:: MESFUNC5 semantic presentation

REAL is non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() set
NAT is epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below Element of bool REAL
bool REAL is non empty set
ExtREAL is non empty ext-real-membered V120() set
[:NAT,ExtREAL:] is non empty V59() set
bool [:NAT,ExtREAL:] is non empty set
bool ExtREAL is non empty set
COMPLEX is non empty non trivial non finite complex-membered V74() set
omega is epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty V58() V59() V60() set
bool [:NAT,REAL:] is non empty set
RAT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() set
INT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V74() set
{} is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
the epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
{{},1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded set
bool (bool REAL) is non empty set
bool RAT is non empty set
[:COMPLEX,COMPLEX:] is non empty V58() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V58() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty V58() V59() V60() set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V58() V59() V60() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is RAT -valued non empty V58() V59() V60() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is RAT -valued non empty V58() V59() V60() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is RAT -valued INT -valued non empty V58() V59() V60() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is RAT -valued INT -valued non empty V58() V59() V60() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is RAT -valued INT -valued non empty V58() V59() V60() V61() set
[:[:NAT,NAT:],NAT:] is RAT -valued INT -valued non empty V58() V59() V60() V61() set
bool [:[:NAT,NAT:],NAT:] is non empty set
K535() is set
2 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
3 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() Element of ExtREAL
+infty is non empty non real ext-real positive non negative Element of ExtREAL
0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() Element of NAT
|.+infty.| is ext-real Element of ExtREAL
-infty is non empty non real ext-real non positive negative Element of ExtREAL
|.-infty.| is ext-real Element of ExtREAL
{-infty} is non empty finite ext-real-membered left_end right_end set
{+infty} is non empty finite ext-real-membered left_end right_end set
Seg 1 is non empty finite V44(1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded set
Seg 2 is non empty finite V44(2) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded set
- 1 is non empty complex real ext-real non positive negative integer rational 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
{+infty,-infty} is non empty finite ext-real-membered left_end right_end set
REAL \/ {+infty,-infty} is non empty ext-real-membered set
bool {} is non empty finite V41() set
{{}} is non empty finite V41() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded set
len {} is V42() set
- -infty is non empty ext-real positive non negative set
X is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
X - S is ext-real Element of ExtREAL
- S is ext-real set
X + (- S) is ext-real set
|.(X - S).| is ext-real Element of ExtREAL
S - X is ext-real Element of ExtREAL
- X is ext-real set
S + (- X) is ext-real set
|.(S - X).| is ext-real Element of ExtREAL
- (X - S) is ext-real Element of ExtREAL
|.(- (X - S)).| is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
X is ext-real Element of ExtREAL
S - X is ext-real Element of ExtREAL
- X is ext-real set
S + (- X) is ext-real set
X - S is ext-real Element of ExtREAL
- S is ext-real set
X + (- S) is ext-real set
|.(X - S).| is ext-real Element of ExtREAL
- |.(X - S).| is ext-real Element of ExtREAL
- (X - S) is ext-real Element of ExtREAL
X is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
X - S is ext-real Element of ExtREAL
- S is ext-real set
X + (- S) is ext-real set
|.(X - S).| is ext-real Element of ExtREAL
M is complex real ext-real set
S - X is ext-real Element of ExtREAL
- X is ext-real set
S + (- X) is ext-real set
R_EAL M is ext-real Element of ExtREAL
X is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
2 |^ X is complex real ext-real Element of REAL
(2 |^ X) * X is complex real ext-real Element of REAL
S is ext-real Element of ExtREAL
M is complex real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
M * (2 |^ f) is complex real ext-real Element of REAL
(M * (2 |^ f)) + 1 is complex real ext-real Element of REAL
[\((M * (2 |^ f)) + 1)/] is complex real ext-real integer rational set
((M * (2 |^ f)) + 1) - 1 is complex real ext-real Element of REAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ f) * f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((2 |^ f) * f) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
x is complex real ext-real integer rational set
[\x/] is complex real ext-real integer rational set
B - 1 is complex real ext-real integer rational Element of REAL
(B - 1) / (2 |^ X) is complex real ext-real Element of REAL
B / (2 |^ X) is complex real ext-real Element of REAL
B - 1 is complex real ext-real integer rational Element of REAL
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
2 |^ X is complex real ext-real Element of REAL
(2 |^ X) * X is complex real ext-real Element of REAL
S / (2 |^ X) is complex real ext-real Element of REAL
M is ext-real Element of ExtREAL
M is ext-real set
X is ext-real set
S is ext-real set
max (X,S) is ext-real set
M * (max (X,S)) is ext-real set
M * X is ext-real set
M * S is ext-real set
max ((M * X),(M * S)) is ext-real set
min (X,S) is ext-real set
M * (min (X,S)) is ext-real set
min ((M * X),(M * S)) is ext-real set
M is ext-real Element of ExtREAL
X is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
min (X,S) is ext-real set
M * (min (X,S)) is ext-real set
M * X is ext-real Element of ExtREAL
M * S is ext-real Element of ExtREAL
max ((M * X),(M * S)) is ext-real set
max (X,S) is ext-real set
M * (max (X,S)) is ext-real set
min ((M * X),(M * S)) is ext-real set
min ((M * S),(M * X)) is ext-real set
X is set
[:X,ExtREAL:] is V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
rng S is ext-real-membered Element of bool ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
M is ext-real Element of ExtREAL
rng S is ext-real-membered set
dom S is Element of bool X
bool X is non empty set
f is set
S . f is ext-real Element of ExtREAL
X is set
[:X,ExtREAL:] is V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom S is Element of bool X
bool X is non empty set
M is ext-real Element of ExtREAL
rng S is ext-real-membered set
rng S is ext-real-membered Element of bool ExtREAL
f is set
S . f is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
rng S is ext-real-membered Element of bool ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
dom S is Element of bool X
bool X is non empty set
M is set
S . M is ext-real Element of ExtREAL
rng S is ext-real-membered Element of bool ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
S . M is ext-real Element of ExtREAL
M is set
dom S is Element of bool X
bool X is non empty set
dom S is Element of bool X
bool X is non empty set
M is set
S . M is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom S is Element of bool X
bool X is non empty set
M is set
S . M is ext-real Element of ExtREAL
M is set
S . M is ext-real Element of ExtREAL
M is set
M is set
S . M is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom S is Element of bool X
bool X is non empty set
M is set
S . M is ext-real Element of ExtREAL
M is set
S . M is ext-real Element of ExtREAL
M is set
M is set
S . M is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is set
S . M is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is set
S . M is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
rng M is ext-real-membered Element of bool ExtREAL
M " {+infty} is Element of bool X
f is set
M . f is ext-real Element of ExtREAL
M " {-infty} is Element of bool X
f is set
M . f is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is set
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M | S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is set
dom (M | S) is Element of bool X
bool X is non empty set
(M | S) . f is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() 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
rng M is ext-real-membered Element of bool ExtREAL
M " {-infty} is Element of bool X
rng S is ext-real-membered Element of bool ExtREAL
S " {-infty} is Element of bool X
S " {+infty} is Element of bool X
(S " {+infty}) /\ (M " {-infty}) is Element of bool X
M " {+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 V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S - M is Relation-like X -defined ExtREAL -valued Function-like V59() 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
