL
(max+ (chi (E,X))) . E0 is ext-real Element of ExtREAL
((max+ (chi (E,X))) | (F \ E)) . E0 is ext-real Element of ExtREAL
integral+ (M,((max+ (chi (E,X))) | (F \ E))) is ext-real Element of ExtREAL
F /\ E is Element of S
(max+ (chi (E,X))) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((max+ (chi (E,X))) | E) is Element of bool X
E \/ (F \ E) is Element of S
(max+ (chi (E,X))) | (E \/ (F \ E)) is Relation-like X -defined E \/ (F \ E) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 is set
(max+ (chi (E,X))) . E0 is ext-real Element of ExtREAL
integral+ (M,((max+ (chi (E,X))) | (E \/ (F \ E)))) is ext-real Element of ExtREAL
integral+ (M,((max+ (chi (E,X))) | E)) is ext-real Element of ExtREAL
(integral+ (M,((max+ (chi (E,X))) | E))) + (integral+ (M,((max+ (chi (E,X))) | (F \ E)))) is ext-real Element of ExtREAL
E0 is set
(chi (E,X)) . E0 is ext-real Element of ExtREAL
(max+ (chi (E,X))) . E0 is ext-real Element of ExtREAL
((max+ (chi (E,X))) | E) . E0 is ext-real Element of ExtREAL
integral' (M,((max+ (chi (E,X))) | E)) is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
M . (dom ((max+ (chi (E,X))) | E)) is ext-real Element of ExtREAL
(R_EAL 1) * (M . (dom ((max+ (chi (E,X))) | E))) is ext-real Element of ExtREAL
dom ((chi (E,X)) | E) is Element of bool X
integral+ (M,((chi (E,X)) | E)) is ext-real Element of ExtREAL
(R_EAL 1) * (M . E) is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom F is Element of bool X
F | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F | E)) is ext-real Element of ExtREAL
max+ (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F | E))) is ext-real Element of ExtREAL
max- (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F | E)))) - (integral+ (M,(max- (F | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F | E))))) is ext-real set
K110((integral+ (M,(max+ (F | E)))),K111((integral+ (M,(max- (F | E)))))) is ext-real set
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom L is Element of bool X
L | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(L | E)) is ext-real Element of ExtREAL
max+ (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (L | E))) is ext-real Element of ExtREAL
max- (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (L | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (L | E)))) - (integral+ (M,(max- (L | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (L | E))))) is ext-real set
K110((integral+ (M,(max+ (L | E)))),K111((integral+ (M,(max- (L | E)))))) is ext-real set
dom (F | E) is Element of bool X
dom (L | E) is Element of bool X
H is Element of X
(F | E) . H is ext-real Element of ExtREAL
(L | E) . H is ext-real Element of ExtREAL
F . H is ext-real Element of ExtREAL
L . H is ext-real Element of ExtREAL
H is set
(L | E) . H is ext-real Element of ExtREAL
(F | E) . H is ext-real Element of ExtREAL
(dom L) /\ E is Element of bool X
(dom F) /\ E is Element of bool X
integral+ (M,(F | E)) is ext-real Element of ExtREAL
integral+ (M,(L | E)) is ext-real Element of ExtREAL
X is Relation-like Function-like ext-real-valued set
S is set
X . S is ext-real set
X is Relation-like Function-like ext-real-valued set
(X,0) is ext-real Element of ExtREAL
S is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(S,0) is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(S,(M + 1)) is ext-real Element of ExtREAL
(S,M) is ext-real Element of ExtREAL
(X,(M + 1)) is ext-real Element of ExtREAL
(S,M) + (X,(M + 1)) is ext-real Element of ExtREAL
S is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(S,0) is ext-real Element of ExtREAL
M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(M,0) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(S,E) is ext-real Element of ExtREAL
(M,E) is ext-real Element of ExtREAL
E + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(S,(E + 1)) is ext-real Element of ExtREAL
(M,(E + 1)) is ext-real Element of ExtREAL
(X,(E + 1)) is ext-real Element of ExtREAL
(M,E) + (X,(E + 1)) is ext-real Element of ExtREAL
X is Relation-like Function-like ext-real-valued set
(X) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (X) is ext-real Element of ExtREAL
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((X),M) is ext-real Element of ExtREAL
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((X),(M + 1)) is ext-real Element of ExtREAL
(X,(M + 1)) is ext-real Element of ExtREAL
((X),M) + (X,(M + 1)) is ext-real Element of ExtREAL
((X),0) is ext-real Element of ExtREAL
(X,0) is ext-real Element of ExtREAL
dom (X) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
M is set
((X),M) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((X),E) is ext-real Element of ExtREAL
((X),M) is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((X),F) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((X),L) is ext-real Element of ExtREAL
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((X),(L + 1)) is ext-real Element of ExtREAL
(X,(L + 1)) is ext-real Element of ExtREAL
((X),L) + (X,(L + 1)) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((X),L) is ext-real Element of ExtREAL
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
L - 1 is V11() real ext-real Element of REAL
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((X),E0) is ext-real Element of ExtREAL
X is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((X),S) is ext-real Element of ExtREAL
S + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((X),(S + 1)) is ext-real Element of ExtREAL
(X,(S + 1)) is ext-real Element of ExtREAL
((X),S) + (X,(S + 1)) is ext-real Element of ExtREAL
((X),0) is ext-real Element of ExtREAL
(X,0) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (M . F) is Element of bool X
bool X is non empty set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . L) is Element of bool X
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . L) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . L) is Element of bool X
(dom (S . L)) /\ E is Element of bool X
S . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . F) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . F) is Element of bool X
(dom (S . F)) /\ E is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
lim S is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E is set
(lim S) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . 0) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . 0) is Element of bool X
dom ((lim S) | E) is Element of bool X
dom (lim S) is Element of bool X
(dom (lim S)) /\ E is Element of bool X
(dom (S . 0)) /\ E is Element of bool X
dom (lim M) is Element of bool X
F is Element of X
(((lim S) | E),F) is ext-real Element of ExtREAL
((lim S),F) is ext-real Element of ExtREAL
S # F is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (S # F) is ext-real Element of ExtREAL
M # F is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (M # F) is ext-real Element of ExtREAL
((lim M),F) is ext-real Element of ExtREAL
M # F is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (M # F) is ext-real Element of ExtREAL
((lim M),F) is ext-real Element of ExtREAL
L is V11() real ext-real set
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((S # F),I) is ext-real Element of ExtREAL
S . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . I),F) is ext-real Element of ExtREAL
(S . I) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((S . I) | E),F) is ext-real Element of ExtREAL
M . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((M . I),F) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((S # F),I) - (lim (S # F)) is ext-real Element of ExtREAL
K111((lim (S # F))) is ext-real set
K110(((S # F),I),K111((lim (S # F)))) is ext-real set
|.(((S # F),I) - (lim (S # F))).| is ext-real Element of ExtREAL
M # F is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((M # F),I) is ext-real Element of ExtREAL
R_EAL L is ext-real Element of ExtREAL
((M # F),I) - (R_EAL L) is ext-real Element of ExtREAL
K111((R_EAL L)) is ext-real set
K110(((M # F),I),K111((R_EAL L))) is ext-real set
|.(((M # F),I) - (R_EAL L)).| is ext-real Element of ExtREAL
lim (M # F) is ext-real Element of ExtREAL
((lim M),F) is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is Element of S
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (E . 0) is Element of bool X
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E . L) is Element of bool X
(dom (E . L)) /\ M is Element of bool X
(E . L) | M is Relation-like X -defined M -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is Element of S
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (E . 0) is Element of bool X
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is Element of X
F # L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(E . 0) | M is Relation-like X -defined M -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((E . 0) | M) is Element of bool X
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . 0) is Element of bool X
E # L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((E # L),E0) is ext-real Element of ExtREAL
((F # L),E0) is ext-real Element of ExtREAL
F . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . E0) is Element of bool X
E . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E . E0) | M is Relation-like X -defined M -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((E . E0) | M) is Element of bool X
(((E . E0) | M),L) is ext-real Element of ExtREAL
((E . E0),L) is ext-real Element of ExtREAL
((F . E0),L) is ext-real Element of ExtREAL
((E # L)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((F # L)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . (M + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E is set
F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L + (S . (M + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F + (S . (M + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 + (S . (M + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is Relation-like Function-like set
dom M is set
M . 0 is set
E is set
rng M is set
F is set
M . F is set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . E0 is set
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (E0 + 1) is set
E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E1 + (S . (E0 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . L is set
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E . F) + (S . (F + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E . L) + (S . (L + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E . F) + (S . (F + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(M . F) + (S . (F + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . E) is Element of bool X
F is set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . E0) is Element of bool X
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . (E0 + 1)) is Element of bool X
S . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E0) + (S . (E0 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . (E0 + 1)) is Element of bool X
(dom ((X,S) . E0)) /\ (dom (S . (E0 + 1))) is Element of bool X
((X,S) . E0) " {-infty} is Element of bool X
(S . (E0 + 1)) " {+infty} is Element of bool X
(((X,S) . E0) " {-infty}) /\ ((S . (E0 + 1)) " {+infty}) is Element of bool X
((X,S) . E0) " {+infty} is Element of bool X
(S . (E0 + 1)) " {-infty} is Element of bool X
(((X,S) . E0) " {+infty}) /\ ((S . (E0 + 1)) " {-infty}) is Element of bool X
((((X,S) . E0) " {-infty}) /\ ((S . (E0 + 1)) " {+infty})) \/ ((((X,S) . E0) " {+infty}) /\ ((S . (E0 + 1)) " {-infty})) is Element of bool X
((dom ((X,S) . E0)) /\ (dom (S . (E0 + 1)))) \ (((((X,S) . E0) " {-infty}) /\ ((S . (E0 + 1)) " {+infty})) \/ ((((X,S) . E0) " {+infty}) /\ ((S . (E0 + 1)) " {-infty}))) is Element of bool X
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . 0) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . E) is Element of bool X
S . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . E) is Element of bool X
F is set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E0) + (S . E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . E0) is Element of bool X
(dom ((X,S) . E0)) /\ (dom (S . E)) is Element of bool X
((X,S) . E0) " {-infty} is Element of bool X
(S . E) " {+infty} is Element of bool X
(((X,S) . E0) " {-infty}) /\ ((S . E) " {+infty}) is Element of bool X
((X,S) . E0) " {+infty} is Element of bool X
(S . E) " {-infty} is Element of bool X
(((X,S) . E0) " {+infty}) /\ ((S . E) " {-infty}) is Element of bool X
((((X,S) . E0) " {-infty}) /\ ((S . E) " {+infty})) \/ ((((X,S) . E0) " {+infty}) /\ ((S . E) " {-infty})) is Element of bool X
((dom ((X,S) . E0)) /\ (dom (S . E))) \ (((((X,S) . E0) " {-infty}) /\ ((S . E) " {+infty})) \/ ((((X,S) . E0) " {+infty}) /\ ((S . E) " {-infty}))) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is set
(((X,S) . M),E) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . L),E) is ext-real Element of ExtREAL
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . (L + 1)),E) is ext-real Element of ExtREAL
dom ((X,S) . L) is Element of bool X
((X,S) . L) " {+infty} is Element of bool X
S . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . (L + 1)) " {-infty} is Element of bool X
(((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty}) is Element of bool X
dom (S . (L + 1)) is Element of bool X
(dom ((X,S) . L)) /\ (dom (S . (L + 1))) is Element of bool X
((X,S) . L) + (S . (L + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . (L + 1)),E) is ext-real Element of ExtREAL
(S . (L + 1)) " {+infty} is Element of bool X
((X,S) . L) " {-infty} is Element of bool X
(((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty}) is Element of bool X
((((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty})) \/ ((((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty})) is Element of bool X
((dom ((X,S) . L)) /\ (dom (S . (L + 1)))) \ (((((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty})) \/ ((((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty}))) is Element of bool X
dom ((X,S) . (L + 1)) is Element of bool X
(((X,S) . L),E) + ((S . (L + 1)),E) is ext-real Element of ExtREAL
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . 0),E) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),E) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is set
(((X,S) . M),F) is ext-real Element of ExtREAL
((S . E),F) is ext-real Element of ExtREAL
dom (S . E) is Element of bool X
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),F) is ext-real Element of ExtREAL
dom (S . L) is Element of bool X
(dom (S . E)) /\ (dom (S . L)) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is set
(((X,S) . M),E) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . L),E) is ext-real Element of ExtREAL
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . (L + 1)),E) is ext-real Element of ExtREAL
dom ((X,S) . L) is Element of bool X
((X,S) . L) " {-infty} is Element of bool X
S . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . (L + 1)) " {+infty} is Element of bool X
(((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty}) is Element of bool X
dom (S . (L + 1)) is Element of bool X
(dom ((X,S) . L)) /\ (dom (S . (L + 1))) is Element of bool X
((X,S) . L) + (S . (L + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . (L + 1)),E) is ext-real Element of ExtREAL
(S . (L + 1)) " {-infty} is Element of bool X
((X,S) . L) " {+infty} is Element of bool X
(((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty}) is Element of bool X
((((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty})) \/ ((((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty})) is Element of bool X
((dom ((X,S) . L)) /\ (dom (S . (L + 1)))) \ (((((X,S) . L) " {-infty}) /\ ((S . (L + 1)) " {+infty})) \/ ((((X,S) . L) " {+infty}) /\ ((S . (L + 1)) " {-infty}))) is Element of bool X
dom ((X,S) . (L + 1)) is Element of bool X
(((X,S) . L),E) + ((S . (L + 1)),E) is ext-real Element of ExtREAL
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . 0),E) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),E) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is set
(((X,S) . M),F) is ext-real Element of ExtREAL
((S . E),F) is ext-real Element of ExtREAL
dom (S . E) is Element of bool X
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),F) is ext-real Element of ExtREAL
dom (S . L) is Element of bool X
(dom (S . E)) /\ (dom (S . L)) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
((X,S) . M) " {-infty} is Element of bool X
bool X is non empty set
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . (M + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . (M + 1)) " {+infty} is Element of bool X
(((X,S) . M) " {-infty}) /\ ((S . (M + 1)) " {+infty}) is Element of bool X
((X,S) . M) " {+infty} is Element of bool X
(S . (M + 1)) " {-infty} is Element of bool X
(((X,S) . M) " {+infty}) /\ ((S . (M + 1)) " {-infty}) is Element of bool X
F is set
F is set
dom ((X,S) . M) is Element of bool X
(((X,S) . M),F) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),F) is ext-real Element of ExtREAL
dom (S . (M + 1)) is Element of bool X
((S . (M + 1)),F) is ext-real Element of ExtREAL
dom (S . L) is Element of bool X
(dom (S . L)) /\ (dom (S . (M + 1))) is Element of bool X
F is set