rng M is ext-real-membered Element of bool ExtREAL
M " {+infty} is Element of bool X
rng S is ext-real-membered Element of bool ExtREAL
S " {-infty} is Element of bool X
S " {+infty} is Element of bool X
(S " {+infty}) /\ (M " {+infty}) is Element of bool X
M " {-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
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:RAT,S:] is non empty set
bool [:RAT,S:] is non empty set
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M + f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like RAT -defined S -valued Function-like V32( RAT ,S) Element of bool [:RAT,S:]
rng c is Element of bool S
bool S is non empty set
union (rng c) is set
B is complex real ext-real Element of REAL
R_EAL B is ext-real Element of ExtREAL
less_dom ((M + f),(R_EAL B)) is Element of bool X
x is Element of S
x /\ (less_dom ((M + f),(R_EAL B))) is Element of bool X
dom (M + f) is Element of bool X
dom M is Element of bool X
dom f is Element of bool X
(dom M) /\ (dom f) is Element of bool X
I1 is set
a is set
dom c is complex-membered ext-real-membered real-membered rational-membered Element of bool RAT
f1 is set
c . f1 is set
E is complex real ext-real rational set
R_EAL E is ext-real Element of ExtREAL
less_dom (M,(R_EAL E)) is Element of bool X
x /\ (less_dom (M,(R_EAL E))) is Element of bool X
B - E is complex real ext-real Element of REAL
R_EAL (B - E) is ext-real Element of ExtREAL
less_dom (f,(R_EAL (B - E))) is Element of bool X
x /\ (less_dom (f,(R_EAL (B - E)))) is Element of bool X
(x /\ (less_dom (M,(R_EAL E)))) /\ (x /\ (less_dom (f,(R_EAL (B - E))))) is Element of bool X
x is Element of X
M . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
NFPG is complex real ext-real Element of REAL
B - NFPG is complex real ext-real Element of REAL
DFPG is complex real ext-real Element of REAL
DFPG + NFPG is complex real ext-real Element of REAL
(M + f) . x is ext-real Element of ExtREAL
(M . x) + (f . x) is ext-real Element of ExtREAL
I1 is set
(M + f) . I1 is ext-real Element of ExtREAL
a is Element of X
M . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(M . a) + (f . a) is ext-real Element of ExtREAL
(R_EAL B) - (f . a) is ext-real Element of ExtREAL
- (f . a) is ext-real set
(R_EAL B) + (- (f . a)) is ext-real set
E is complex real ext-real Element of REAL
B - E is complex real ext-real Element of REAL
f1 is complex real ext-real Element of REAL
x is complex real ext-real rational set
B - x is complex real ext-real Element of REAL
R_EAL (B - x) is ext-real Element of ExtREAL
less_dom (f,(R_EAL (B - x))) is Element of bool X
x /\ (less_dom (f,(R_EAL (B - x)))) is Element of bool X
dom c is complex-membered ext-real-membered real-membered rational-membered Element of bool RAT
c . x is set
R_EAL x is ext-real Element of ExtREAL
less_dom (M,(R_EAL x)) is Element of bool X
x /\ (less_dom (M,(R_EAL x))) is Element of bool X
(x /\ (less_dom (M,(R_EAL x)))) /\ (x /\ (less_dom (f,(R_EAL (B - x))))) is Element of bool X
X is non empty set
[:X,REAL:] is non empty V58() V59() V60() set
bool [:X,REAL:] is non empty set
S is Relation-like X -defined REAL -valued Function-like V58() V59() V60() Element of bool [:X,REAL:]
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is set
(f + c) . B is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
(f . B) + (c . B) is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is complex real ext-real Element of REAL
M (#) S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (M (#) S) is Element of bool X
bool X is non empty set
B is set
(M (#) S) . B is ext-real Element of ExtREAL
S . B is ext-real Element of ExtREAL
R_EAL M is ext-real Element of ExtREAL
(R_EAL M) * (S . B) is ext-real Element of ExtREAL
c is set
S . c is ext-real Element of ExtREAL
R_EAL M is ext-real Element of ExtREAL
(R_EAL M) * (S . c) is ext-real Element of ExtREAL
(M (#) S) . c is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() 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 V59() Element of bool [:X,ExtREAL:]
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
S - M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is set
dom (S - M) is Element of bool X
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
M " {-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)) \ (((S " {+infty}) /\ (M " {+infty})) \/ ((S " {-infty}) /\ (M " {-infty}))) is Element of bool X
S . f is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
(S . f) - (M . f) is ext-real Element of ExtREAL
- (M . f) is ext-real set
(S . f) + (- (M . f)) is ext-real set
(S - 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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
|.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- f) is Element of bool X
c is set
(max- f) . c is ext-real Element of ExtREAL
dom (max+ f) is Element of bool X
c is set
(max+ f) . c is ext-real Element of ExtREAL
c is set
dom |.f.| is Element of bool X
|.f.| . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
|.(f . c).| is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() 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
f is set
M . f is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(S . f) + (M . f) is ext-real Element of ExtREAL
(S + M) . f is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(S + M) + f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((S + M) + f) 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
dom f is Element of bool X
((dom S) /\ (dom M)) /\ (dom f) is Element of bool X
dom (S + M) is Element of bool X
(dom (S + M)) /\ (dom f) is Element of bool X
c is set
((S + M) + f) . c is ext-real Element of ExtREAL
(S + M) . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
((S + M) . c) + (f . c) is ext-real Element of ExtREAL
S . c is ext-real Element of ExtREAL
M . c is ext-real Element of ExtREAL
(S . c) + (M . c) is ext-real Element of ExtREAL
((S . c) + (M . c)) + (f . c) is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (S + M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ (S + M)) + (max- S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max+ (S + M)) + (max- S)) 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
max- (S + M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- (S + M)) + (max+ S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max- (S + M)) + (max+ S)) is Element of bool X
max- M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max+ (S + M)) + (max- S)) + (max- M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (((max+ (S + M)) + (max- S)) + (max- M)) is Element of bool X
max+ M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max- (S + M)) + (max+ S)) + (max+ M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (((max- (S + M)) + (max+ S)) + (max+ M)) is Element of bool X
dom (S + M) is Element of bool X
dom (max- (S + M)) is Element of bool X
dom (max- S) is Element of bool X
dom (max+ S) is Element of bool X
dom (max+ M) is Element of bool X
dom (max- M) is Element of bool X
f is set
(max- S) . f is ext-real Element of ExtREAL
c is set
(max+ S) . c is ext-real Element of ExtREAL
B is set
(max+ M) . B is ext-real Element of ExtREAL
x is set
(max- M) . x is ext-real Element of ExtREAL
dom (max+ (S + M)) is Element of bool X
f is set
(max+ (S + M)) . f is ext-real Element of ExtREAL
c is set
(max- (S + M)) . c is ext-real Element of ExtREAL
(dom (max+ (S + M))) /\ (dom (max- S)) is Element of bool X
(dom (max- (S + M))) /\ (dom (max+ S)) is Element of bool X
(dom S) /\ (dom S) is Element of bool X
(dom M) /\ ((dom S) /\ (dom S)) is Element of bool X
f is set
((max+ (S + M)) + (max- S)) . f is ext-real Element of ExtREAL
((max- (S + M)) + (max+ S)) . f is ext-real Element of ExtREAL
(max- S) . f is ext-real Element of ExtREAL
(max+ (S + M)) . f is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(max+ S) . f is ext-real Element of ExtREAL
(max- (S + M)) . f is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
f is set
((max+ (S + M)) + (max- S)) . f is ext-real Element of ExtREAL
c is set
((max- (S + M)) + (max+ S)) . c is ext-real Element of ExtREAL
((dom S) /\ (dom M)) /\ (dom M) is Element of bool X
(dom M) /\ (dom M) is Element of bool X
(dom S) /\ ((dom M) /\ (dom M)) is Element of bool X
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (S + M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ (S + M)) + (max- S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max+ (S + M)) + (max- S)) + (max- M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- (S + M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- (S + M)) + (max+ S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max- (S + M)) + (max+ S)) + (max+ M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- (S + M)) is Element of bool X
bool X is non empty set
dom (S + M) is Element of bool X
dom (max- M) is Element of bool X
f is set
(max- M) . f is ext-real Element of ExtREAL
dom (max+ M) is Element of bool X
f is set
(max+ M) . f is ext-real Element of ExtREAL
dom (max- S) is Element of bool X
dom S is Element of bool X
dom (max+ (S + M)) is Element of bool X
f is set
(max+ (S + M)) . f is ext-real Element of ExtREAL
dom (max+ S) is Element of bool X
f is set
(max+ S) . f is ext-real Element of ExtREAL
dom M is Element of bool X
f is set
(max- S) . f is ext-real Element of ExtREAL
dom (((max+ (S + M)) + (max- S)) + (max- M)) is Element of bool X
(dom (S + M)) /\ (dom S) is Element of bool X
((dom (S + M)) /\ (dom S)) /\ (dom M) is Element of bool X
(dom S) /\ (dom M) is Element of bool X
(dom (S + M)) /\ ((dom S) /\ (dom M)) is Element of bool X
f is set
(max- (S + M)) . f is ext-real Element of ExtREAL
f is set
(((max+ (S + M)) + (max- S)) + (max- M)) . f is ext-real Element of ExtREAL
(((max- (S + M)) + (max+ S)) + (max+ M)) . f is ext-real Element of ExtREAL
(max+ M) . f is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
max ((M . f),0) is ext-real set
(max+ (S + M)) . f is ext-real Element of ExtREAL
(S + M) . f is ext-real Element of ExtREAL
max (((S + M) . f),0) is ext-real set
S . f is ext-real Element of ExtREAL
(S . f) + (M . f) is ext-real Element of ExtREAL
max (((S . f) + (M . f)),0) is ext-real set
(max+ S) . f is ext-real Element of ExtREAL
max ((S . f),0) is ext-real set
(max- (S + M)) . f is ext-real Element of ExtREAL
- ((S + M) . f) is ext-real Element of ExtREAL
max ((- ((S + M) . f)),0) is ext-real set
- ((S . f) + (M . f)) is ext-real Element of ExtREAL
max ((- ((S . f) + (M . f))),0) is ext-real set
- (S . f) is ext-real Element of ExtREAL
(max- S) . f is ext-real Element of ExtREAL
max ((- (S . f)),0) is ext-real set
(max- M) . f is ext-real Element of ExtREAL
- (M . f) is ext-real Element of ExtREAL
max ((- (M . f)),0) is ext-real set
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((S . f) + (M . f)) + 0 is ext-real set
(((S . f) + (M . f)) + 0) + 0 is ext-real set
R_EAL 0 is ext-real Element of ExtREAL
((S . f) + (M . f)) + (R_EAL 0) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
(R_EAL 0) + (S . f) is ext-real Element of ExtREAL
((R_EAL 0) + (S . f)) + (M . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
(R_EAL 0) + (S . f) is ext-real Element of ExtREAL
((R_EAL 0) + (S . f)) + (R_EAL 0) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((S . f) + (M . f)) + (R_EAL 0) is ext-real Element of ExtREAL
(((S . f) + (M . f)) + (R_EAL 0)) + (- (M . f)) is ext-real Element of ExtREAL
((S . f) + (M . f)) - (M . f) is ext-real Element of ExtREAL
- (M . f) is ext-real set
((S . f) + (M . f)) + (- (M . f)) is ext-real set
(M . f) - (M . f) is ext-real Element of ExtREAL
(M . f) + (- (M . f)) is ext-real set
(S . f) + ((M . f) - (M . f)) is ext-real Element of ExtREAL
(S . f) + (R_EAL 0) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
(R_EAL 0) + (R_EAL 0) is ext-real Element of ExtREAL
((R_EAL 0) + (R_EAL 0)) + (- (M . f)) is ext-real Element of ExtREAL
0 + (- (M . f)) is ext-real set
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
(- ((S . f) + (M . f))) + (S . f) is ext-real Element of ExtREAL
((- ((S . f) + (M . f))) + (S . f)) + (R_EAL 0) is ext-real Element of ExtREAL
(- (M . f)) - (S . f) is ext-real Element of ExtREAL
- (S . f) is ext-real set
(- (M . f)) + (- (S . f)) is ext-real set
((- (M . f)) - (S . f)) + (S . f) is ext-real Element of ExtREAL
(- (S . f)) + (S . f) is ext-real Element of ExtREAL
(- (M . f)) + ((- (S . f)) + (S . f)) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
(R_EAL 0) + (R_EAL 0) is ext-real Element of ExtREAL
((R_EAL 0) + (R_EAL 0)) + (M . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((S . f) + (M . f)) + (- (S . f)) is ext-real Element of ExtREAL
(((S . f) + (M . f)) + (- (S . f))) + (R_EAL 0) is ext-real Element of ExtREAL
(S . f) - (S . f) is ext-real Element of ExtREAL
- (S . f) is ext-real set
(S . f) + (- (S . f)) is ext-real set
(M . f) + ((S . f) - (S . f)) is ext-real Element of ExtREAL
(M . f) + 0 is ext-real set
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
(- ((S . f) + (M . f))) + (R_EAL 0) is ext-real Element of ExtREAL
((- ((S . f) + (M . f))) + (R_EAL 0)) + (M . f) is ext-real Element of ExtREAL
(- ((S . f) + (M . f))) + (M . f) is ext-real Element of ExtREAL
(- (S . f)) - (M . f) is ext-real Element of ExtREAL
- (M . f) is ext-real set
(- (S . f)) + (- (M . f)) is ext-real set
((- (S . f)) - (M . f)) + (M . f) is ext-real Element of ExtREAL
(- (M . f)) + (M . f) is ext-real Element of ExtREAL
(- (S . f)) + ((- (M . f)) + (M . f)) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
(R_EAL 0) + (- (S . f)) is ext-real Element of ExtREAL
((R_EAL 0) + (- (S . f))) + (R_EAL 0) is ext-real Element of ExtREAL
0 + (- (S . f)) is ext-real set
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
(R_EAL 0) + (- (S . f)) is ext-real Element of ExtREAL
((R_EAL 0) + (- (S . f))) + (- (M . f)) is ext-real Element of ExtREAL
(- (S . f)) - (M . f) is ext-real Element of ExtREAL
- (M . f) is ext-real set
(- (S . f)) + (- (M . f)) is ext-real set
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
(- ((S . f) + (M . f))) + (R_EAL 0) is ext-real Element of ExtREAL
((- ((S . f) + (M . f))) + (R_EAL 0)) + (R_EAL 0) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
((max+ (S + M)) . f) + ((max- S) . f) is ext-real Element of ExtREAL
(((max+ (S + M)) . f) + ((max- S) . f)) + ((max- M) . f) is ext-real Element of ExtREAL
((max- (S + M)) . f) + ((max+ S) . f) is ext-real Element of ExtREAL
(((max- (S + M)) . f) + ((max+ S) . f)) + ((max+ M) . f) is ext-real Element of ExtREAL
(dom (max+ (S + M))) /\ (dom (max- S)) is Element of bool X
((dom (max+ (S + M))) /\ (dom (max- S))) /\ (dom (max- M)) is Element of bool X
dom (((max- (S + M)) + (max+ S)) + (max+ M)) is Element of bool X
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is complex real ext-real Element of REAL
M (#) S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (M (#) S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M (#) (max+ S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- (M (#) S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M (#) (max- S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ (M (#) S)) is Element of bool X
bool X is non empty set
dom (M (#) S) is Element of bool X
dom S is Element of bool X
dom (max+ S) is Element of bool X
dom (M (#) (max+ S)) is Element of bool X
f is Element of X
(max+ (M (#) S)) . f is ext-real Element of ExtREAL
(M (#) (max+ S)) . f is ext-real Element of ExtREAL
(M (#) S) . f is ext-real Element of ExtREAL
max (((M (#) S) . f),0) is ext-real set
R_EAL M is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(R_EAL M) * (S . f) is ext-real Element of ExtREAL
max (((R_EAL M) * (S . f)),0) is ext-real set
(max+ S) . f is ext-real Element of ExtREAL
(R_EAL M) * ((max+ S) . f) is ext-real Element of ExtREAL
max ((S . f),0) is ext-real set
M * (max ((S . f),0)) is ext-real set
M * (S . f) is ext-real set
M * 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
M * 0 is complex real ext-real set
max ((M * (S . f)),(M * 0)) is ext-real set
dom (max- (M (#) S)) is Element of bool X
dom (max- S) is Element of bool X
dom (M (#) (max- S)) is Element of bool X
f is Element of X
(max- (M (#) S)) . f is ext-real Element of ExtREAL
(M (#) (max- S)) . f is ext-real Element of ExtREAL
(M (#) S) . f is ext-real Element of ExtREAL
- ((M (#) S) . f) is ext-real Element of ExtREAL
max ((- ((M (#) S) . f)),0) is ext-real set
R_EAL M is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(R_EAL M) * (S . f) is ext-real Element of ExtREAL
- ((R_EAL M) * (S . f)) is ext-real Element of ExtREAL
max ((- ((R_EAL M) * (S . f))),0) is ext-real set
(max- S) . f is ext-real Element of ExtREAL
(R_EAL M) * ((max- S) . f) is ext-real Element of ExtREAL
- (S . f) is ext-real Element of ExtREAL
max ((- (S . f)),0) is ext-real set
M * (max ((- (S . f)),0)) is ext-real set
M * (- (S . f)) is ext-real set
M * 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
M * 0 is complex real ext-real set
max ((M * (- (S . f))),(M * 0)) is ext-real set
M * (S . f) is ext-real set
- (M * (S . f)) is ext-real set
max ((- (M * (S . f))),(M * 0)) is ext-real set
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is complex real ext-real Element of REAL
- M is complex real ext-real Element of REAL
(- M) (#) S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ ((- M) (#) S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M (#) (max- S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- ((- M) (#) S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M (#) (max+ S) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