F is set
(((X,S) . M),F) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . L),F) is ext-real Element of ExtREAL
((S . (M + 1)),F) is ext-real Element of ExtREAL
dom (S . L) is Element of bool X
(dom (S . L)) /\ (dom (S . (M + 1))) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= M } is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= M } is set
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= a₁ } is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= a₁ } is set
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . 0) is Element of bool X
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . 0) is Element of bool X
F is set
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= 0 } is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . L) is Element of bool X
{(dom (S . 0))} is non empty set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . F) is Element of bool X
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= F } is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= F } is set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . (F + 1)) is Element of bool X
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= F + 1 } is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= F + 1 } is set
S . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . F) + (S . (F + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . F) " {+infty} is Element of bool X
(S . (F + 1)) " {-infty} is Element of bool X
(((X,S) . F) " {+infty}) /\ ((S . (F + 1)) " {-infty}) is Element of bool X
L is set
meet H₁(F) is set
dom (S . (F + 1)) is Element of bool X
(meet H₁(F)) /\ (dom (S . (F + 1))) is Element of bool X
E0 is set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . E1) is Element of bool X
meet H₁(F + 1) is set
L is set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . E0) is Element of bool X
L is set
((X,S) . F) " {-infty} is Element of bool X
(S . (F + 1)) " {+infty} is Element of bool X
(((X,S) . F) " {-infty}) /\ ((S . (F + 1)) " {+infty}) is Element of bool X
(dom ((X,S) . F)) /\ (dom (S . (F + 1))) is Element of bool X
{} \/ {} is Relation-like RAT -valued complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() set
((dom ((X,S) . F)) /\ (dom (S . (F + 1)))) \ ({} \/ {}) is Element of bool X
F is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= 0 } is set
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . M) is Element of bool X
{ (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= M } is set
E is set
meet { (dom (S . b₁)) where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT : b₁ <= M } is set
{(dom (S . 0))} is non empty set
meet {(dom (S . 0))} is set
E is set
F is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . L) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . M) is Element of bool X
bool X is non empty set
S . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . E) is Element of bool X
(dom (S . M)) /\ (dom (S . E)) is Element of bool X
F is set
((S . M),F) is ext-real Element of ExtREAL
((S . E),F) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (M . F) is Element of bool X
bool X is non empty set
M . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . L) is Element of bool X
(dom (M . F)) /\ (dom (M . L)) is Element of bool X
S . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . L) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . L) is Element of bool X
E0 is set
((M . F),E0) is ext-real Element of ExtREAL
((M . L),E0) is ext-real Element of ExtREAL
S . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(S . F) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . F) is Element of bool X
(dom (S . F)) /\ (dom (S . L)) is Element of bool X
((S . F),E0) is ext-real Element of ExtREAL
((S . L),E0) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E is Element of X
S # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((S # E)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(((S # E)),M) is ext-real Element of ExtREAL
(X,S) # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(((X,S) # E),M) is ext-real Element of ExtREAL
F is set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # E)),E1) is ext-real Element of ExtREAL
(((X,S) # E),E1) is ext-real Element of ExtREAL
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((S # E)),(E1 + 1)) is ext-real Element of ExtREAL
(((X,S) # E),(E1 + 1)) is ext-real Element of ExtREAL
((S # E),(E1 + 1)) is ext-real Element of ExtREAL
(((S # E)),E1) + ((S # E),(E1 + 1)) is ext-real Element of ExtREAL
S . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . (E1 + 1)),E) is ext-real Element of ExtREAL
(((X,S) # E),E1) + ((S . (E1 + 1)),E) is ext-real Element of ExtREAL
(X,S) . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E1),E) is ext-real Element of ExtREAL
(((X,S) . E1),E) + ((S . (E1 + 1)),E) is ext-real Element of ExtREAL
(X,S) . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E1) + (S . (E1 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . (E1 + 1)) is Element of bool X
(((X,S) . (E1 + 1)),E) is ext-real Element of ExtREAL
(((S # E)),0) is ext-real Element of ExtREAL
((S # E),0) is ext-real Element of ExtREAL
((S . 0),E) is ext-real Element of ExtREAL
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . 0),E) is ext-real Element of ExtREAL
(((X,S) # E),0) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is Element of X
S # M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((S # M)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,S) # M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E is set
E0 is V11() real ext-real set
R_EAL E0 is ext-real Element of ExtREAL
E1 is V11() real ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # M)),I) is ext-real Element of ExtREAL
(((X,S) # M),I) is ext-real Element of ExtREAL
(((X,S) # M),I) - (R_EAL E0) is ext-real Element of ExtREAL
K111((R_EAL E0)) is ext-real set
K110((((X,S) # M),I),K111((R_EAL E0))) is ext-real set
|.((((X,S) # M),I) - (R_EAL E0)).| is ext-real Element of ExtREAL
E0 is V11() real ext-real set
R_EAL E0 is ext-real Element of ExtREAL
E1 is V11() real ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # M)),I) is ext-real Element of ExtREAL
(((X,S) # M),I) is ext-real Element of ExtREAL
(((S # M)),I) - (R_EAL E0) is ext-real Element of ExtREAL
K111((R_EAL E0)) is ext-real set
K110((((S # M)),I),K111((R_EAL E0))) is ext-real set
|.((((S # M)),I) - (R_EAL E0)).| is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # M)),I) is ext-real Element of ExtREAL
(((X,S) # M),I) is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,S) # M),I) is ext-real Element of ExtREAL
(((S # M)),I) is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # M)),I) is ext-real Element of ExtREAL
(((X,S) # M),I) is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,S) # M),I) is ext-real Element of ExtREAL
(((S # M)),I) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom M is Element of bool X
E is Element of X
S # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(M,E) is ext-real Element of ExtREAL
((S # E)) is ext-real Element of ExtREAL
((S # E)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((S # E)) is ext-real Element of ExtREAL
(X,S) # E is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,S) # E) is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E0 is V11() real ext-real set
E1 is V11() real ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((S # E)),I) is ext-real Element of ExtREAL
(((X,S) # E),I) is ext-real Element of ExtREAL
(((X,S) # E),I) - (M,E) is ext-real Element of ExtREAL
K111((M,E)) is ext-real set
K110((((X,S) # E),I),K111((M,E))) is ext-real set
|.((((X,S) # E),I) - (M,E)).| is ext-real Element of ExtREAL
E0 is V11() real ext-real set
E1 is V11() real ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,M) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,M) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . F) is Element of bool X
M . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . L) is Element of bool X
(dom (M . F)) /\ (dom (M . L)) is Element of bool X
E0 is set
((M . F),E0) is ext-real Element of ExtREAL
((M . L),E0) is ext-real Element of ExtREAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,M) . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,M) . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M . (F + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,M) . F) + (M . (F + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,M) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
M . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E) + (S . (E + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is Element of X
(((X,S) . M),F) is ext-real Element of ExtREAL
(((X,S) . E),F) is ext-real Element of ExtREAL
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E0),F) is ext-real Element of ExtREAL
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . (E0 + 1)),F) is ext-real Element of ExtREAL
S . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E0) + (S . (E0 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . (E0 + 1)),F) is ext-real Element of ExtREAL
dom ((X,S) . (E0 + 1)) is Element of bool X
(((X,S) . E0),F) + ((S . (E0 + 1)),F) is ext-real Element of ExtREAL
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E0),F) is ext-real Element of ExtREAL
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 - 1 is V11() real ext-real Element of REAL
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E1),F) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is Element of X
(X,S) # M is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,S) # M),F) is ext-real Element of ExtREAL
(((X,S) # M),E) is ext-real Element of ExtREAL
(X,S) . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . F),M) is ext-real Element of ExtREAL
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E),M) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E) + (S . (E + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . (E + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E) + (S . (E + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is Element of S
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,E) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,E) . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,E) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,E) . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . (E0 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,E) . E0) + (E . (E0 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,E) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . 0) is Element of bool X
(dom (S . 0)) /\ (dom (M . 0)) is Element of bool X
(X,M) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,M) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F is Element of X
(((X,S) . E),F) is ext-real Element of ExtREAL
(((X,M) . E),F) is ext-real Element of ExtREAL
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . E1),F) is ext-real Element of ExtREAL
(X,M) . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,M) . E1),F) is ext-real Element of ExtREAL
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,S) . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . (E1 + 1)),F) is ext-real Element of ExtREAL
(X,M) . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,M) . (E1 + 1)),F) is ext-real Element of ExtREAL
dom ((X,S) . (E1 + 1)) is Element of bool X
dom ((X,M) . (E1 + 1)) is Element of bool X
M . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,M) . E1) + (M . (E1 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((M . (E1 + 1)),F) is ext-real Element of ExtREAL
(((X,M) . E1),F) + ((M . (E1 + 1)),F) is ext-real Element of ExtREAL
S . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,S) . E1) + (S . (E1 + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((S . (E1 + 1)),F) is ext-real Element of ExtREAL
(((X,S) . E1),F) + ((S . (E1 + 1)),F) is ext-real Element of ExtREAL
(X,M) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,S) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,S) . 0),F) is ext-real Element of ExtREAL
(((X,M) . 0),F) is ext-real Element of ExtREAL
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,S) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom ((X,S) . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,S) . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,S) . E) is Element of bool X
S . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . 0) is Element of bool X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is Element of S
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (E . 0) is Element of bool X
(X,E) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (X,E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
L is Element of X
E # L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((E # L)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F # L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,E) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,E) . 0) is Element of bool X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,E) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,E) . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
L + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,E) . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E . (L + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,E) . L) + (E . (L + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,E) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (F . 0) is Element of bool X
(X,F) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(L) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F) . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . E0)) is ext-real Element of ExtREAL
max+ ((X,F) . E0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . E0))) is ext-real Element of ExtREAL
max- ((X,F) . E0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . E0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . E0)))) - (integral+ (M,(max- ((X,F) . E0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . E0))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . E0)))),K111((integral+ (M,(max- ((X,F) . E0)))))) is ext-real set
((L),E0) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . I)) is ext-real Element of ExtREAL
max+ ((X,F) . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . I))) is ext-real Element of ExtREAL
max- ((X,F) . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . I)))) - (integral+ (M,(max- ((X,F) . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . I))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . I)))),K111((integral+ (M,(max- ((X,F) . I)))))) is ext-real set
((L),I) is ext-real Element of ExtREAL
I + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,F) . (I + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . (I + 1))) is ext-real Element of ExtREAL
max+ ((X,F) . (I + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . (I + 1)))) is ext-real Element of ExtREAL
max- ((X,F) . (I + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . (I + 1)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . (I + 1))))) - (integral+ (M,(max- ((X,F) . (I + 1))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . (I + 1)))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . (I + 1))))),K111((integral+ (M,(max- ((X,F) . (I + 1))))))) is ext-real set
((L),(I + 1)) is ext-real Element of ExtREAL
F . (I + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . (I + 1)) is Element of bool X
dom ((X,F) . I) is Element of bool X
((X,F) . I) + (F . (I + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (((X,F) . I) + (F . (I + 1))) is Element of bool X
integral+ (M,(((X,F) . I) + (F . (I + 1)))) is ext-real Element of ExtREAL
I is Element of S
((X,F) . I) | I is Relation-like X -defined I -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(((X,F) . I) | I)) is ext-real Element of ExtREAL
(F . (I + 1)) | I is Relation-like X -defined I -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,((F . (I + 1)) | I)) is ext-real Element of ExtREAL
(integral+ (M,(((X,F) . I) | I))) + (integral+ (M,((F . (I + 1)) | I))) is ext-real Element of ExtREAL
E /\ E is Element of S
dom ((X,F) . (I + 1)) is Element of bool X
integral+ (M,((X,F) . (I + 1))) is ext-real Element of ExtREAL
(Integral (M,((X,F) . I))) + (integral+ (M,((F . (I + 1)) | I))) is ext-real Element of ExtREAL
Integral (M,(F . (I + 1))) is ext-real Element of ExtREAL
max+ (F . (I + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . (I + 1)))) is ext-real Element of ExtREAL
max- (F . (I + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . (I + 1)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . (I + 1))))) - (integral+ (M,(max- (F . (I + 1))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . (I + 1)))))) is ext-real set
K110((integral+ (M,(max+ (F . (I + 1))))),K111((integral+ (M,(max- (F . (I + 1))))))) is ext-real set
(Integral (M,((X,F) . I))) + (Integral (M,(F . (I + 1)))) is ext-real Element of ExtREAL
(L,(I + 1)) is ext-real Element of ExtREAL
((L),I) + (L,(I + 1)) is ext-real Element of ExtREAL
(X,F) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . 0)) is ext-real Element of ExtREAL
max+ ((X,F) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . 0))) is ext-real Element of ExtREAL
max- ((X,F) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . 0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . 0)))) - (integral+ (M,(max- ((X,F) . 0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . 0))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . 0)))),K111((integral+ (M,(max- ((X,F) . 0)))))) is ext-real set
Integral (M,(F . 0)) is ext-real Element of ExtREAL
max+ (F . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . 0))) is ext-real Element of ExtREAL
max- (F . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . 0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . 0)))) - (integral+ (M,(max- (F . 0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . 0))))) is ext-real set
K110((integral+ (M,(max+ (F . 0)))),K111((integral+ (M,(max- (F . 0)))))) is ext-real set
(L,0) is ext-real Element of ExtREAL
((L),0) is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom L is Element of bool X
L | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(L | E)) is ext-real Element of ExtREAL
max+ (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (L | E))) is ext-real Element of ExtREAL
max- (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (L | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (L | E)))) - (integral+ (M,(max- (L | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (L | E))))) is ext-real set
K110((integral+ (M,(max+ (L | E)))),K111((integral+ (M,(max- (L | E)))))) is ext-real set
NAT --> 0. is non empty T-Sequence-like Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:NAT,ExtREAL:]
E0 is non empty T-Sequence-like Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:NAT,ExtREAL:]
(E0) is ext-real Element of ExtREAL
(E0) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (E0) is ext-real Element of ExtREAL
(M,E) is ext-real Element of ExtREAL
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(E0,E1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative Element of ExtREAL
F . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . E1) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . E1) | E)) is ext-real Element of ExtREAL
max+ ((F . E1) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . E1) | E))) is ext-real Element of ExtREAL
max- ((F . E1) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . E1) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . E1) | E)))) - (integral+ (M,(max- ((F . E1) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . E1) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . E1) | E)))),K111((integral+ (M,(max- ((F . E1) | E)))))) is ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
I is Element of S
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . H) | E)) is ext-real Element of ExtREAL
max+ ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . H) | E))) is ext-real Element of ExtREAL
max- ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . H) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . H) | E)))) - (integral+ (M,(max- ((F . H) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . H) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . H) | E)))),K111((integral+ (M,(max- ((F . H) | E)))))) is ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((E0),E1) is ext-real Element of ExtREAL
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((E0),(E1 + 1)) is ext-real Element of ExtREAL
(E0,(E1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative Element of ExtREAL
((E0),E1) + (E0,(E1 + 1)) is ext-real Element of ExtREAL
((E0),0) is ext-real Element of ExtREAL
(E0,0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative Element of ExtREAL
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((E0),E1) is ext-real Element of ExtREAL
dom (L | E) is Element of bool X
(dom L) /\ E is Element of bool X
(L | E) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((L | E) | E)) is ext-real Element of ExtREAL
max+ ((L | E) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((L | E) | E))) is ext-real Element of ExtREAL
max- ((L | E) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((L | E) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((L | E) | E)))) - (integral+ (M,(max- ((L | E) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((L | E) | E))))) is ext-real set
K110((integral+ (M,(max+ ((L | E) | E)))),K111((integral+ (M,(max- ((L | E) | E)))))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom L is Element of bool X
L | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(L | E)) is ext-real Element of ExtREAL
max+ (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (L | E))) is ext-real Element of ExtREAL
max- (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (L | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (L | E)))) - (integral+ (M,(max- (L | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (L | E))))) is ext-real set
K110((integral+ (M,(max+ (L | E)))),K111((integral+ (M,(max- (L | E)))))) is ext-real set
E0 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . E1) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . E1) | E) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . E1) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . H) is Element of bool X
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . H) | E) is Element of bool X
F . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . E1) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . E1) | E) is Element of bool X
(X,E0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(H) is ext-real Element of ExtREAL
(H) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (H) is ext-real Element of ExtREAL
dom (L | E) is Element of bool X
(dom L) /\ E is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(H,I) is ext-real Element of ExtREAL
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . I) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . I) | E)) is ext-real Element of ExtREAL
max+ ((F . I) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . I) | E))) is ext-real Element of ExtREAL
max- ((F . I) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . I) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . I) | E)))) - (integral+ (M,(max- ((F . I) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . I) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . I) | E)))),K111((integral+ (M,(max- ((F . I) | E)))))) is ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(H,H) is ext-real Element of ExtREAL
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(E0 . H)) is ext-real Element of ExtREAL
max+ (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (E0 . H))) is ext-real Element of ExtREAL
max- (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (E0 . H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (E0 . H)))) - (integral+ (M,(max- (E0 . H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (E0 . H))))) is ext-real set
K110((integral+ (M,(max+ (E0 . H)))),K111((integral+ (M,(max- (E0 . H)))))) is ext-real set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . I) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,E0) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,E0) . I) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . 0) is Element of bool X
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . 0) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . 0) | E) is Element of bool X
(X,E0) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . 0) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,E0) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,((X,E0) . I)) is ext-real Element of ExtREAL
((H),I) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((H),H) is ext-real Element of ExtREAL
(X,E0) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,((X,E0) . H)) is ext-real Element of ExtREAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((H),(H + 1)) is ext-real Element of ExtREAL
(X,E0) . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,((X,E0) . (H + 1))) is ext-real Element of ExtREAL
(H,(H + 1)) is ext-real Element of ExtREAL
((H),H) + (H,(H + 1)) is ext-real Element of ExtREAL
F . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . (H + 1)) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . (H + 1)) | E)) is ext-real Element of ExtREAL
max+ ((F . (H + 1)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . (H + 1)) | E))) is ext-real Element of ExtREAL
max- ((F . (H + 1)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . (H + 1)) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . (H + 1)) | E)))) - (integral+ (M,(max- ((F . (H + 1)) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . (H + 1)) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . (H + 1)) | E)))),K111((integral+ (M,(max- ((F . (H + 1)) | E)))))) is ext-real set
(integral' (M,((X,E0) . H))) + (Integral (M,((F . (H + 1)) | E))) is ext-real Element of ExtREAL
dom ((F . (H + 1)) | E) is Element of bool X
E0 . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,E0) . H) + (E0 . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,E0) . H) + ((F . (H + 1)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,E0) . H) is Element of bool X
(dom ((X,E0) . H)) /\ (dom ((F . (H + 1)) | E)) is Element of bool X
dom (((X,E0) . H) + ((F . (H + 1)) | E)) is Element of bool X
integral' (M,(((X,E0) . H) + ((F . (H + 1)) | E))) is ext-real Element of ExtREAL
((X,E0) . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(((X,E0) . H) | E)) is ext-real Element of ExtREAL
((F . (H + 1)) | E) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(((F . (H + 1)) | E) | E)) is ext-real Element of ExtREAL
(integral' (M,(((X,E0) . H) | E))) + (integral' (M,(((F . (H + 1)) | E) | E))) is ext-real Element of ExtREAL
((H),0) is ext-real Element of ExtREAL
(H,0) is ext-real Element of ExtREAL
Integral (M,((X,E0) . 0)) is ext-real Element of ExtREAL
max+ ((X,E0) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,E0) . 0))) is ext-real Element of ExtREAL
max- ((X,E0) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,E0) . 0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,E0) . 0)))) - (integral+ (M,(max- ((X,E0) . 0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,E0) . 0))))) is ext-real set
K110((integral+ (M,(max+ ((X,E0) . 0)))),K111((integral+ (M,(max- ((X,E0) . 0)))))) is ext-real set
integral' (M,((X,E0) . 0)) is ext-real Element of ExtREAL
dom (E0 . 0) is Element of bool X
I is Element of X
E0 # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((L | E),I) is ext-real Element of ExtREAL
((E0 # I)) is ext-real Element of ExtREAL
((E0 # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((E0 # I)) is ext-real Element of ExtREAL
(L,I) is ext-real Element of ExtREAL
F # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is set
((F # I),H) is ext-real Element of ExtREAL
((E0 # I),H) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . G),I) is ext-real Element of ExtREAL
(F . G) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . G) | E) is Element of bool X
E0 . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E0 . G),I) is ext-real Element of ExtREAL
((F # I)) is ext-real Element of ExtREAL
((F # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((F # I)) is ext-real Element of ExtREAL
integral+ (M,(L | E)) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim H is ext-real Element of ExtREAL
G is Element of X
I # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,E0) # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (I # G) is ext-real Element of ExtREAL
lim ((X,E0) # G) is ext-real Element of ExtREAL
E0 # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((E0 # G)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((L | E),G) is ext-real Element of ExtREAL
((E0 # G)) is ext-real Element of ExtREAL
lim ((E0 # G)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is Element of X
(X,E0) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,E0) . G),G) is ext-real Element of ExtREAL
(X,E0) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,E0) . G),G) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(H,G) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . G) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . G) | E)) is ext-real Element of ExtREAL
max+ ((F . G) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . G) | E))) is ext-real Element of ExtREAL
max- ((F . G) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . G) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . G) | E)))) - (integral+ (M,(max- ((F . G) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . G) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . G) | E)))),K111((integral+ (M,(max- ((F . G) | E)))))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom L is Element of bool X
L | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(L | E)) is ext-real Element of ExtREAL
max+ (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (L | E))) is ext-real Element of ExtREAL
max- (L | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (L | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (L | E)))) - (integral+ (M,(max- (L | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (L | E))))) is ext-real set
K110((integral+ (M,(max+ (L | E)))),K111((integral+ (M,(max- (L | E)))))) is ext-real set
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom F is Element of bool X
F | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F | E)) is ext-real Element of ExtREAL
max+ (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F | E))) is ext-real Element of ExtREAL
max- (F | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F | E)))) - (integral+ (M,(max- (F | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F | E))))) is ext-real set
K110((integral+ (M,(max+ (F | E)))),K111((integral+ (M,(max- (F | E)))))) is ext-real set
dom (F | E) is Element of bool X
(dom F) /\ E is Element of bool X
E0 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 . (E1 + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . (E1 + 1)) - (E0 . E1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E1 is Relation-like Function-like set
dom E1 is set
E1 . 0 is set
H is set
rng E1 is set
I is set
E1 . I is set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 . H is set
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 . (H + 1) is set
E0 . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . (H + 1)) - (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 . I is set
H is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . I) - (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
- (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . I) + (- (E0 . H)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . I) is Element of bool X
(- 1) (#) (E0 . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((E0 . I) - (E0 . H)) is Element of bool X
dom (E0 . H) is Element of bool X
(dom (E0 . I)) /\ (dom (E0 . H)) is Element of bool X
G is set
((E0 . H),G) is ext-real Element of ExtREAL
((E0 . I),G) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is Element of X
E0 # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((H # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),I) is ext-real Element of ExtREAL
((E0 # I),I) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),G) is ext-real Element of ExtREAL
((E0 # I),G) is ext-real Element of ExtREAL
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((H # I)),(G + 1)) is ext-real Element of ExtREAL
((E0 # I),(G + 1)) is ext-real Element of ExtREAL
E0 . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E0 . G),I) is ext-real Element of ExtREAL
E0 . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . (G + 1)) is Element of bool X
((E0 . (G + 1)),I) is ext-real Element of ExtREAL
dom (E0 . G) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E0 . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E0 . (G + 1)),I) is ext-real Element of ExtREAL
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . (G + 1)) - (E0 . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E0 . G),I) is ext-real Element of ExtREAL
((E0 . (G + 1)),I) - ((E0 . G),I) is ext-real Element of ExtREAL
K111(((E0 . G),I)) is ext-real set
K110(((E0 . (G + 1)),I),K111(((E0 . G),I))) is ext-real set
G is V11() real ext-real Element of REAL
G is V11() real ext-real Element of REAL
G - G is V11() real ext-real Element of REAL
(((E0 . (G + 1)),I) - ((E0 . G),I)) + ((E0 . G),I) is ext-real Element of ExtREAL
(G - G) + G is V11() real ext-real Element of REAL
(((H # I)),(G + 1)) is ext-real Element of ExtREAL
(((H # I)),G) is ext-real Element of ExtREAL
((H # I),(G + 1)) is ext-real Element of ExtREAL
(((H # I)),G) + ((H # I),(G + 1)) is ext-real Element of ExtREAL
((H . (G + 1)),I) is ext-real Element of ExtREAL
((E0 . G),I) + ((H . (G + 1)),I) is ext-real Element of ExtREAL
dom ((E0 . (G + 1)) - (E0 . G)) is Element of bool X
dom (E0 . (G + 1)) is Element of bool X
dom (E0 . G) is Element of bool X
(dom (E0 . (G + 1))) /\ (dom (E0 . G)) is Element of bool X
(((H # I)),0) is ext-real Element of ExtREAL
((H # I),0) is ext-real Element of ExtREAL
H . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((H . 0),I) is ext-real Element of ExtREAL