R_EAL M is ext-real Element of ExtREAL
- (R_EAL M) is ext-real Element of ExtREAL
dom (max+ ((- M) (#) S)) is Element of bool X
bool X is non empty set
dom ((- M) (#) S) is Element of bool X
dom S is Element of bool X
dom (max- S) is Element of bool X
dom (M (#) (max- S)) is Element of bool X
f is Element of X
(max+ ((- M) (#) S)) . f is ext-real Element of ExtREAL
(M (#) (max- S)) . f is ext-real Element of ExtREAL
((- M) (#) S) . f is ext-real Element of ExtREAL
max ((((- M) (#) S) . f),0) is ext-real set
R_EAL (- M) is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(R_EAL (- M)) * (S . f) is ext-real Element of ExtREAL
max (((R_EAL (- M)) * (S . f)),0) is ext-real set
(R_EAL M) * (S . f) is ext-real Element of ExtREAL
- ((R_EAL M) * (S . f)) is ext-real Element of ExtREAL
max ((- ((R_EAL M) * (S . f))),0) is ext-real set
(max- S) . f is ext-real Element of ExtREAL
(R_EAL M) * ((max- S) . f) is ext-real Element of ExtREAL
- (S . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
max ((- (S . f)),(R_EAL 0)) is ext-real set
(R_EAL M) * (max ((- (S . f)),(R_EAL 0))) is ext-real set
(R_EAL M) * (- (S . f)) is ext-real Element of ExtREAL
(R_EAL M) * (R_EAL 0) is ext-real Element of ExtREAL
max (((R_EAL M) * (- (S . f))),((R_EAL M) * (R_EAL 0))) is ext-real set
M * 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
M * 0 is complex real ext-real set
max ((- ((R_EAL M) * (S . f))),(M * 0)) is ext-real set
dom (max- ((- M) (#) S)) is Element of bool X
dom (max+ S) is Element of bool X
dom (M (#) (max+ S)) is Element of bool X
f is Element of X
(max- ((- M) (#) S)) . f is ext-real Element of ExtREAL
(M (#) (max+ S)) . f is ext-real Element of ExtREAL
((- M) (#) S) . f is ext-real Element of ExtREAL
- (((- M) (#) S) . f) is ext-real Element of ExtREAL
max ((- (((- M) (#) S) . f)),0) is ext-real set
R_EAL (- M) is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(R_EAL (- M)) * (S . f) is ext-real Element of ExtREAL
- ((R_EAL (- M)) * (S . f)) is ext-real Element of ExtREAL
max ((- ((R_EAL (- M)) * (S . f))),0) is ext-real set
- (- (R_EAL M)) is ext-real Element of ExtREAL
(- (- (R_EAL M))) * (S . f) is ext-real Element of ExtREAL
max (((- (- (R_EAL M))) * (S . f)),0) is ext-real set
(max+ S) . f is ext-real Element of ExtREAL
(R_EAL M) * ((max+ S) . f) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
max ((S . f),(R_EAL 0)) is ext-real set
(R_EAL M) * (max ((S . f),(R_EAL 0))) is ext-real set
(R_EAL M) * (S . f) is ext-real Element of ExtREAL
M * 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
M * 0 is complex real ext-real set
max (((R_EAL M) * (S . f)),(M * 0)) is ext-real set
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is set
S | M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (S | M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ S) | M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- (S | M) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- S) | M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ (S | M)) is Element of bool X
bool X is non empty set
dom (S | M) is Element of bool X
dom S is Element of bool X
(dom S) /\ M is Element of bool X
dom (max+ S) is Element of bool X
(dom (max+ S)) /\ M is Element of bool X
dom ((max+ S) | M) is Element of bool X
f is Element of X
(max+ (S | M)) . f is ext-real Element of ExtREAL
((max+ S) | M) . f is ext-real Element of ExtREAL
(max+ S) . f is ext-real Element of ExtREAL
(S | M) . f is ext-real Element of ExtREAL
max (((S | M) . f),0) is ext-real set
S . f is ext-real Element of ExtREAL
max ((S . f),0) is ext-real set
dom (max- (S | M)) is Element of bool X
dom (max- S) is Element of bool X
(dom (max- S)) /\ M is Element of bool X
dom ((max- S) | M) is Element of bool X
f is Element of X
(max- (S | M)) . f is ext-real Element of ExtREAL
((max- S) | M) . f is ext-real Element of ExtREAL
(max- S) . f is ext-real Element of ExtREAL
(S | M) . f is ext-real Element of ExtREAL
- ((S | M) . f) is ext-real Element of ExtREAL
max ((- ((S | M) . f)),0) is ext-real set
S . f is ext-real Element of ExtREAL
- (S . f) is ext-real Element of ExtREAL
max ((- (S . f)),0) is ext-real set
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
S + M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (S + M) is Element of bool X
bool X is non empty set
f is set
(S + M) | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((S + M) | f) is Element of bool X
S | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(S | f) + (M | f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((S | f) + (M | f)) is Element of bool X
dom M is Element of bool X
c is set
M . c is ext-real Element of ExtREAL
[:(dom M),ExtREAL:] is V59() set
bool [:(dom M),ExtREAL:] is non empty set
dom (M | f) is Element of bool X
B is set
(M | f) . B is ext-real Element of ExtREAL
[:(dom (M | f)),ExtREAL:] is V59() set
bool [:(dom (M | f)),ExtREAL:] is non empty set
x is set
M " {+infty} is Element of bool X
(M " {+infty}) /\ f is Element of bool X
c is Relation-like dom M -defined ExtREAL -valued Function-like V32( dom M, ExtREAL ) V59() Element of bool [:(dom M),ExtREAL:]
dom c is Element of bool (dom M)
bool (dom M) is non empty set
(dom c) /\ f is Element of bool (dom M)
c | f is Relation-like dom M -defined ExtREAL -valued Function-like V59() Element of bool [:(dom M),ExtREAL:]
dom (c | f) is Element of bool (dom M)
c . x is ext-real Element of ExtREAL
B is Relation-like dom (M | f) -defined ExtREAL -valued Function-like V32( dom (M | f), ExtREAL ) V59() Element of bool [:(dom (M | f)),ExtREAL:]
B . x is ext-real Element of ExtREAL
(M | f) " {+infty} is Element of bool X
x is set
dom B is Element of bool (dom (M | f))
bool (dom (M | f)) is non empty set
(dom M) /\ f is Element of bool X
B . x is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
c " {+infty} is Element of bool (dom M)
x is set
M " {-infty} is Element of bool X
(M " {-infty}) /\ f is Element of bool X
c . x is ext-real Element of ExtREAL
B . x is ext-real Element of ExtREAL
(M | f) " {-infty} is Element of bool X
x is set
B . x is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
c " {-infty} is Element of bool (dom M)
dom S is Element of bool X
x is set
S . x is ext-real Element of ExtREAL
[:(dom S),ExtREAL:] is V59() set
bool [:(dom S),ExtREAL:] is non empty set
dom (S | f) is Element of bool X
I1 is set
(S | f) . I1 is ext-real Element of ExtREAL
[:(dom (S | f)),ExtREAL:] is V59() set
bool [:(dom (S | f)),ExtREAL:] is non empty set
a is set
S " {+infty} is Element of bool X
(S " {+infty}) /\ f is Element of bool X
x is Relation-like dom S -defined ExtREAL -valued Function-like V32( dom S, ExtREAL ) V59() Element of bool [:(dom S),ExtREAL:]
dom x is Element of bool (dom S)
bool (dom S) is non empty set
(dom x) /\ f is Element of bool (dom S)
x | f is Relation-like dom S -defined ExtREAL -valued Function-like V59() Element of bool [:(dom S),ExtREAL:]
dom (x | f) is Element of bool (dom S)
x . a is ext-real Element of ExtREAL
I1 is Relation-like dom (S | f) -defined ExtREAL -valued Function-like V32( dom (S | f), ExtREAL ) V59() Element of bool [:(dom (S | f)),ExtREAL:]
I1 . a is ext-real Element of ExtREAL
(S | f) " {+infty} is Element of bool X
a is set
S " {-infty} is Element of bool X
(S " {-infty}) /\ f is Element of bool X
x . a is ext-real Element of ExtREAL
I1 . a is ext-real Element of ExtREAL
(S | f) " {-infty} is Element of bool X
a is set
dom I1 is Element of bool (dom (S | f))
bool (dom (S | f)) is non empty set
(dom S) /\ f is Element of bool X
I1 . a is ext-real Element of ExtREAL
S . a is ext-real Element of ExtREAL
x " {-infty} is Element of bool (dom S)
((S | f) " {-infty}) /\ ((M | f) " {+infty}) is Element of bool X
((S " {-infty}) /\ f) /\ (M " {+infty}) is Element of bool X
(((S " {-infty}) /\ f) /\ (M " {+infty})) /\ f is Element of bool X
(S " {-infty}) /\ (M " {+infty}) is Element of bool X
((S " {-infty}) /\ (M " {+infty})) /\ f is Element of bool X
(((S " {-infty}) /\ (M " {+infty})) /\ f) /\ f is Element of bool X
f /\ f is set
((S " {-infty}) /\ (M " {+infty})) /\ (f /\ f) is Element of bool X
a is set
I1 . a is ext-real Element of ExtREAL
S . a is ext-real Element of ExtREAL
x " {+infty} is Element of bool (dom S)
((S | f) " {+infty}) /\ ((M | f) " {-infty}) is Element of bool X
((S " {+infty}) /\ f) /\ (M " {-infty}) is Element of bool X
(((S " {+infty}) /\ f) /\ (M " {-infty})) /\ f is Element of bool X
(S " {+infty}) /\ (M " {-infty}) is Element of bool X
((S " {+infty}) /\ (M " {-infty})) /\ f is Element of bool X
(((S " {+infty}) /\ (M " {-infty})) /\ f) /\ f is Element of bool X
((S " {+infty}) /\ (M " {-infty})) /\ (f /\ f) is Element of bool X
(((S | f) " {-infty}) /\ ((M | f) " {+infty})) \/ (((S | f) " {+infty}) /\ ((M | f) " {-infty})) is Element of bool X