((E0 # I),0) is ext-real Element of ExtREAL
lim (E0 # I) is ext-real Element of ExtREAL
((F | E),I) is ext-real Element of ExtREAL
I is V11() real ext-real set
R_EAL I is ext-real Element of ExtREAL
H is V11() real ext-real set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),G) is ext-real Element of ExtREAL
(((H # I)),G) - (R_EAL I) is ext-real Element of ExtREAL
K111((R_EAL I)) is ext-real set
K110((((H # I)),G),K111((R_EAL I))) is ext-real set
|.((((H # I)),G) - (R_EAL I)).| is ext-real Element of ExtREAL
((E0 # I),G) is ext-real Element of ExtREAL
((E0 # I),G) - (lim (E0 # I)) is ext-real Element of ExtREAL
K111((lim (E0 # I))) is ext-real set
K110(((E0 # I),G),K111((lim (E0 # I)))) is ext-real set
|.(((E0 # I),G) - (lim (E0 # I))).| is ext-real Element of ExtREAL
H is V11() real ext-real set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),G) is ext-real Element of ExtREAL
(((H # I)),G) - (lim (E0 # I)) is ext-real Element of ExtREAL
K111((lim (E0 # I))) is ext-real set
K110((((H # I)),G),K111((lim (E0 # I)))) is ext-real set
|.((((H # I)),G) - (lim (E0 # I))).| is ext-real Element of ExtREAL
((E0 # I),G) is ext-real Element of ExtREAL
((E0 # I),G) - (lim (E0 # I)) is ext-real Element of ExtREAL
K110(((E0 # I),G),K111((lim (E0 # I)))) is ext-real set
|.(((E0 # I),G) - (lim (E0 # I))).| is ext-real Element of ExtREAL
lim ((H # I)) is ext-real Element of ExtREAL
((H # I)) is ext-real Element of ExtREAL
(F,I) is ext-real Element of ExtREAL
I is V11() real ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),G) is ext-real Element of ExtREAL
((E0 # I),G) is ext-real Element of ExtREAL
lim ((H # I)) is ext-real Element of ExtREAL
((H # I)) is ext-real Element of ExtREAL
(F,I) is ext-real Element of ExtREAL
I is V11() real ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((H # I)),G) is ext-real Element of ExtREAL
((E0 # I),G) is ext-real Element of ExtREAL
lim ((H # I)) is ext-real Element of ExtREAL
((H # I)) is ext-real Element of ExtREAL
(F,I) is ext-real Element of ExtREAL
I is Element of X
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(F,I) is ext-real Element of ExtREAL
((H # I)) is ext-real Element of ExtREAL
((H # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((H # I)) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
<:{},X,ExtREAL:> is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
S is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . M is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . M) is Element of bool X
bool X is non empty set
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (S . E) is Element of bool X
M is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L is set
M . E is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . E) is Element of bool X
M . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (M . F) is Element of bool X
(dom (M . E)) /\ (dom (M . F)) is Element of bool X
dom {} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() set
(dom {}) /\ (dom (M . F)) is Relation-like V73() V74() V75() V76() V77() V78() Element of bool X
((M . E),L) is ext-real Element of ExtREAL
((M . F),L) is ext-real Element of ExtREAL
X is non empty set
S is non empty set
[:X,S:] is non empty Relation-like set
M is non empty set
PFuncs (M,ExtREAL) is non empty functional set
[:[:X,S:],(PFuncs (M,ExtREAL)):] is non empty Relation-like set
bool [:[:X,S:],(PFuncs (M,ExtREAL)):] is non empty set
E is non empty Relation-like [:X,S:] -defined PFuncs (M,ExtREAL) -valued Function-like total quasi_total Element of bool [:[:X,S:],(PFuncs (M,ExtREAL)):]
F is Element of X
L is Element of S
E . (F,L) is set
[F,L] is set
{F,L} is non empty set
{F} is non empty set
{{F,L},{F}} is non empty set
E . [F,L] is Relation-like Function-like set
[:M,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:M,ExtREAL:] is non empty set
E . (F,L) is Relation-like Function-like Element of PFuncs (M,ExtREAL)
X is non empty set
S is non empty set
[:X,S:] is non empty Relation-like set
M is non empty set
[:[:X,S:],M:] is non empty Relation-like set
bool [:[:X,S:],M:] is non empty set
[:S,M:] is non empty Relation-like set
bool [:S,M:] is non empty set
E is non empty Relation-like [:X,S:] -defined M -valued Function-like total quasi_total Element of bool [:[:X,S:],M:]
F is Element of X
L is Relation-like Function-like set
dom L is set
E0 is set
[F,E0] is set
{F,E0} is non empty set
{F} is non empty set
{{F,E0},{F}} is non empty set
L . E0 is set
E . (F,E0) is set
E . [F,E0] is set
E0 is non empty Relation-like S -defined M -valued Function-like total quasi_total Element of bool [:S,M:]
E1 is Element of S
E0 . E1 is Element of M
E . (F,E1) is Element of M
[F,E1] is set
{F,E1} is non empty set
{{F,E1},{F}} is non empty set
E . [F,E1] is set
L is non empty Relation-like S -defined M -valued Function-like total quasi_total Element of bool [:S,M:]
E0 is non empty Relation-like S -defined M -valued Function-like total quasi_total Element of bool [:S,M:]
E1 is set
H is Element of S
L . H is Element of M
E . (F,H) is Element of M
[F,H] is set
{F,H} is non empty set
{F} is non empty set
{{F,H},{F}} is non empty set
E . [F,H] is set
L . E1 is set
E0 . E1 is set
X is non empty set
S is non empty set
[:X,S:] is non empty Relation-like set
M is non empty set
[:[:X,S:],M:] is non empty Relation-like set
bool [:[:X,S:],M:] is non empty set
[:X,M:] is non empty Relation-like set
bool [:X,M:] is non empty set
E is non empty Relation-like [:X,S:] -defined M -valued Function-like total quasi_total Element of bool [:[:X,S:],M:]
F is Element of S
L is Relation-like Function-like set
dom L is set
E0 is set
[E0,F] is set
{E0,F} is non empty set
{E0} is non empty set
{{E0,F},{E0}} is non empty set
L . E0 is set
E . (E0,F) is set
E . [E0,F] is set
E0 is non empty Relation-like X -defined M -valued Function-like total quasi_total Element of bool [:X,M:]
E1 is Element of X
E0 . E1 is Element of M
E . (E1,F) is Element of M
[E1,F] is set
{E1,F} is non empty set
{E1} is non empty set
{{E1,F},{E1}} is non empty set
E . [E1,F] is set
L is non empty Relation-like X -defined M -valued Function-like total quasi_total Element of bool [:X,M:]
E0 is non empty Relation-like X -defined M -valued Function-like total quasi_total Element of bool [:X,M:]
E1 is set
H is Element of X
L . H is Element of M
E . (H,F) is Element of M
[H,F] is set
{H,F} is non empty set
{H} is non empty set
{{H,F},{H}} is non empty set
E . [H,F] is set
L . E1 is set
E0 . E1 is set
X is set
S is set
PFuncs (X,S) is non empty functional set
[:[:NAT,NAT:],(PFuncs (X,S)):] is non empty Relation-like set
bool [:[:NAT,NAT:],(PFuncs (X,S)):] is non empty set
[:NAT,(PFuncs (X,S)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,S)):] is non empty set
M is non empty Relation-like [:NAT,NAT:] -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:[:NAT,NAT:],(PFuncs (X,S)):]
E is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F is Relation-like Function-like set
dom F is set
E0 is set
rng F is set
E1 is set
F . E1 is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (L,H) is Relation-like Function-like Element of PFuncs (X,S)
[L,H] is set
{L,H} is non empty V73() V74() V75() V76() V77() V78() set
{L} is non empty V73() V74() V75() V76() V77() V78() set
{{L,H},{L}} is non empty set
M . [L,H] is Relation-like Function-like set
L is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
[:X,S:] is Relation-like set
bool [:X,S:] is non empty set
M . (E,E0) is set
[E,E0] is set
{E,E0} is non empty V73() V74() V75() V76() V77() V78() set
{E} is non empty V73() V74() V75() V76() V77() V78() set
{{E,E0},{E}} is non empty set
M . [E,E0] is Relation-like Function-like set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (E1,H) is Relation-like Function-like Element of PFuncs (X,S)
[E1,H] is set
{E1,H} is non empty V73() V74() V75() V76() V77() V78() set
{E1} is non empty V73() V74() V75() V76() V77() V78() set
{{E1,H},{E1}} is non empty set
M . [E1,H] is Relation-like Function-like set
F is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
L is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
[:X,S:] is Relation-like set
bool [:X,S:] is non empty set
L . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . (E,E1) is set
[E,E1] is set
{E,E1} is non empty V73() V74() V75() V76() V77() V78() set
{E} is non empty V73() V74() V75() V76() V77() V78() set
{{E,E1},{E}} is non empty set
M . [E,E1] is Relation-like Function-like set
F is Relation-like Function-like set
dom F is set
E0 is set
rng F is set
E1 is set
F . E1 is set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (H,L) is Relation-like Function-like Element of PFuncs (X,S)
[H,L] is set
{H,L} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,L},{H}} is non empty set
M . [H,L] is Relation-like Function-like set
L is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
[:X,S:] is Relation-like set
bool [:X,S:] is non empty set
M . (E0,E) is set
[E0,E] is set
{E0,E} is non empty V73() V74() V75() V76() V77() V78() set
{E0} is non empty V73() V74() V75() V76() V77() V78() set
{{E0,E},{E0}} is non empty set
M . [E0,E] is Relation-like Function-like set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
M . (H,E1) is Relation-like Function-like Element of PFuncs (X,S)
[H,E1] is set
{H,E1} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,E1},{H}} is non empty set
M . [H,E1] is Relation-like Function-like set
F is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
L is non empty Relation-like NAT -defined PFuncs (X,S) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,S)):]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
[:X,S:] is Relation-like set
bool [:X,S:] is non empty set
L . E0 is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
M . (E1,E) is set
[E1,E] is set
{E1,E} is non empty V73() V74() V75() V76() V77() V78() set
{E1} is non empty V73() V74() V75() V76() V77() V78() set
{{E1,E},{E1}} is non empty set
M . [E1,E] is Relation-like Function-like set
X is non empty set
PFuncs (X,ExtREAL) is non empty functional set
Funcs (NAT,(PFuncs (X,ExtREAL))) is non empty functional FUNCTION_DOMAIN of NAT , PFuncs (X,ExtREAL)
[:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty Relation-like set
bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty set
S is non empty Relation-like NAT -defined Funcs (NAT,(PFuncs (X,ExtREAL))) -valued Function-like total quasi_total Element of bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):]
M is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
S . M is Relation-like Function-like set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S . M is Relation-like Function-like Element of Funcs (NAT,(PFuncs (X,ExtREAL)))
E is Relation-like Function-like set
dom E is set
rng E is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
Funcs (NAT,(PFuncs (X,ExtREAL))) is non empty functional FUNCTION_DOMAIN of NAT , PFuncs (X,ExtREAL)
[:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty Relation-like set
bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
M is Element of S
E is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (E . 0) is Element of bool X
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E . F) is Element of bool X
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E . F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E . F) is Element of bool X
L is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E0 is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of Funcs (NAT,(PFuncs (X,ExtREAL)))
E1 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E1 . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is Element of X
E1 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E1 . H),G) is ext-real Element of ExtREAL
E1 . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E1 . G),G) is ext-real Element of ExtREAL
G is Element of X
E1 # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (E1 # G) is ext-real Element of ExtREAL
((E . F),G) is ext-real Element of ExtREAL
F is non empty Relation-like NAT -defined Funcs (NAT,(PFuncs (X,ExtREAL))) -valued Function-like total quasi_total Element of bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):]
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F,L) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E . L) is Element of bool X
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,F,E0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E . E0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E . E0) is Element of bool X
E1 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E1 . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is Element of X
E1 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E1 . H),G) is ext-real Element of ExtREAL
E1 . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((E1 . G),G) is ext-real Element of ExtREAL
G is Element of X
E1 # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (E1 # G) is ext-real Element of ExtREAL
((E . E0),G) is ext-real Element of ExtREAL
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F,L) . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F,L) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,F,L) . H) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F,L) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is Element of X
(X,F,L) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,F,L) . I),G) is ext-real Element of ExtREAL
(X,F,L) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,F,L) . H),G) is ext-real Element of ExtREAL
G is Element of X
(X,F,L) # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,F,L) # G) is ext-real Element of ExtREAL
((E . L),G) is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (F . 0) is Element of bool X
(X,F) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E0 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(E1) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(E1,I) is ext-real Element of ExtREAL
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . I)) is ext-real Element of ExtREAL
max+ (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . I))) is ext-real Element of ExtREAL
max- (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . I)))) - (integral+ (M,(max- (F . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . I))))) is ext-real set
K110((integral+ (M,(max+ (F . I)))),K111((integral+ (M,(max- (F . I)))))) is ext-real set
(X,F) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . I)) is ext-real Element of ExtREAL
max+ ((X,F) . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . I))) is ext-real Element of ExtREAL
max- ((X,F) . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . I)))) - (integral+ (M,(max- ((X,F) . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . I))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . I)))),K111((integral+ (M,(max- ((X,F) . I)))))) is ext-real set
((E1),I) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . I)) is ext-real Element of ExtREAL
max+ (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . I))) is ext-real Element of ExtREAL
max- (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . I)))) - (integral+ (M,(max- (F . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . I))))) is ext-real set
K110((integral+ (M,(max+ (F . I)))),K111((integral+ (M,(max- (F . I)))))) is ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . H)) is ext-real Element of ExtREAL
max+ ((X,F) . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . H))) is ext-real Element of ExtREAL
max- ((X,F) . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . H)))) - (integral+ (M,(max- ((X,F) . H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . H))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . H)))),K111((integral+ (M,(max- ((X,F) . H)))))) is ext-real set
((E1),H) is ext-real Element of ExtREAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,F) . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . (H + 1))) is ext-real Element of ExtREAL
max+ ((X,F) . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . (H + 1)))) is ext-real Element of ExtREAL
max- ((X,F) . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . (H + 1)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . (H + 1))))) - (integral+ (M,(max- ((X,F) . (H + 1))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . (H + 1)))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . (H + 1))))),K111((integral+ (M,(max- ((X,F) . (H + 1))))))) is ext-real set
((E1),(H + 1)) is ext-real Element of ExtREAL
F . (H + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . (H + 1)) is Element of bool X
dom ((X,F) . H) is Element of bool X
((X,F) . H) + (F . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (((X,F) . H) + (F . (H + 1))) is Element of bool X
integral+ (M,(((X,F) . H) + (F . (H + 1)))) is ext-real Element of ExtREAL
G is Element of S
((X,F) . H) | G is Relation-like X -defined G -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(((X,F) . H) | G)) is ext-real Element of ExtREAL
(F . (H + 1)) | G is Relation-like X -defined G -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,((F . (H + 1)) | G)) is ext-real Element of ExtREAL
(integral+ (M,(((X,F) . H) | G))) + (integral+ (M,((F . (H + 1)) | G))) is ext-real Element of ExtREAL
(dom ((X,F) . H)) /\ (dom (F . (H + 1))) is Element of bool X
dom ((X,F) . (H + 1)) is Element of bool X
integral+ (M,((X,F) . (H + 1))) is ext-real Element of ExtREAL
(Integral (M,((X,F) . H))) + (integral+ (M,((F . (H + 1)) | G))) is ext-real Element of ExtREAL
Integral (M,(F . (H + 1))) is ext-real Element of ExtREAL
max+ (F . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . (H + 1)))) is ext-real Element of ExtREAL
max- (F . (H + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . (H + 1)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . (H + 1))))) - (integral+ (M,(max- (F . (H + 1))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . (H + 1)))))) is ext-real set
K110((integral+ (M,(max+ (F . (H + 1))))),K111((integral+ (M,(max- (F . (H + 1))))))) is ext-real set
(Integral (M,((X,F) . H))) + (Integral (M,(F . (H + 1)))) is ext-real Element of ExtREAL
(E1,(H + 1)) is ext-real Element of ExtREAL
((E1),H) + (E1,(H + 1)) is ext-real Element of ExtREAL
(X,F) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,F) . 0)) is ext-real Element of ExtREAL
max+ ((X,F) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . 0))) is ext-real Element of ExtREAL
max- ((X,F) . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . 0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . 0)))) - (integral+ (M,(max- ((X,F) . 0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . 0))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . 0)))),K111((integral+ (M,(max- ((X,F) . 0)))))) is ext-real set
Integral (M,(F . 0)) is ext-real Element of ExtREAL
max+ (F . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . 0))) is ext-real Element of ExtREAL
max- (F . 0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . 0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . 0)))) - (integral+ (M,(max- (F . 0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . 0))))) is ext-real set
K110((integral+ (M,(max+ (F . 0)))),K111((integral+ (M,(max- (F . 0)))))) is ext-real set
(E1,0) is ext-real Element of ExtREAL
((E1),0) is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (F . 0) is Element of bool X
(X,F) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (X,F) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(lim (X,F)) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((lim (X,F)) | E)) is ext-real Element of ExtREAL
max+ ((lim (X,F)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((lim (X,F)) | E))) is ext-real Element of ExtREAL
max- ((lim (X,F)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((lim (X,F)) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((lim (X,F)) | E)))) - (integral+ (M,(max- ((lim (X,F)) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((lim (X,F)) | E))))) is ext-real set
K110((integral+ (M,(max+ ((lim (X,F)) | E)))),K111((integral+ (M,(max- ((lim (X,F)) | E)))))) is ext-real set
Funcs (NAT,(PFuncs (X,ExtREAL))) is non empty functional FUNCTION_DOMAIN of NAT , PFuncs (X,ExtREAL)
[:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty Relation-like set
bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):] is non empty set
L is non empty Relation-like NAT -defined Funcs (NAT,(PFuncs (X,ExtREAL))) -valued Function-like total quasi_total Element of bool [:NAT,(Funcs (NAT,(PFuncs (X,ExtREAL)))):]
E0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,L,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,L,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is Relation-like Function-like Element of PFuncs (X,ExtREAL)
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,L,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,H) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:[:NAT,NAT:],(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:[:NAT,NAT:],(PFuncs (X,ExtREAL)):] is non empty set
E0 is non empty Relation-like [:NAT,NAT:] -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:[:NAT,NAT:],(PFuncs (X,ExtREAL)):]
E1 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . (H,I) is set
[H,I] is set
{H,I} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,I},{H}} is non empty set
E0 . [H,I] is Relation-like Function-like set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,I,H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[I,H] is set
{I,H} is non empty V73() V74() V75() V76() V77() V78() set
{I} is non empty V73() V74() V75() V76() V77() V78() set
{{I,H},{I}} is non empty set
E0 . [I,H] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . (I,H) is set
[I,H] is set
{I,H} is non empty V73() V74() V75() V76() V77() V78() set
{I} is non empty V73() V74() V75() V76() V77() V78() set
{{I,H},{I}} is non empty set
E0 . [I,H] is Relation-like Function-like set
(NAT,NAT,X,E0,H,I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[H,I] is set
{H,I} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,I},{H}} is non empty set
E0 . [H,I] is Relation-like Function-like set
(X,L,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,H) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . I) is Element of bool X
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . I) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,H) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E1 . H) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E1 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E1 . I) is Element of bool X
(X,ExtREAL,E0,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,H)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,H)) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,(X,ExtREAL,E0,H)) . H) is Element of bool X
(X,ExtREAL,E0,H) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,H) . 0) is Element of bool X
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,(X,ExtREAL,E0,I)) . I) is Element of bool X
H is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (lim H) is Element of bool X
H . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . 0) is Element of bool X
(X,ExtREAL,E0,0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,0)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,0)) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,(X,ExtREAL,E0,0)) . 0) is Element of bool X
(X,ExtREAL,E0,0) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,0) . 0) is Element of bool X
(NAT,NAT,X,E0,0,0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[0,0] is set
{0,0} is non empty functional V73() V74() V75() V76() V77() V78() set
{0} is non empty functional V73() V74() V75() V76() V77() V78() set
{{0,0},{0}} is non empty set
E0 . [0,0] is Relation-like Function-like set
dom (NAT,NAT,X,E0,0,0) is Element of bool X
(X,L,0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,0) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,L,0) . 0) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(dom (F . 0)) /\ (dom ((X,ExtREAL,E0,I) . 0)) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is Element of X
(((X,ExtREAL,E0,I) . I),H) is ext-real Element of ExtREAL
((F . I),H) is ext-real Element of ExtREAL
dom (F . I) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,L,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,G) # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is ext-real set
dom ((X,L,G) # H) is non empty set
G is ext-real set
((X,L,G) # H) . G is ext-real set
((X,L,G) # H) . G is ext-real set
dom ((X,L,G) # H) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,L,G) # H),n) is ext-real Element of ExtREAL
(X,L,G) . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,G) . n),H) is ext-real Element of ExtREAL
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,L,G) # H),J) is ext-real Element of ExtREAL
(X,L,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,G) . J),H) is ext-real Element of ExtREAL
lim ((X,L,G) # H) is ext-real Element of ExtREAL
sup ((X,L,G) # H) is ext-real Element of ExtREAL
rng ((X,L,G) # H) is non empty V55() V74() Element of bool ExtREAL
sup (rng ((X,L,G) # H)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,L,G) # H),G) is ext-real Element of ExtREAL
(((X,L,G) # H),I) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . G),H) is ext-real Element of ExtREAL
(NAT,NAT,X,E0,G,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[G,G] is set
{G,G} is non empty V73() V74() V75() V76() V77() V78() set
{G} is non empty V73() V74() V75() V76() V77() V78() set
{{G,G},{G}} is non empty set
E0 . [G,G] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is Element of X
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,F) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((H # I),I) is ext-real Element of ExtREAL
(((X,F) # I),I) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(dom (F . 0)) /\ (dom ((X,ExtREAL,E0,I) . 0)) is Element of bool X
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((H . I),I) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,I)) . I),I) is ext-real Element of ExtREAL
G is Element of X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) . H),G) is ext-real Element of ExtREAL
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . H),G) is ext-real Element of ExtREAL
(X,F) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,F) . I),I) is ext-real Element of ExtREAL
dom (lim (X,F)) is Element of bool X
(X,F) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,F) . 0) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,H,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[H,G] is set
{H,G} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,G},{H}} is non empty set
E0 . [H,G] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(NAT,NAT,X,E0,G,H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[G,H] is set
{G,H} is non empty V73() V74() V75() V76() V77() V78() set
{G} is non empty V73() V74() V75() V76() V77() V78() set
{{G,H},{G}} is non empty set
E0 . [G,H] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is Element of X
(X,ExtREAL,E0,I) # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((F . I),H) is ext-real Element of ExtREAL
lim ((X,ExtREAL,E0,I) # H) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,ExtREAL,E0,I) # H),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) . G),H) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . I) is Element of bool X
G is ext-real set
dom ((X,ExtREAL,E0,I) # H) is non empty set
G is ext-real set
((X,ExtREAL,E0,I) # H) . G is ext-real set
((X,ExtREAL,E0,I) # H) . G is ext-real set
dom ((X,ExtREAL,E0,I) # H) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,ExtREAL,E0,I) # H),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) . G),H) is ext-real Element of ExtREAL
(NAT,NAT,X,E0,I,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[I,G] is set
{I,G} is non empty V73() V74() V75() V76() V77() V78() set
{I} is non empty V73() V74() V75() V76() V77() V78() set
{{I,G},{I}} is non empty set
E0 . [I,G] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,ExtREAL,E0,I) # H),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) . G),H) is ext-real Element of ExtREAL
(NAT,NAT,X,E0,I,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[I,G] is set
{I,G} is non empty V73() V74() V75() V76() V77() V78() set
{{I,G},{I}} is non empty set
E0 . [I,G] is Relation-like Function-like set
(((X,L,I) . G),H) is ext-real Element of ExtREAL
(X,L,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,I) . G),H) is ext-real Element of ExtREAL
G is V11() real ext-real set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
R_EAL G is ext-real Element of ExtREAL
G is V11() real ext-real set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,L,I) # H),G) is ext-real Element of ExtREAL
(((X,L,I) # H),G) - (R_EAL G) is ext-real Element of ExtREAL
K111((R_EAL G)) is ext-real set
K110((((X,L,I) # H),G),K111((R_EAL G))) is ext-real set
|.((((X,L,I) # H),G) - (R_EAL G)).| is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . (I,G) is set
[I,G] is set
{I,G} is non empty V73() V74() V75() V76() V77() V78() set
{I} is non empty V73() V74() V75() V76() V77() V78() set
{{I,G},{I}} is non empty set
E0 . [I,G] is Relation-like Function-like set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,L,I) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) # H),G) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,I) . G),H) is ext-real Element of ExtREAL
lim ((X,L,I) # H) is ext-real Element of ExtREAL
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is V11() real ext-real set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,L,I) # H),G) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,I) # H),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,I,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[I,G] is set
{I,G} is non empty V73() V74() V75() V76() V77() V78() set
{I} is non empty V73() V74() V75() V76() V77() V78() set
{{I,G},{I}} is non empty set
E0 . [I,G] is Relation-like Function-like set
(X,L,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,I) . G),H) is ext-real Element of ExtREAL
lim ((X,L,I) # H) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . I) is Element of bool X
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,(X,ExtREAL,E0,I)) . I) is Element of bool X
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,I) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
(((X,ExtREAL,E0,I) . H),G) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,I) . H),G) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,G,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[G,G] is set
{G,G} is non empty V73() V74() V75() V76() V77() V78() set
{G} is non empty V73() V74() V75() V76() V77() V78() set
{{G,G},{G}} is non empty set
E0 . [G,G] is Relation-like Function-like set
((NAT,NAT,X,E0,G,G),G) is ext-real Element of ExtREAL
(X,L,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,H) . I),G) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,G,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[G,G] is set
{G,G} is non empty V73() V74() V75() V76() V77() V78() set
{{G,G},{G}} is non empty set
E0 . [G,G] is Relation-like Function-like set
((NAT,NAT,X,E0,G,G),G) is ext-real Element of ExtREAL
(X,L,H) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,H) . I),G) is ext-real Element of ExtREAL
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(dom ((X,ExtREAL,E0,I) . 0)) /\ (dom ((X,ExtREAL,E0,I) . 0)) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,I) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
(((X,ExtREAL,E0,I) . H),G) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,I) . H),G) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,(X,ExtREAL,E0,I)) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,I)) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
(((X,(X,ExtREAL,E0,I)) . H),G) is ext-real Element of ExtREAL
(((X,(X,ExtREAL,E0,I)) . H),G) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
((H . I),G) is ext-real Element of ExtREAL
((H . I),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,I) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,I) . 0) is Element of bool X
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,(X,ExtREAL,E0,I)) # G),H) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,(X,ExtREAL,E0,I)) # G),G) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,I)) . I),G) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,I)) . I),G) is ext-real Element of ExtREAL
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,I)) . I),G) is ext-real Element of ExtREAL
I is Element of X
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((H # I),H) is ext-real Element of ExtREAL
((H # I),I) is ext-real Element of ExtREAL
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((H . H),I) is ext-real Element of ExtREAL
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((H . I),I) is ext-real Element of ExtREAL
I is Element of X
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (H # I) is ext-real Element of ExtREAL
(X,F) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,F) # I) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,(X,ExtREAL,E0,G)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Relation-like Function-like Element of PFuncs (X,ExtREAL)
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is non empty Relation-like [:NAT,NAT:] -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:[:NAT,NAT:],(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,G) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,H)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,H)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(NAT,NAT,X,I,H,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[H,G] is set
{H,G} is non empty V73() V74() V75() V76() V77() V78() set
{H} is non empty V73() V74() V75() V76() V77() V78() set
{{H,G},{H}} is non empty set
I . [H,G] is Relation-like Function-like set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,H) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((X,ExtREAL,I,H) # I) ^\ H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,ExtREAL,I,H) # I) is ext-real Element of ExtREAL
lim (((X,ExtREAL,I,H) # I) ^\ H) is ext-real Element of ExtREAL
G is ext-real set
dom ((X,ExtREAL,I,H) # I) is non empty set
G is ext-real set
((X,ExtREAL,I,H) # I) . G is ext-real set
((X,ExtREAL,I,H) # I) . G is ext-real set