((S " {-infty}) /\ (M " {+infty})) \/ ((S " {+infty}) /\ (M " {-infty})) is Element of bool X
(((S " {-infty}) /\ (M " {+infty})) \/ ((S " {+infty}) /\ (M " {-infty}))) /\ f is Element of bool X
(dom (S | f)) /\ (dom (M | f)) is Element of bool X
((dom S) /\ f) /\ (dom (M | f)) is Element of bool X
((dom S) /\ f) /\ ((dom M) /\ f) is Element of bool X
((dom S) /\ f) /\ (dom M) is Element of bool X
(((dom S) /\ f) /\ (dom M)) /\ f is Element of bool X
(dom S) /\ (dom M) is Element of bool X
((dom S) /\ (dom M)) /\ f is Element of bool X
(((dom S) /\ (dom M)) /\ f) /\ f is Element of bool X
((dom S) /\ (dom M)) /\ (f /\ f) is Element of bool X
(((dom S) /\ (dom M)) /\ f) \ ((((S | f) " {-infty}) /\ ((M | f) " {+infty})) \/ (((S | f) " {+infty}) /\ ((M | f) " {-infty}))) is Element of bool X
((dom S) /\ (dom M)) \ (((S " {-infty}) /\ (M " {+infty})) \/ ((S " {+infty}) /\ (M " {-infty}))) is Element of bool X
(((dom S) /\ (dom M)) \ (((S " {-infty}) /\ (M " {+infty})) \/ ((S " {+infty}) /\ (M " {-infty})))) /\ f is Element of bool X
(dom (S + M)) /\ f is Element of bool X
a is set
((S + M) | f) . a is ext-real Element of ExtREAL
(S + M) . a is ext-real Element of ExtREAL
S . a is ext-real Element of ExtREAL
M . a is ext-real Element of ExtREAL
(S . a) + (M . a) is ext-real Element of ExtREAL
(S | f) . a is ext-real Element of ExtREAL
((S | f) . a) + (M . a) is ext-real Element of ExtREAL
(M | f) . a is ext-real Element of ExtREAL
((S | f) . a) + ((M | f) . a) is ext-real Element of ExtREAL
((S | f) + (M | f)) . a is ext-real Element of ExtREAL
X is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M is ext-real Element of ExtREAL
eq_dom (S,M) is Element of bool X
bool X is non empty set
{M} is non empty finite ext-real-membered left_end right_end set
S " {M} is Element of bool X
f is set
S . f is ext-real Element of ExtREAL
dom S is Element of bool X
f is set
S . 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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M + f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
B is complex real ext-real set
R_EAL B is ext-real Element of ExtREAL
less_dom ((M + f),(R_EAL B)) is Element of bool X
c /\ (less_dom ((M + f),(R_EAL B))) is Element of bool X
[:RAT,S:] is non empty set
bool [:RAT,S:] is non empty set
x is complex real ext-real Element of REAL
I1 is Relation-like RAT -defined S -valued Function-like V32( RAT ,S) Element of bool [:RAT,S:]
[:NAT,S:] is non empty set
bool [:NAT,S:] is non empty set
rng I1 is Element of bool S
bool S is non empty set
a is Relation-like NAT -defined S -valued Function-like V32( NAT ,S) Element of bool [:NAT,S:]
rng a is non empty Element of bool (bool X)
R_EAL x is ext-real Element of ExtREAL
less_dom ((M + f),(R_EAL x)) is Element of bool X
c /\ (less_dom ((M + f),(R_EAL x))) is Element of bool X
union (rng I1) is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is set
f . c is ext-real Element of ExtREAL
c is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
B is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
B . 1 is ext-real Element of ExtREAL
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * c is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg (len c) is finite V44( len c) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len c ) } is set
dom c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Sum x is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
I1 . (len x) is ext-real Element of ExtREAL
I1 . 0 is ext-real Element of ExtREAL
rng c is finite Element of bool S
bool S is non empty set
union (rng c) is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c . a is set
f1 is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is ext-real Element of ExtREAL
a + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
I1 . (a + 1) is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
I1 . (f1 + 1) is ext-real Element of ExtREAL
I1 . f1 is ext-real Element of ExtREAL
x . (f1 + 1) is ext-real Element of ExtREAL
(I1 . f1) + (x . (f1 + 1)) is ext-real Element of ExtREAL
c . (f1 + 1) is set
M . (c . (f1 + 1)) is ext-real Element of ExtREAL
(M * c) . (f1 + 1) is ext-real Element of ExtREAL
B . (f1 + 1) is ext-real Element of ExtREAL
(B . (f1 + 1)) * ((M * c) . (f1 + 1)) is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . a is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . f1 is ext-real Element of ExtREAL
B . f1 is ext-real Element of ExtREAL
(M * c) . f1 is ext-real Element of ExtREAL
(B . f1) * ((M * c) . f1) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom M is Element of bool X
f is Element of S
c is complex real ext-real Element of REAL
R_EAL c is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL c)) is Element of bool X
f /\ (great_eq_dom (M,(R_EAL c))) is Element of bool X
B is complex real ext-real Element of REAL
R_EAL B is ext-real Element of ExtREAL
less_dom (M,(R_EAL B)) is Element of bool X
(f /\ (great_eq_dom (M,(R_EAL c)))) /\ (less_dom (M,(R_EAL B))) is Element of bool X
f /\ (less_dom (M,(R_EAL B))) is Element of bool X
(f /\ (great_eq_dom (M,(R_EAL c)))) /\ (f /\ (less_dom (M,(R_EAL B)))) is Element of bool X
(f /\ (great_eq_dom (M,(R_EAL c)))) /\ f is Element of bool X
((f /\ (great_eq_dom (M,(R_EAL c)))) /\ f) /\ (less_dom (M,(R_EAL B))) is Element of bool X
f /\ f is M13(X,S)
(great_eq_dom (M,(R_EAL c))) /\ (f /\ f) is Element of bool X
((great_eq_dom (M,(R_EAL c))) /\ (f /\ f)) /\ (less_dom (M,(R_EAL B))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng B is finite Element of bool S
bool S is non empty set
union (rng B) is set
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
len B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
rng x is finite set
I1 is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . a is set
Seg (len B) is finite V44( len B) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
B . a is set
(B . a) /\ c is Element of bool X
Seg (len B) is finite V44( len B) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is set
I1 . f1 is set
B . a is set
B . f1 is set
(B . f1) /\ c is Element of bool X
(B . a) /\ c is Element of bool X
(I1 . a) /\ (I1 . f1) is set
((B . a) /\ c) /\ (B . f1) is Element of bool X
(((B . a) /\ c) /\ (B . f1)) /\ c is Element of bool X
(B . a) /\ (B . f1) is set
((B . a) /\ (B . f1)) /\ c is Element of bool X
(((B . a) /\ (B . f1)) /\ c) /\ c is Element of bool X
{} /\ c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool X
({} /\ c) /\ c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool X
a is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng a is finite Element of bool S
union (rng a) is set
dom (f | c) is Element of bool X
f1 is set
E is set
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . x is set
B . x is set
(B . x) /\ c is Element of bool X
(dom f) /\ c is Element of bool X
f1 is set
(dom f) /\ c is Element of bool X
E is set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is set
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
a . x is set
(B . x) /\ c is Element of bool X
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . f1 is set
E is Element of X
x is Element of X
(f | c) . E is ext-real Element of ExtREAL
(f | c) . x is ext-real Element of ExtREAL
B . f1 is set
(B . f1) /\ c is Element of bool X
f . E is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
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 V109() V110() V111() sigma-additive Element of bool (bool X)
M is Element of S
f is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
c is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
rng c is finite set
B is set
x is set
c . x is set
f . x is set
rng f is finite Element of bool S
bool S is non empty set
(f . x) /\ M is Element of bool X
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f . x is set
f . I1 is set
B . I1 is set
(f . I1) /\ M is Element of bool X
B . x is set
(f . x) /\ M 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is Element of S
M | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is set
rng B is finite Element of bool S
bool S is non empty set
union (rng B) is set
a is set
f1 is set
B . f1 is set
c . f1 is set
rng c is finite Element of bool S
(c . f1) /\ f is Element of bool X
union (rng c) is set
dom M is Element of bool X
(dom M) /\ f is Element of bool X
dom (M | f) is Element of bool X
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . I1 is set
x . I1 is ext-real Element of ExtREAL
a is set
M . a is ext-real Element of ExtREAL
c . I1 is set
(c . I1) /\ f is Element of bool X
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . I1 is set
x . I1 is ext-real Element of ExtREAL
a is set
(M | f) . a is ext-real Element of ExtREAL
M . a is ext-real Element of ExtREAL
I1 is set
a is set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c . f1 is set