dom ((X,ExtREAL,I,H) # I) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,G) . 0) is Element of bool X
(((X,ExtREAL,I,H) # I),G) is ext-real Element of ExtREAL
(X,ExtREAL,I,H) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,H) . G),I) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,G)) . H),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,H) # I),G) is ext-real Element of ExtREAL
(X,ExtREAL,I,H) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,H) . G),I) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,G)) . H),I) is ext-real Element of ExtREAL
(X,ExtREAL,E0,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,G) . 0) is Element of bool X
(dom ((X,ExtREAL,E0,G) . 0)) /\ (dom ((X,ExtREAL,E0,G) . 0)) is Element of bool X
n is Element of X
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,G) . J),n) is ext-real Element of ExtREAL
(X,ExtREAL,E0,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,G) . J),n) is ext-real Element of ExtREAL
(X,ExtREAL,I,0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,0) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,ExtREAL,E0,0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,0) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is set
(((X,ExtREAL,I,0) # I),H) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,0) # I),H) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,0) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(NAT,NAT,X,E0,0,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[0,G] is set
{0,G} is non empty V73() V74() V75() V76() V77() V78() set
{{0,G},{0}} is non empty set
E0 . [0,G] is Relation-like Function-like set
(X,ExtREAL,E0,0) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,0) . G),I) is ext-real Element of ExtREAL
lim ((X,ExtREAL,I,0) # I) is ext-real Element of ExtREAL
((F . 0),I) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,F) # I),H) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(H # I) ^\ G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((H # I) ^\ G) is ext-real Element of ExtREAL
(X,ExtREAL,I,H) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,H) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((X,ExtREAL,I,H) # I) ^\ G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,I,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,G) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,ExtREAL,I,G) # I) is ext-real Element of ExtREAL
(((X,F) # I),G) is ext-real Element of ExtREAL
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,(G + 1)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,(G + 1)) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,ExtREAL,I,(G + 1)) # I) is ext-real Element of ExtREAL
(((X,F) # I),(G + 1)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,I,G) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is set
(X,ExtREAL,E0,(G + 1)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,(G + 1)) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
dom ((X,ExtREAL,E0,(G + 1)) # I) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,(G + 1)) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,E0,(G + 1)) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,(G + 1)) # I),G) is ext-real Element of ExtREAL
G is set
dom ((X,ExtREAL,I,G) # I) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,I,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,J) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,J)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,J)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,J) . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,G) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) # I),G) is ext-real Element of ExtREAL
(X,F) . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,F) . (G + 1)) is Element of bool X
(X,F) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,F) . G) + (F . (G + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (((X,F) . G) + (F . (G + 1))) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,ExtREAL,I,(G + 1)) # I),G) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) # I),G) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,(G + 1)) # I),G) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) # I),G) + (((X,ExtREAL,E0,(G + 1)) # I),G) is ext-real Element of ExtREAL
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,J) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,J) . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(NAT,NAT,X,E0,(G + 1),J) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[(G + 1),J] is set
{(G + 1),J} is non empty V73() V74() V75() V76() V77() V78() set
{(G + 1)} is non empty V73() V74() V75() V76() V77() V78() set
{{(G + 1),J},{(G + 1)}} is non empty set
E0 . [(G + 1),J] is Relation-like Function-like set
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,E0,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,G) . 0) is Element of bool X
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,(X,ExtREAL,E0,G)) . (G + 1)) is Element of bool X
(X,ExtREAL,I,(G + 1)) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,I,(G + 1)) . J) is Element of bool X
(X,(X,ExtREAL,E0,J)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,J)) . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,J)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,(X,ExtREAL,E0,J)) . G) + ((X,ExtREAL,E0,J) . (G + 1)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,J)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,I,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,(G + 1)) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((X,ExtREAL,I,G) . J) + ((X,ExtREAL,E0,(G + 1)) . J) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,(G + 1)) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,E0,(G + 1)) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) . J),I) + (((X,ExtREAL,E0,(G + 1)) . J),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,G) # I),G) + (((X,ExtREAL,E0,(G + 1)) . J),I) is ext-real Element of ExtREAL
lim ((X,ExtREAL,E0,(G + 1)) # I) is ext-real Element of ExtREAL
((F . (G + 1)),I) is ext-real Element of ExtREAL
(lim ((X,ExtREAL,I,G) # I)) + (lim ((X,ExtREAL,E0,(G + 1)) # I)) is ext-real Element of ExtREAL
(((X,F) . G),I) is ext-real Element of ExtREAL
(((X,F) . G),I) + ((F . (G + 1)),I) is ext-real Element of ExtREAL
((((X,F) . G) + (F . (G + 1))),I) is ext-real Element of ExtREAL
(((X,F) . (G + 1)),I) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((((X,ExtREAL,I,H) # I) ^\ G),G) is ext-real Element of ExtREAL
(((H # I) ^\ G),G) is ext-real Element of ExtREAL
H + G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,ExtREAL,E0,(H + G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,(H + G)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,(H + G)) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,(H + G)) . 0) is Element of bool X
(X,(X,ExtREAL,E0,(H + G))) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,(H + G))) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(((X,(X,ExtREAL,E0,(H + G))) # I),G) is ext-real Element of ExtREAL
G + G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,(X,ExtREAL,E0,(H + G))) # I),(G + G)) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,(H + G))) . (H + G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,(H + G))) . (H + G)),I) is ext-real Element of ExtREAL
((H # I),(H + G)) is ext-real Element of ExtREAL
H . (H + G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((H . (H + G)),I) is ext-real Element of ExtREAL
(((X,ExtREAL,I,H) # I),(H + G)) is ext-real Element of ExtREAL
(X,ExtREAL,I,H) . (H + G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,ExtREAL,I,H) . (H + G)),I) is ext-real Element of ExtREAL
(X,(X,ExtREAL,E0,(H + G))) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,(X,ExtREAL,E0,(H + G))) . H),I) is ext-real Element of ExtREAL
(((X,F) # I),0) is ext-real Element of ExtREAL
(((X,F) . 0),I) is ext-real Element of ExtREAL
lim (((X,ExtREAL,I,H) # I) ^\ G) is ext-real Element of ExtREAL
lim ((X,ExtREAL,I,H) # I) is ext-real Element of ExtREAL
F # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((F # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((H # I),H) is ext-real Element of ExtREAL
(((X,F) # I),H) is ext-real Element of ExtREAL
I is Element of X
H # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (H # I) is ext-real Element of ExtREAL
((lim (X,F)),I) is ext-real Element of ExtREAL
(X,F) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,F) # I) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,I) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,I) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,I)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,I)) . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
I is set
((lim (X,F)),I) is ext-real Element of ExtREAL
I is Element of X
(X,F) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(((X,F) # I),H) is ext-real Element of ExtREAL
(X,F) . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,F) . H),I) is ext-real Element of ExtREAL
lim ((X,F) # I) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(I) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
dom I is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
I is set
(I,I) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . H) is Element of bool X
Integral (M,(F . H)) is ext-real Element of ExtREAL
max+ (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . H))) is ext-real Element of ExtREAL
max- (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . H)))) - (integral+ (M,(max- (F . H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . H))))) is ext-real set
K110((integral+ (M,(max+ (F . H)))),K111((integral+ (M,(max- (F . H)))))) is ext-real set
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(H,G) is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(H . G)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,(H . G)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(H,G) is ext-real Element of ExtREAL
(H,G) is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . G) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((H . G) - (H . G)) is Element of bool X
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . J) is Element of bool X
dom (H . G) is Element of bool X
dom (H . G) is Element of bool X
(dom (H . G)) /\ (dom (H . G)) is Element of bool X
(dom (lim (X,F))) /\ (dom (H . G)) is Element of bool X
(dom (lim (X,F))) /\ (dom (lim (X,F))) is Element of bool X
(H . G) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((H . G) - (H . G)) is Element of bool X
dom (H . G) is Element of bool X
(H . G) | (dom ((H . G) - (H . G))) is Relation-like X -defined dom ((H . G) - (H . G)) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is set
((H . G),G) is ext-real Element of ExtREAL
((H . G),G) is ext-real Element of ExtREAL
dom (H . G) is Element of bool X
(H . G) | (dom ((H . G) - (H . G))) is Relation-like X -defined dom ((H . G) - (H . G)) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral' (M,((H . G) | (dom ((H . G) - (H . G))))) is ext-real Element of ExtREAL
integral' (M,((H . G) | (dom ((H . G) - (H . G))))) is ext-real Element of ExtREAL
integral' (M,(H . G)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,F) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(lim (X,F))) is ext-real Element of ExtREAL
lim H is ext-real Element of ExtREAL
Integral (M,(lim (X,F))) is ext-real Element of ExtREAL
max+ (lim (X,F)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (lim (X,F)))) is ext-real Element of ExtREAL
max- (lim (X,F)) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (lim (X,F)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (lim (X,F))))) - (integral+ (M,(max- (lim (X,F))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (lim (X,F)))))) is ext-real set
K110((integral+ (M,(max+ (lim (X,F))))),K111((integral+ (M,(max- (lim (X,F))))))) is ext-real set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . G) is Element of bool X
(X,L,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is Element of X
(X,L,G) # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is ext-real set
dom ((X,L,G) # G) is non empty set
G is ext-real set
((X,L,G) # G) . G is ext-real set
((X,L,G) # G) . G is ext-real set
dom ((X,L,G) # G) is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,L,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,G) . J),G) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(X,L,G) . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,G) . n),G) is ext-real Element of ExtREAL
(((X,L,G) # G),J) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,(X,ExtREAL,E0,G)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,(X,ExtREAL,E0,G)) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,F) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
J is Element of X
(((X,(X,ExtREAL,E0,G)) . G),J) is ext-real Element of ExtREAL
((H . G),J) is ext-real Element of ExtREAL
(((X,(X,ExtREAL,E0,G)) . G),J) is ext-real Element of ExtREAL
(((X,F) . G),J) is ext-real Element of ExtREAL
((lim (X,F)),J) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,ExtREAL,E0,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E0,G) . 0) is Element of bool X
(X,(X,ExtREAL,E0,G)) # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,(X,ExtREAL,E0,G)) # J),G) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,(X,ExtREAL,E0,G)) # J),G) is ext-real Element of ExtREAL
(((X,(X,ExtREAL,E0,G)) # J),G) is ext-real Element of ExtREAL
(dom ((X,ExtREAL,E0,G) . 0)) /\ (dom (F . 0)) is Element of bool X
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(X,ExtREAL,E0,G) . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
m is Element of X
(((X,ExtREAL,E0,G) . n),m) is ext-real Element of ExtREAL
((F . n),m) is ext-real Element of ExtREAL
dom (F . n) is Element of bool X
(X,L,n) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,L,n) # m is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,L,n) # m) is ext-real Element of ExtREAL
sup ((X,L,n) # m) is ext-real Element of ExtREAL
rng ((X,L,n) # m) is non empty V55() V74() Element of bool ExtREAL
sup (rng ((X,L,n) # m)) is ext-real Element of ExtREAL
(((X,L,n) # m),G) is ext-real Element of ExtREAL
(((X,L,n) # m),G) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(NAT,NAT,X,E0,n,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[n,G] is set
{n,G} is non empty V73() V74() V75() V76() V77() V78() set
{n} is non empty V73() V74() V75() V76() V77() V78() set
{{n,G},{n}} is non empty set
E0 . [n,G] is Relation-like Function-like set
((NAT,NAT,X,E0,n,G),m) is ext-real Element of ExtREAL
(X,L,n) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((X,L,n) . G),m) is ext-real Element of ExtREAL
(X,F) # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim ((X,F) # J) is ext-real Element of ExtREAL
sup ((X,F) # J) is ext-real Element of ExtREAL
rng ((X,F) # J) is non empty V55() V74() Element of bool ExtREAL
sup (rng ((X,F) # J)) is ext-real Element of ExtREAL
(((X,F) # J),G) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((I),G) is ext-real Element of ExtREAL
(X,F) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,F) . G) is Element of bool X
G is Element of X
(((X,F) . G),G) is ext-real Element of ExtREAL
((lim (X,F)),G) is ext-real Element of ExtREAL
integral+ (M,((X,F) . G)) is ext-real Element of ExtREAL
Integral (M,((X,F) . G)) is ext-real Element of ExtREAL
max+ ((X,F) . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . G))) is ext-real Element of ExtREAL
max- ((X,F) . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . G))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . G)))) - (integral+ (M,(max- ((X,F) . G)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . G))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . G)))),K111((integral+ (M,(max- ((X,F) . G)))))) is ext-real set
lim (I) is ext-real Element of ExtREAL
(I) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(I,G) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . G) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . G) | E)) is ext-real Element of ExtREAL
max+ ((F . G) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . G) | E))) is ext-real Element of ExtREAL
max- ((F . G) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . G) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . G) | E)))) - (integral+ (M,(max- ((F . G) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . G) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . G) | E)))),K111((integral+ (M,(max- ((F . G) | E)))))) is ext-real set
dom (F . G) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(H,G) is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(H . G)) is ext-real Element of ExtREAL
max+ (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (H . G))) is ext-real Element of ExtREAL
max- (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (H . G))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (H . G)))) - (integral+ (M,(max- (H . G)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (H . G))))) is ext-real set
K110((integral+ (M,(max+ (H . G)))),K111((integral+ (M,(max- (H . G)))))) is ext-real set
integral' (M,(H . G)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(H,G) is ext-real Element of ExtREAL