B . f1 is set
(c . f1) /\ f 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng c is finite Element of bool S
bool S is non empty set
union (rng c) is set
dom c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
X is set
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M is set
rng S is finite set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . f is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
rng B is finite Element of bool S
bool S is non empty set
union (rng B) is set
a is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
f1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len B) * (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg ((len B) * (len a)) is finite V44((len B) * (len a)) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (len B) * (len a) ) } is set
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(NFPG -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((NFPG -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(NFPG -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((NFPG -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
((len B) * (len a)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len B) * (len a)) -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len B) * (len a)) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len B) * (len a)) div (len a)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len B) is finite V44( len B) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len B ) } is set
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
B . (((NFPG -' 1) div (len a)) + 1) is set
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
a . (((NFPG -' 1) mod (len a)) + 1) is set
rng a is finite Element of bool S
x . NFPG is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ (a . (((NFPG -' 1) mod (len a)) + 1)) is set
DFPG is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom DFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . NFPG is set
DFPG . x is set
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(x -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(x -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
NFPG -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(NFPG -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((NFPG -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(NFPG -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((NFPG -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
B . (((NFPG -' 1) div (len a)) + 1) is set
a . (((NFPG -' 1) mod (len a)) + 1) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ (a . (((NFPG -' 1) mod (len a)) + 1)) is set
(x -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((x -' 1) div (len a)) + 1) - 1 is complex real ext-real integer rational Element of REAL
((((x -' 1) div (len a)) + 1) - 1) * (len a) is complex real ext-real integer rational Element of REAL
(((x -' 1) mod (len a)) + 1) - 1 is complex real ext-real integer rational Element of REAL
(((((x -' 1) div (len a)) + 1) - 1) * (len a)) + ((((x -' 1) mod (len a)) + 1) - 1) is complex real ext-real integer rational Element of REAL
((((((x -' 1) div (len a)) + 1) - 1) * (len a)) + ((((x -' 1) mod (len a)) + 1) - 1)) + 1 is complex real ext-real integer rational Element of REAL
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) + 1 is complex real ext-real integer rational Element of REAL
(((((x -' 1) div (len a)) + 1) - 1) * (len a)) + (((x -' 1) mod (len a)) + 1) is complex real ext-real integer rational Element of REAL
(NFPG -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((NFPG -' 1) div (len a)) + 1) - 1 is complex real ext-real integer rational Element of REAL
((((NFPG -' 1) div (len a)) + 1) - 1) * (len a) is complex real ext-real integer rational Element of REAL
(((NFPG -' 1) mod (len a)) + 1) - 1 is complex real ext-real integer rational Element of REAL
(((((NFPG -' 1) div (len a)) + 1) - 1) * (len a)) + ((((NFPG -' 1) mod (len a)) + 1) - 1) is complex real ext-real integer rational Element of REAL
((((((NFPG -' 1) div (len a)) + 1) - 1) * (len a)) + ((((NFPG -' 1) mod (len a)) + 1) - 1)) + 1 is complex real ext-real integer rational Element of REAL
NFPG - 1 is complex real ext-real integer rational Element of REAL
(NFPG - 1) + 1 is complex real ext-real integer rational Element of REAL
(((((NFPG -' 1) div (len a)) + 1) - 1) * (len a)) + (((NFPG -' 1) mod (len a)) + 1) is complex real ext-real integer rational Element of REAL
(DFPG . NFPG) /\ (DFPG . x) is set
B . (((x -' 1) div (len a)) + 1) is set
a . (((x -' 1) mod (len a)) + 1) is set
(B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)) is set
((B . (((NFPG -' 1) div (len a)) + 1)) /\ (a . (((NFPG -' 1) mod (len a)) + 1))) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(a . (((NFPG -' 1) mod (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)))) is set
(a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)) is set
(B . (((x -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ (B . (((x -' 1) div (len a)) + 1)) is set
((B . (((NFPG -' 1) div (len a)) + 1)) /\ (B . (((x -' 1) div (len a)) + 1))) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
{} /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded set
(DFPG . NFPG) /\ (DFPG . x) is set
B . (((x -' 1) div (len a)) + 1) is set
a . (((x -' 1) mod (len a)) + 1) is set
(B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)) is set
((B . (((NFPG -' 1) div (len a)) + 1)) /\ (a . (((NFPG -' 1) mod (len a)) + 1))) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(a . (((NFPG -' 1) mod (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)))) is set
(a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)) is set
(B . (((x -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ ((B . (((x -' 1) div (len a)) + 1)) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)))) is set
(B . (((NFPG -' 1) div (len a)) + 1)) /\ (B . (((x -' 1) div (len a)) + 1)) is set
((B . (((NFPG -' 1) div (len a)) + 1)) /\ (B . (((x -' 1) div (len a)) + 1))) /\ ((a . (((NFPG -' 1) mod (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1))) is set
((B . (((NFPG -' 1) div (len a)) + 1)) /\ (B . (((x -' 1) div (len a)) + 1))) /\ {} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded set
(dom f) /\ (dom c) is Element of bool X
union (rng a) is set
NFPG is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng NFPG is finite Element of bool S
union (rng NFPG) is set
x is set
ff is set
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . KB is set
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
1 + KAB is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
n is set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . m is set
Seg (len a) is finite V44( len a) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len a ) } is set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
1 + n is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
KAB * (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(KAB * (len a)) + m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB - 1 is complex real ext-real integer rational Element of REAL
(KB - 1) * (len a) is complex real ext-real integer rational Element of REAL
((KB - 1) * (len a)) + m is complex real ext-real integer rational Element of REAL
(len B) - 1 is complex real ext-real integer rational Element of REAL
((len B) - 1) * (len a) is complex real ext-real integer rational Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len B) - 1) * (len a)) + m is complex real ext-real integer rational Element of REAL
0 + m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m - 1 is complex real ext-real integer rational Element of REAL
(KAB * (len a)) + n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((m -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((len B) - 1) * (len a)) + (len a) is complex real ext-real integer rational Element of REAL
dom NFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
NFPG . m is set
(m -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((m -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(B . KB) /\ (a . m) is set
ff is set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB is set
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG . KAB is set
len NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len NFPG) is finite V44( len NFPG) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len NFPG ) } is set
KAB -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(KAB -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KAB -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(KAB -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KAB -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((len B) * (len a)) -' 1) div x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