((I),G) is ext-real Element of ExtREAL
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,F) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
G is Element of X
((H . G),G) is ext-real Element of ExtREAL
(((X,F) . G),G) is ext-real Element of ExtREAL
H # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((H # G),G) is ext-real Element of ExtREAL
(X,F) # G is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(((X,F) # G),G) is ext-real Element of ExtREAL
dom ((X,F) . G) is Element of bool X
integral+ (M,(H . G)) is ext-real Element of ExtREAL
integral+ (M,((X,F) . G)) is ext-real Element of ExtREAL
Integral (M,(H . G)) is ext-real Element of ExtREAL
max+ (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (H . G))) is ext-real Element of ExtREAL
max- (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (H . G))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (H . G)))) - (integral+ (M,(max- (H . G)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (H . G))))) is ext-real set
K110((integral+ (M,(max+ (H . G)))),K111((integral+ (M,(max- (H . G)))))) is ext-real set
Integral (M,((X,F) . G)) is ext-real Element of ExtREAL
max+ ((X,F) . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,F) . G))) is ext-real Element of ExtREAL
max- ((X,F) . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,F) . G))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,F) . G)))) - (integral+ (M,(max- ((X,F) . G)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,F) . G))))) is ext-real set
K110((integral+ (M,(max+ ((X,F) . G)))),K111((integral+ (M,(max- ((X,F) . G)))))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (F . 0) is Element of bool X
(X,F) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (X,F) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(lim (X,F)) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((lim (X,F)) | E)) is ext-real Element of ExtREAL
max+ ((lim (X,F)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((lim (X,F)) | E))) is ext-real Element of ExtREAL
max- ((lim (X,F)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((lim (X,F)) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((lim (X,F)) | E)))) - (integral+ (M,(max- ((lim (X,F)) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((lim (X,F)) | E))))) is ext-real set
K110((integral+ (M,(max+ ((lim (X,F)) | E)))),K111((integral+ (M,(max- ((lim (X,F)) | E)))))) is ext-real set
L is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E0 is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom (X) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . E1 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . E1) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . 0) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . 0) | E) is Element of bool X
E0 . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . 0) is Element of bool X
E1 is Element of X
F # E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E0 # E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is set
((F # E1),H) is ext-real Element of ExtREAL
((E0 # E1),H) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
E0 . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (E0 . I) is Element of bool X
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . I) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . I) | E) is Element of bool X
(((F . I) | E),E1) is ext-real Element of ExtREAL
((F . I),E1) is ext-real Element of ExtREAL
((E0 . I),E1) is ext-real Element of ExtREAL
((F # E1),I) is ext-real Element of ExtREAL
(X,E0) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (X,E0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(lim (X,E0)) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (lim (X,E0)) is Element of bool X
(X,E0) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,E0) . 0) is Element of bool X
(X,F) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((X,F) . 0) is Element of bool X
dom (lim (X,F)) is Element of bool X
I is Element of X
((lim (X,E0)),I) is ext-real Element of ExtREAL
((lim (X,F)),I) is ext-real Element of ExtREAL
F # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((F # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
E0 # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((E0 # I)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,E0) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,F) # I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((X,E0) # I),G) is ext-real Element of ExtREAL
(((X,F) # I),G) is ext-real Element of ExtREAL
(((E0 # I)),G) is ext-real Element of ExtREAL
(((F # I)),G) is ext-real Element of ExtREAL
lim ((X,E0) # I) is ext-real Element of ExtREAL
lim ((X,F) # I) is ext-real Element of ExtREAL
dom ((lim (X,E0)) | E) is Element of bool X
dom ((lim (X,F)) | E) is Element of bool X
I is Element of X
(((lim (X,E0)) | E),I) is ext-real Element of ExtREAL
(((lim (X,F)) | E),I) is ext-real Element of ExtREAL
((lim (X,F)),I) is ext-real Element of ExtREAL
((lim (X,E0)),I) is ext-real Element of ExtREAL
E1 is Element of X
E0 # E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
Integral (M,((lim (X,E0)) | E)) is ext-real Element of ExtREAL
max+ ((lim (X,E0)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((lim (X,E0)) | E))) is ext-real Element of ExtREAL
max- ((lim (X,E0)) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((lim (X,E0)) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((lim (X,E0)) | E)))) - (integral+ (M,(max- ((lim (X,E0)) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((lim (X,E0)) | E))))) is ext-real set
K110((integral+ (M,(max+ ((lim (X,E0)) | E)))),K111((integral+ (M,(max- ((lim (X,E0)) | E)))))) is ext-real set
E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(E1) is ext-real Element of ExtREAL
(E1) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (E1) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . H) | E) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
E0 . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E0 . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(E1,H) is ext-real Element of ExtREAL
Integral (M,((F . H) | E)) is ext-real Element of ExtREAL
max+ ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . H) | E))) is ext-real Element of ExtREAL
max- ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . H) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . H) | E)))) - (integral+ (M,(max- ((F . H) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . H) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . H) | E)))),K111((integral+ (M,(max- ((F . H) | E)))))) is ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(E1,H) is ext-real Element of ExtREAL
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . H) | E)) is ext-real Element of ExtREAL
max+ ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . H) | E))) is ext-real Element of ExtREAL
max- ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . H) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . H) | E)))) - (integral+ (M,(max- ((F . H) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . H) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . H) | E)))),K111((integral+ (M,(max- ((F . H) | E)))))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty Relation-like set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:S,ExtREAL:] is non empty set
M is non empty Relation-like S -defined ExtREAL -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive Element of bool [:S,ExtREAL:]
E is Element of S
F is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
F . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty Relation-like ext-real-valued set
bool [:X,ExtREAL:] is non empty set
dom (F . 0) is Element of bool X
lim F is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(lim F)) is ext-real Element of ExtREAL
max+ (lim F) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (lim F))) is ext-real Element of ExtREAL
max- (lim F) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (lim F))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (lim F)))) - (integral+ (M,(max- (lim F)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (lim F))))) is ext-real set
K110((integral+ (M,(max+ (lim F)))),K111((integral+ (M,(max- (lim F)))))) is ext-real set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . L) is Element of bool X
E0 is set
((F . L),E0) is ext-real Element of ExtREAL
((F . 0),E0) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . L),+infty) is Element of bool X
E /\ (eq_dom ((F . L),+infty)) is Element of bool X
(M,(E /\ (eq_dom ((F . L),+infty)))) is ext-real Element of ExtREAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . L),+infty) is Element of bool X
E /\ (eq_dom ((F . L),+infty)) is Element of bool X
(M,(E /\ (eq_dom ((F . L),+infty)))) is ext-real Element of ExtREAL
dom (F . L) is Element of bool X
E0 is Element of S
(F . L) | E0 is Relation-like X -defined E0 -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . L) | E0) is Element of bool X
(dom (F . L)) /\ E0 is Element of bool X
E1 is set
((F . L),E1) is ext-real Element of ExtREAL
(((F . L) | E0),E1) is ext-real Element of ExtREAL
eq_dom (((F . L) | E0),+infty) is Element of bool X
E1 is set
E0 /\ (eq_dom (((F . L) | E0),+infty)) is Element of bool X
(M,(E0 /\ (eq_dom (((F . L) | E0),+infty)))) is ext-real Element of ExtREAL
Integral (M,((F . L) | E0)) is ext-real Element of ExtREAL
max+ ((F . L) | E0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . L) | E0))) is ext-real Element of ExtREAL
max- ((F . L) | E0) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . L) | E0))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . L) | E0)))) - (integral+ (M,(max- ((F . L) | E0)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . L) | E0))))) is ext-real set
K110((integral+ (M,(max+ ((F . L) | E0)))),K111((integral+ (M,(max- ((F . L) | E0)))))) is ext-real set
(F . L) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . L) | E)) is ext-real Element of ExtREAL
max+ ((F . L) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . L) | E))) is ext-real Element of ExtREAL
max- ((F . L) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . L) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . L) | E)))) - (integral+ (M,(max- ((F . L) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . L) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . L) | E)))),K111((integral+ (M,(max- ((F . L) | E)))))) is ext-real set
H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(I,I) is ext-real Element of ExtREAL
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . I)) is ext-real Element of ExtREAL
max+ (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . I))) is ext-real Element of ExtREAL
max- (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . I)))) - (integral+ (M,(max- (F . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . I))))) is ext-real set
K110((integral+ (M,(max+ (F . I)))),K111((integral+ (M,(max- (F . I)))))) is ext-real set
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . H)) is ext-real Element of ExtREAL
max+ (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . H))) is ext-real Element of ExtREAL
max- (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . H)))) - (integral+ (M,(max- (F . H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . H))))) is ext-real set
K110((integral+ (M,(max+ (F . H)))),K111((integral+ (M,(max- (F . H)))))) is ext-real set
lim I is ext-real Element of ExtREAL
dom (lim F) is Element of bool X
I is set
((lim F),I) is ext-real Element of ExtREAL
H is Element of X
F # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
((F # H),G) is ext-real Element of ExtREAL
((F . 0),H) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . G),H) is ext-real Element of ExtREAL
lim (F # H) is ext-real Element of ExtREAL
eq_dom ((lim F),+infty) is Element of bool X
E /\ (eq_dom ((lim F),+infty)) is Element of bool X
I is set
H is Element of X
((F . L),H) is ext-real Element of ExtREAL
F # H is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((F # H),L) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((F # H),G) is ext-real Element of ExtREAL
((F # H),G) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . G),H) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
((F . G),H) is ext-real Element of ExtREAL
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(F # H) ^\ E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(((F # H) ^\ E1),0) is ext-real Element of ExtREAL
0 + L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((F # H),(0 + L)) is ext-real Element of ExtREAL
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(((F # H) ^\ E1),G) is ext-real Element of ExtREAL
lim ((F # H) ^\ E1) is ext-real Element of ExtREAL
sup (F # H) is ext-real Element of ExtREAL
rng (F # H) is non empty V55() V74() Element of bool ExtREAL
sup (rng (F # H)) is ext-real Element of ExtREAL
sup ((F # H) ^\ E1) is ext-real Element of ExtREAL
rng ((F # H) ^\ E1) is non empty V55() V74() Element of bool ExtREAL
sup (rng ((F # H) ^\ E1)) is ext-real Element of ExtREAL
lim (F # H) is ext-real Element of ExtREAL
((lim F),H) is ext-real Element of ExtREAL
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(I,H) is ext-real Element of ExtREAL
(I,I) is ext-real Element of ExtREAL
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Element of X
((F . H),G) is ext-real Element of ExtREAL
((F . I),G) is ext-real Element of ExtREAL
dom (F . H) is Element of bool X
dom (F . I) is Element of bool X
(F . H) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . H) | E)) is ext-real Element of ExtREAL
max+ ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . H) | E))) is ext-real Element of ExtREAL
max- ((F . H) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . H) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . H) | E)))) - (integral+ (M,(max- ((F . H) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . H) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . H) | E)))),K111((integral+ (M,(max- ((F . H) | E)))))) is ext-real set
(F . I) | E is Relation-like X -defined E -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . I) | E)) is ext-real Element of ExtREAL
max+ ((F . I) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . I) | E))) is ext-real Element of ExtREAL
max- ((F . I) | E) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . I) | E))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . I) | E)))) - (integral+ (M,(max- ((F . I) | E)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . I) | E))))) is ext-real set
K110((integral+ (M,(max+ ((F . I) | E)))),K111((integral+ (M,(max- ((F . I) | E)))))) is ext-real set
Integral (M,(F . H)) is ext-real Element of ExtREAL
max+ (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . H))) is ext-real Element of ExtREAL
max- (F . H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . H)))) - (integral+ (M,(max- (F . H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . H))))) is ext-real set
K110((integral+ (M,(max+ (F . H)))),K111((integral+ (M,(max- (F . H)))))) is ext-real set
Integral (M,(F . I)) is ext-real Element of ExtREAL
max+ (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . I))) is ext-real Element of ExtREAL
max- (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . I)))) - (integral+ (M,(max- (F . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . I))))) is ext-real set
K110((integral+ (M,(max+ (F . I)))),K111((integral+ (M,(max- (F . I)))))) is ext-real set
E1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
I ^\ E1 is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
Integral (M,(F . L)) is ext-real Element of ExtREAL
max+ (F . L) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . L))) is ext-real Element of ExtREAL
max- (F . L) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . L))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . L)))) - (integral+ (M,(max- (F . L)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . L))))) is ext-real set
K110((integral+ (M,(max+ (F . L)))),K111((integral+ (M,(max- (F . L)))))) is ext-real set
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((I ^\ E1),I) is ext-real Element of ExtREAL
(I,E1) is ext-real Element of ExtREAL
E1 + I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(I,(E1 + I)) is ext-real Element of ExtREAL
lim (I ^\ E1) is ext-real Element of ExtREAL
(M,(E /\ (eq_dom ((lim F),+infty)))) is ext-real Element of ExtREAL
sup I is ext-real Element of ExtREAL
rng I is non empty V55() V74() Element of bool ExtREAL
sup (rng I) is ext-real Element of ExtREAL
sup (I ^\ E1) is ext-real Element of ExtREAL
rng (I ^\ E1) is non empty V55() V74() Element of bool ExtREAL
sup (rng (I ^\ E1)) is ext-real Element of ExtREAL
I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(I,I) is ext-real Element of ExtREAL
F . I is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . I)) is ext-real Element of ExtREAL
max+ (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . I))) is ext-real Element of ExtREAL
max- (F . I) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . I))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . I)))) - (integral+ (M,(max- (F . I)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . I))))) is ext-real set
K110((integral+ (M,(max+ (F . I)))),K111((integral+ (M,(max- (F . I)))))) is ext-real set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . L),+infty) is Element of bool X
E /\ (eq_dom ((F . L),+infty)) is Element of bool X
dom (F . L) is Element of bool X
E0 is Element of S
[:NAT,S:] is non empty Relation-like set
bool [:NAT,S:] is non empty set
L is non empty Relation-like NAT -defined S -valued Function-like total quasi_total Element of bool [:NAT,S:]
rng L is non empty Element of bool (bool X)
E0 is non empty V55() N_Measure_fam of S
union E0 is Element of S
E \ (union E0) is Element of S
H is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . H) is Element of bool X
F . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . H) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((F . H) | (E \ (union E0))) is Element of bool X
dom (F . H) is Element of bool X
G is set
((H . H),G) is ext-real Element of ExtREAL
L . H is Element of S
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . G),+infty) is Element of bool X
E /\ (eq_dom ((F . G),+infty)) is Element of bool X
dom L is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
(((F . H) | (E \ (union E0))),G) is ext-real Element of ExtREAL
((F . H),G) is ext-real Element of ExtREAL
eq_dom ((F . H),+infty) is Element of bool X
H is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . H) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . G) is Element of bool X
H is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (G + 1)) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is Relation-like Function-like set
dom G is set
G . 0 is set
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G . (G + 1) is set
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (G + 1)) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (G + 1)) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is set
rng G is set
G is set
G . G is set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . J is set
J + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G . (J + 1) is set
H . (J + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (J + 1)) - (H . J) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . G is set
G is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . G) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (G + 1)) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . (G + 1)) is Element of bool X
dom (H . (G + 1)) is Element of bool X
dom (H . G) is Element of bool X
(dom (H . (G + 1))) /\ (dom (H . G)) is Element of bool X
(E \ (union E0)) /\ (dom (H . G)) is Element of bool X
(E \ (union E0)) /\ (E \ (union E0)) is Element of S
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . G) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . G) is Element of bool X
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
G . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (G + 1)) - (H . G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . (G + 1)) is Element of bool X
J is set
((G . (G + 1)),J) is ext-real Element of ExtREAL
((H . G),J) is ext-real Element of ExtREAL
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . G) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((F . G) | (E \ (union E0))),J) is ext-real Element of ExtREAL
((F . G),J) is ext-real Element of ExtREAL
((H . (G + 1)),J) is ext-real Element of ExtREAL
F . (G + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . (G + 1)) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(((F . (G + 1)) | (E \ (union E0))),J) is ext-real Element of ExtREAL
((F . (G + 1)),J) is ext-real Element of ExtREAL
((H . (G + 1)),J) - ((H . G),J) is ext-real Element of ExtREAL
K111(((H . G),J)) is ext-real set
K110(((H . (G + 1)),J),K111(((H . G),J))) is ext-real set
F . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . G) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
G is set
H . G is Relation-like Function-like set
(X,G) . G is Relation-like Function-like set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,G) . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
J + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . (J + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,G) . (J + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G . (J + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . (J + 1)) is Element of bool X
(G . (J + 1)) + (H . J) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom ((G . (J + 1)) + (H . J)) is Element of bool X
dom (H . J) is Element of bool X
(dom (G . (J + 1))) /\ (dom (H . J)) is Element of bool X
(E \ (union E0)) /\ (E \ (union E0)) is Element of S
dom (H . (J + 1)) is Element of bool X
(H . (J + 1)) - (H . J) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
n is Element of X
((H . (J + 1)),n) is ext-real Element of ExtREAL
(((G . (J + 1)) + (H . J)),n) is ext-real Element of ExtREAL
((H . J),n) is ext-real Element of ExtREAL
((H . (J + 1)),n) - ((H . J),n) is ext-real Element of ExtREAL
K111(((H . J),n)) is ext-real set
K110(((H . (J + 1)),n),K111(((H . J),n))) is ext-real set
(((H . (J + 1)),n) - ((H . J),n)) + ((H . J),n) is ext-real Element of ExtREAL
((H . J),n) - ((H . J),n) is ext-real Element of ExtREAL
K110(((H . J),n),K111(((H . J),n))) is ext-real set
((H . (J + 1)),n) - (((H . J),n) - ((H . J),n)) is ext-real Element of ExtREAL
K111((((H . J),n) - ((H . J),n))) is ext-real set
K110(((H . (J + 1)),n),K111((((H . J),n) - ((H . J),n)))) is ext-real set
((H . (J + 1)),n) - 0. is ext-real Element of ExtREAL
K111(0.) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() set
- 0. is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() set
K110(((H . (J + 1)),n),K111(0.)) is ext-real set
((G . (J + 1)),n) is ext-real Element of ExtREAL
((G . (J + 1)),n) + ((H . J),n) is ext-real Element of ExtREAL
(X,G) . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,G) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
lim H is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
lim (X,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(X,G) . G is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
H . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . J) is Element of bool X
n is Element of X
((H . J),n) is ext-real Element of ExtREAL
|.((H . J),n).| is ext-real Element of ExtREAL
G is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total with_the_same_dom (X) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
G . J is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
H . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (H . n) is Element of bool X
G . (n + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
H . (n + 1) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(H . (n + 1)) - (H . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
G . 0 is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (G . 0) is Element of bool X
J is Element of X
G # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
F # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
H # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,G) # J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((G # J)) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(X,G) is non empty Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like total quasi_total Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (X,G) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(lim (X,G)) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((lim (X,G)) | (E \ (union E0)))) is ext-real Element of ExtREAL
max+ ((lim (X,G)) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((lim (X,G)) | (E \ (union E0))))) is ext-real Element of ExtREAL
max- ((lim (X,G)) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((lim (X,G)) | (E \ (union E0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((lim (X,G)) | (E \ (union E0)))))) - (integral+ (M,(max- ((lim (X,G)) | (E \ (union E0)))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((lim (X,G)) | (E \ (union E0))))))) is ext-real set
K110((integral+ (M,(max+ ((lim (X,G)) | (E \ (union E0)))))),K111((integral+ (M,(max- ((lim (X,G)) | (E \ (union E0)))))))) is ext-real set
J is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
(J) is ext-real Element of ExtREAL
(J) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
lim (J) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
n is set
(I,n) is ext-real Element of ExtREAL
((J),n) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(J,n) is ext-real Element of ExtREAL
G . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(G . n)) is ext-real Element of ExtREAL
max+ (G . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (G . n))) is ext-real Element of ExtREAL
max- (G . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (G . n))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (G . n)))) - (integral+ (M,(max- (G . n)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (G . n))))) is ext-real set
K110((integral+ (M,(max+ (G . n)))),K111((integral+ (M,(max- (G . n)))))) is ext-real set
dom (G . n) is Element of bool X
(G . n) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((J),m) is ext-real Element of ExtREAL
(X,G) . m is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((X,G) . m)) is ext-real Element of ExtREAL
max+ ((X,G) . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((X,G) . m))) is ext-real Element of ExtREAL
max- ((X,G) . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((X,G) . m))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((X,G) . m)))) - (integral+ (M,(max- ((X,G) . m)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((X,G) . m))))) is ext-real set
K110((integral+ (M,(max+ ((X,G) . m)))),K111((integral+ (M,(max- ((X,G) . m)))))) is ext-real set
H . m is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(H . m)) is ext-real Element of ExtREAL
max+ (H . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (H . m))) is ext-real Element of ExtREAL
max- (H . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (H . m))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (H . m)))) - (integral+ (M,(max- (H . m)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (H . m))))) is ext-real set
K110((integral+ (M,(max+ (H . m)))),K111((integral+ (M,(max- (H . m)))))) is ext-real set
F . m is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . m) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . m) | (E \ (union E0)))) is ext-real Element of ExtREAL
max+ ((F . m) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . m) | (E \ (union E0))))) is ext-real Element of ExtREAL
max- ((F . m) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . m) | (E \ (union E0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . m) | (E \ (union E0)))))) - (integral+ (M,(max- ((F . m) | (E \ (union E0)))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . m) | (E \ (union E0))))))) is ext-real set
K110((integral+ (M,(max+ ((F . m) | (E \ (union E0)))))),K111((integral+ (M,(max- ((F . m) | (E \ (union E0)))))))) is ext-real set
dom (lim (X,G)) is Element of bool X
dom (H . 0) is Element of bool X
lim I is ext-real Element of ExtREAL
Integral (M,(lim H)) is ext-real Element of ExtREAL
max+ (lim H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (lim H))) is ext-real Element of ExtREAL
max- (lim H) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (lim H))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (lim H)))) - (integral+ (M,(max- (lim H)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (lim H))))) is ext-real set
K110((integral+ (M,(max+ (lim H)))),K111((integral+ (M,(max- (lim H)))))) is ext-real set
n is Element of X
F # n is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
M * L is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
((M * L),n) is ext-real Element of ExtREAL
dom L is non empty V73() V74() V75() V76() V77() V78() Element of bool NAT
L . n is Element of S
(M,(L . n)) is ext-real Element of ExtREAL
F . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . n),+infty) is Element of bool X
E /\ (eq_dom ((F . n),+infty)) is Element of bool X
(M,(E /\ (eq_dom ((F . n),+infty)))) is ext-real Element of ExtREAL
SUM (M * L) is ext-real Element of ExtREAL
Ser (M * L) is non empty Relation-like NAT -defined ExtREAL -valued Function-like total quasi_total ext-real-valued Element of bool [:NAT,ExtREAL:]
((Ser (M * L)),0) is ext-real Element of ExtREAL
((M * L),0) is ext-real Element of ExtREAL
(M,(union E0)) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(I,n) is ext-real Element of ExtREAL
F . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . n)) is ext-real Element of ExtREAL
max+ (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . n))) is ext-real Element of ExtREAL
max- (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . n))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . n)))) - (integral+ (M,(max- (F . n)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . n))))) is ext-real set
K110((integral+ (M,(max+ (F . n)))),K111((integral+ (M,(max- (F . n)))))) is ext-real set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
F . m is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
(F . m) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((F . m) | (E \ (union E0)))) is ext-real Element of ExtREAL
max+ ((F . m) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((F . m) | (E \ (union E0))))) is ext-real Element of ExtREAL
max- ((F . m) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((F . m) | (E \ (union E0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((F . m) | (E \ (union E0)))))) - (integral+ (M,(max- ((F . m) | (E \ (union E0)))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((F . m) | (E \ (union E0))))))) is ext-real set
K110((integral+ (M,(max+ ((F . m) | (E \ (union E0)))))),K111((integral+ (M,(max- ((F . m) | (E \ (union E0)))))))) is ext-real set
dom (F . n) is Element of bool X
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() Element of NAT
(I,m) is ext-real Element of ExtREAL
(I,n) is ext-real Element of ExtREAL
F . m is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
dom (F . m) is Element of bool X
F . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
n is Element of X
((F . m),n) is ext-real Element of ExtREAL
((F . n),n) is ext-real Element of ExtREAL
dom (F . n) is Element of bool X
integral+ (M,(F . m)) is ext-real Element of ExtREAL
integral+ (M,(F . n)) is ext-real Element of ExtREAL
Integral (M,(F . m)) is ext-real Element of ExtREAL
max+ (F . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . m))) is ext-real Element of ExtREAL
max- (F . m) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . m))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . m)))) - (integral+ (M,(max- (F . m)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . m))))) is ext-real set
K110((integral+ (M,(max+ (F . m)))),K111((integral+ (M,(max- (F . m)))))) is ext-real set
Integral (M,(F . n)) is ext-real Element of ExtREAL
max+ (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . n))) is ext-real Element of ExtREAL
max- (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . n))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . n)))) - (integral+ (M,(max- (F . n)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . n))))) is ext-real set
K110((integral+ (M,(max+ (F . n)))),K111((integral+ (M,(max- (F . n)))))) is ext-real set
dom (lim F) is Element of bool X
(lim F) | (E \ (union E0)) is Relation-like X -defined E \ (union E0) -defined X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,((lim F) | (E \ (union E0)))) is ext-real Element of ExtREAL
max+ ((lim F) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ ((lim F) | (E \ (union E0))))) is ext-real Element of ExtREAL
max- ((lim F) | (E \ (union E0))) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- ((lim F) | (E \ (union E0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ ((lim F) | (E \ (union E0)))))) - (integral+ (M,(max- ((lim F) | (E \ (union E0)))))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- ((lim F) | (E \ (union E0))))))) is ext-real set
K110((integral+ (M,(max+ ((lim F) | (E \ (union E0)))))),K111((integral+ (M,(max- ((lim F) | (E \ (union E0)))))))) is ext-real set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(I,n) is ext-real Element of ExtREAL
F . n is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
Integral (M,(F . n)) is ext-real Element of ExtREAL
max+ (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (F . n))) is ext-real Element of ExtREAL
max- (F . n) is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
integral+ (M,(max- (F . n))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (F . n)))) - (integral+ (M,(max- (F . n)))) is ext-real Element of ExtREAL
K111((integral+ (M,(max- (F . n))))) is ext-real set
K110((integral+ (M,(max+ (F . n)))),K111((integral+ (M,(max- (F . n)))))) is ext-real set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
F . L is Relation-like X -defined ExtREAL -valued Function-like ext-real-valued Element of bool [:X,ExtREAL:]
eq_dom ((F . L),+infty) is Element of bool X
E /\ (eq_dom ((F . L),+infty)) is Element of bool X
(M,(E /\ (eq_dom ((F . L),+infty)))) is ext-real Element of ExtREAL