B . (((KAB -' 1) div (len a)) + 1) is set
a . (((KAB -' 1) mod (len a)) + 1) is set
(B . (((KAB -' 1) div (len a)) + 1)) /\ (a . (((KAB -' 1) mod (len a)) + 1)) is set
dom NFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG . x is set
ff is Element of X
KB is Element of X
(f + c) . ff is ext-real Element of ExtREAL
(f + c) . KB is ext-real Element of ExtREAL
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(x -' 1) mod (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) mod (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(x -' 1) div (len a) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) div (len a)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
B . (((x -' 1) div (len a)) + 1) is set
a . (((x -' 1) mod (len a)) + 1) is set
(B . (((x -' 1) div (len a)) + 1)) /\ (a . (((x -' 1) mod (len a)) + 1)) is set
len NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len NFPG) is finite V44( len NFPG) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len NFPG ) } is set
f . ff is ext-real Element of ExtREAL
x . (((x -' 1) div (len a)) + 1) is ext-real Element of ExtREAL
Seg (len a) is finite V44( len a) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len a ) } is set
f . KB is ext-real Element of ExtREAL
c . ff is ext-real Element of ExtREAL
(f . ff) + (c . ff) is ext-real Element of ExtREAL
f1 . (((x -' 1) mod (len a)) + 1) is ext-real Element of ExtREAL
(x . (((x -' 1) div (len a)) + 1)) + (f1 . (((x -' 1) mod (len a)) + 1)) is ext-real Element of ExtREAL
c . KB is ext-real Element of ExtREAL
(f . KB) + (c . KB) is ext-real Element of ExtREAL
x is Element of X
c . x is ext-real Element of ExtREAL
|.(c . x).| is ext-real Element of ExtREAL
(f + c) . x is ext-real Element of ExtREAL
|.((f + c) . x).| is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
(f . x) + (c . x) is ext-real Element of ExtREAL
|.((f . x) + (c . x)).| is ext-real Element of ExtREAL
|.(f . x).| is ext-real Element of ExtREAL
|.(f . x).| + |.(c . x).| 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
M + f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (M + f) is Element of bool X
c is Element of S
<*c*> is Relation-like NAT -defined S -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like FinSequence of S
[1,c] is set
{1,c} is non empty finite set
{{1,c},{1}} is non empty finite V41() set
{[1,c]} is non empty finite set
dom <*c*> is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
<*c*> . x is set
<*c*> . I1 is set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
<*c*> . x is set
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
union (bool {}) is finite set
I1 is Element of X
(M + f) . I1 is ext-real Element of ExtREAL
|.((M + f) . I1).| is ext-real Element of ExtREAL
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is Element of X
x . I1 is set
f1 is Element of X
(M + f) . a is ext-real Element of ExtREAL
(M + f) . f1 is ext-real Element of ExtREAL
dom (M + f) is Element of bool X
M | (dom (M + f)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(M | (dom (M + f))) " {+infty} is Element of bool X
M " {+infty} is Element of bool X
(dom (M + f)) /\ (M " {+infty}) is Element of bool X
rng f is ext-real-membered Element of bool ExtREAL
f " {+infty} is Element of bool X
f | (dom (M + f)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | (dom (M + f))) " {+infty} is Element of bool X
(dom (M + f)) /\ (f " {+infty}) is Element of bool X
rng M is ext-real-membered Element of bool ExtREAL
dom M is Element of bool X
dom f is Element of bool X
(dom M) /\ (dom f) is Element of bool X
f " {-infty} is Element of bool X
(M " {+infty}) /\ (f " {-infty}) is Element of bool X
M " {-infty} is Element of bool X
(M " {-infty}) /\ (f " {+infty}) is Element of bool X
((M " {+infty}) /\ (f " {-infty})) \/ ((M " {-infty}) /\ (f " {+infty})) is Element of bool X
((dom M) /\ (dom f)) \ (((M " {+infty}) /\ (f " {-infty})) \/ ((M " {-infty}) /\ (f " {+infty}))) is Element of bool X
dom (M | (dom (M + f))) is Element of bool X
(dom M) /\ (dom (M + f)) is Element of bool X
(dom M) /\ (dom M) is Element of bool X
((dom M) /\ (dom M)) /\ (dom f) is Element of bool X
dom (f | (dom (M + f))) is Element of bool X
(dom f) /\ (dom (M + f)) is Element of bool X
(dom f) /\ (dom f) is Element of bool X
((dom f) /\ (dom f)) /\ (dom M) is Element of bool X
(M | (dom (M + f))) + (f | (dom (M + f))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((M | (dom (M + f))) + (f | (dom (M + f)))) is Element of bool X
(dom (M | (dom (M + f)))) /\ (dom (f | (dom (M + f)))) is Element of bool X
(f | (dom (M + f))) " {-infty} is Element of bool X
((M | (dom (M + f))) " {+infty}) /\ ((f | (dom (M + f))) " {-infty}) is Element of bool X
(M | (dom (M + f))) " {-infty} is Element of bool X
((M | (dom (M + f))) " {-infty}) /\ ((f | (dom (M + f))) " {+infty}) is Element of bool X
(((M | (dom (M + f))) " {+infty}) /\ ((f | (dom (M + f))) " {-infty})) \/ (((M | (dom (M + f))) " {-infty}) /\ ((f | (dom (M + f))) " {+infty})) is Element of bool X
((dom (M | (dom (M + f)))) /\ (dom (f | (dom (M + f))))) \ ((((M | (dom (M + f))) " {+infty}) /\ ((f | (dom (M + f))) " {-infty})) \/ (((M | (dom (M + f))) " {-infty}) /\ ((f | (dom (M + f))) " {+infty}))) is Element of bool X
c is Element of X
((M | (dom (M + f))) + (f | (dom (M + f)))) . c is ext-real Element of ExtREAL
(M + f) . c is ext-real Element of ExtREAL
(M | (dom (M + f))) . c is ext-real Element of ExtREAL
(f | (dom (M + f))) . c is ext-real Element of ExtREAL
((M | (dom (M + f))) . c) + ((f | (dom (M + f))) . c) is ext-real Element of ExtREAL
M . c is ext-real Element of ExtREAL
(M . c) + ((f | (dom (M + f))) . c) is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
(M . c) + (f . c) is ext-real Element of ExtREAL
dom (M + f) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is complex real ext-real Element of REAL
c (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is Element of X
dom (c (#) f) is Element of bool X
R_EAL c is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
(R_EAL c) * (f . I1) is ext-real Element of ExtREAL
(c (#) f) . I1 is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
|.((c (#) f) . I1).| is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is Element of X
x . I1 is set
f1 is Element of X
(c (#) f) . f1 is ext-real Element of ExtREAL
f . f1 is ext-real Element of ExtREAL
(R_EAL c) * (f . f1) is ext-real Element of ExtREAL
(c (#) f) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(R_EAL c) * (f . a) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - c) is Element of bool X
rng c is ext-real-membered Element of bool ExtREAL
c " {-infty} is Element of bool X
rng f is ext-real-membered Element of bool ExtREAL
f " {+infty} is Element of bool X
dom f is Element of bool X
dom c is Element of bool X
(dom f) /\ (dom c) is Element of bool X
c " {+infty} is Element of bool X
(f " {+infty}) /\ (c " {+infty}) is Element of bool X
f " {-infty} is Element of bool X
(f " {-infty}) /\ (c " {-infty}) is Element of bool X
((f " {+infty}) /\ (c " {+infty})) \/ ((f " {-infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {+infty}) /\ (c " {+infty})) \/ ((f " {-infty}) /\ (c " {-infty}))) is Element of bool X
B is set
c . B is ext-real Element of ExtREAL
f . B 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Element of S
c is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
B is set
B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom B is Element of bool X
|.c.| is ext-real Element of ExtREAL
x is Element of X
B . x is ext-real Element of ExtREAL
|.(B . x).| is ext-real Element of ExtREAL
x is set
<*(dom B)*> is Relation-like NAT -defined bool X -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like FinSequence of bool X
[1,(dom B)] is set
{1,(dom B)} is non empty finite set
{{1,(dom B)},{1}} is non empty finite V41() set
{[1,(dom B)]} is non empty finite set
I1 is set
rng <*(dom B)*> is finite Element of bool (bool X)
{(dom B)} is non empty finite set
{f} is non empty finite set
I1 is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is set
I1 . f1 is set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
E is Element of X
a . f1 is set
x is Element of X
B . E is ext-real Element of ExtREAL
B . x is ext-real Element of ExtREAL
rng a is finite Element of bool S
bool S is non empty set
union (rng a) is set
f1 is set
B . f1 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
(dom f) /\ c is Element of bool X
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
I1 is set
dom (f | c) is Element of bool X
(f | c) . I1 is ext-real Element of ExtREAL
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
a is ext-real Element of ExtREAL
f1 is ext-real Element of ExtREAL
less_dom ((f | c),(R_EAL x)) is Element of bool X
a is ext-real Element of ExtREAL
f1 is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
B /\ (less_dom ((f | c),(R_EAL x))) is Element of bool X
less_dom (f,(R_EAL x)) is Element of bool X
c /\ (less_dom (f,(R_EAL x))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Element of S
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c + B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (c + B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ (c + B)) + (max- c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Element of S
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom B is Element of bool X
(dom c) /\ (dom B) is Element of bool X
c + B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- (c + B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- (c + B)) + (max+ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c + B) is Element of bool X
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 V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is set
M . f is ext-real Element of ExtREAL
c is Element of S
M . c is ext-real Element of ExtREAL
B is Element of S
M . B 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is complex real ext-real Element of REAL
B is Element of S
<*B*> is Relation-like NAT -defined S -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like FinSequence of S
[1,B] is set
{1,B} is non empty finite set
{{1,B},{1}} is non empty finite V41() set
{[1,B]} is non empty finite set
dom <*B*> is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
<*B*> . I1 is set
<*B*> . a is set
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
<*B*> . I1 is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
f1 is Element of X
I1 . a is set
E is Element of X
f . f1 is ext-real Element of ExtREAL
R_EAL c is ext-real Element of ExtREAL
f . E is ext-real Element of ExtREAL
I1 . 1 is set
<*(I1 . 1)*> is Relation-like NAT -defined non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like set
[1,(I1 . 1)] is set
{1,(I1 . 1)} is non empty finite set
{{1,(I1 . 1)},{1}} is non empty finite V41() set
{[1,(I1 . 1)]} is non empty finite set
rng I1 is finite Element of bool S
bool S is non empty set
{(I1 . 1)} is non empty finite set
a is Element of X
f . a is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
|.(f . a).| is ext-real Element of ExtREAL
union (rng I1) is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f " {+infty} is Element of bool X
f " {-infty} is Element of bool X
c " {+infty} is Element of bool X
c " {-infty} is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
X \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
B is Element of S
B is Element of S
x is Element of S
x is Element of S
x /\ B is M13(X,S)
(x /\ B) /\ (X \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})))) is Element of bool X
(x /\ B) /\ X is Element of bool X
((x /\ B) /\ X) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
(x /\ B) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Element of S
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is complex real ext-real set
R_EAL c is ext-real Element of ExtREAL
less_dom (f,(R_EAL c)) is Element of bool X
M /\ (less_dom (f,(R_EAL c))) is Element of bool X
less_dom ((f | M),(R_EAL c)) is Element of bool X
B is set
f . B is ext-real Element of ExtREAL
(f | M) . B is ext-real Element of ExtREAL
dom f is Element of bool X
M /\ (dom f) is Element of bool X
dom (f | M) is Element of bool X
B is set
(dom f) /\ M is Element of bool X
(f | M) . B is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
x 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Element of S
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
B is complex real ext-real set
R_EAL B is ext-real Element of ExtREAL
less_dom ((c | f),(R_EAL B)) is Element of bool X
f /\ (less_dom ((c | f),(R_EAL B))) is Element of bool X
less_dom (c,(R_EAL B)) is Element of bool X
f /\ (less_dom (c,(R_EAL B))) is Element of bool X
f /\ (f /\ (less_dom (c,(R_EAL B)))) is Element of bool X
f /\ f is M13(X,S)
(f /\ f) /\ (less_dom (c,(R_EAL B))) is Element of bool X
B is complex real ext-real set
R_EAL B is ext-real Element of ExtREAL
less_dom (c,(R_EAL B)) is Element of bool X
(f /\ f) /\ (less_dom (c,(R_EAL B))) is Element of bool X
f /\ (less_dom (c,(R_EAL B))) is Element of bool X
f /\ (f /\ (less_dom (c,(R_EAL B)))) is Element of bool X
less_dom ((c | f),(R_EAL B)) is Element of bool X
f /\ (less_dom ((c | f),(R_EAL B))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is Element of S
B is Element of S
x is set
c " {-infty} is Element of bool X
c . x is ext-real Element of ExtREAL
eq_dom (c,-infty) is Element of bool X
B /\ (eq_dom (c,-infty)) is Element of bool X
x is set
c . x is ext-real Element of ExtREAL
x is set
c " {+infty} is Element of bool X
c . x is ext-real Element of ExtREAL
eq_dom (c,+infty) is Element of bool X
B /\ (eq_dom (c,+infty)) is Element of bool X
x is set
c . x is ext-real Element of ExtREAL
f " {+infty} is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
f " {-infty} is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
(dom f) /\ (dom c) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
(c " {+infty}) \/ (c " {-infty}) is Element of bool X
I1 is Element of S
I1 is Element of S
eq_dom (f,+infty) is Element of bool X
I1 /\ (eq_dom (f,+infty)) is Element of bool X
a is set
f . a is ext-real Element of ExtREAL
a is set
f . a is ext-real Element of ExtREAL
a is set
eq_dom (f,-infty) is Element of bool X
I1 /\ (eq_dom (f,-infty)) is Element of bool X
f . a is ext-real Element of ExtREAL
a is set
f . a is ext-real Element of ExtREAL
(f " {+infty}) \/ (f " {-infty}) is Element of bool X
a is Element of S
x is Element of S
a \/ x is M13(X,S)
f1 is Element of S
I1 \ f1 is M13(X,S)
DFPG is Element of S
NFPG is Element of S
E is Element of S
c | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(dom f) /\ E is Element of bool X
(f + c) | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | E) + (c | E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | E) + (c | E)) is Element of bool X
x is Element of S
x \ (c " {+infty}) is Element of bool X
(f " {-infty}) /\ (x \ (c " {+infty})) is Element of bool X
x \ (f " {+infty}) is Element of bool X
(c " {-infty}) /\ (x \ (f " {+infty})) is Element of bool X
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
less_dom ((f + c),(R_EAL x)) is Element of bool X
x /\ (less_dom ((f + c),(R_EAL x))) is Element of bool X
less_dom (((f | E) + (c | E)),(R_EAL x)) is Element of bool X
E /\ (less_dom (((f | E) + (c | E)),(R_EAL x))) is Element of bool X
(E /\ (less_dom (((f | E) + (c | E)),(R_EAL x)))) \/ ((f " {-infty}) /\ (x \ (c " {+infty}))) is Element of bool X
((E /\ (less_dom (((f | E) + (c | E)),(R_EAL x)))) \/ ((f " {-infty}) /\ (x \ (c " {+infty})))) \/ ((c " {-infty}) /\ (x \ (f " {+infty}))) is Element of bool X
KAB is set
((f | E) + (c | E)) . KAB is ext-real Element of ExtREAL
(f + c) . KAB is ext-real Element of ExtREAL
c . KAB is ext-real Element of ExtREAL
f . KAB is ext-real Element of ExtREAL
(f . KAB) + (c . KAB) is ext-real Element of ExtREAL
(f + c) . KAB is ext-real Element of ExtREAL
c . KAB is ext-real Element of ExtREAL
f . KAB is ext-real Element of ExtREAL
(f . KAB) + (c . KAB) is ext-real Element of ExtREAL
(f + c) . KAB is ext-real Element of ExtREAL
KAB is set
((f " {+infty}) /\ (c " {-infty})) \/ ((f " {-infty}) /\ (c " {+infty})) is Element of bool X
(f + c) . KAB is ext-real Element of ExtREAL
f . KAB is ext-real Element of ExtREAL
c . KAB is ext-real Element of ExtREAL
(f . KAB) + (c . KAB) is ext-real Element of ExtREAL
(f " {-infty}) \/ (f " {+infty}) is Element of bool X
((f " {-infty}) \/ (f " {+infty})) \/ (c " {-infty}) is Element of bool X
(((f " {-infty}) \/ (f " {+infty})) \/ (c " {-infty})) \/ (c " {+infty}) is Element of bool X
((f | E) + (c | E)) . KAB is ext-real Element of ExtREAL
ff is set
f . ff is ext-real Element of ExtREAL
[:(dom f),ExtREAL:] is V59() set
bool [:(dom f),ExtREAL:] is non empty set
x is set
dom (f | E) is Element of bool X
(f | E) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
ff is Relation-like dom f -defined ExtREAL -valued Function-like V32( dom f, ExtREAL ) V59() Element of bool [:(dom f),ExtREAL:]
ff . x is ext-real Element of ExtREAL
ff " {-infty} is Element of bool (dom f)
bool (dom f) is non empty set
ff " {+infty} is Element of bool (dom f)
- +infty is non empty ext-real non positive negative Element of ExtREAL
x is Element of X
(f | E) . x is ext-real Element of ExtREAL
|.((f | E) . x).| is ext-real Element of ExtREAL
ff is set
c . ff is ext-real Element of ExtREAL
[:(dom c),ExtREAL:] is V59() set
bool [:(dom c),ExtREAL:] is non empty set
x is set
dom (c | E) is Element of bool X
(dom c) /\ E is Element of bool X
(c | E) . x is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
ff is Relation-like dom c -defined ExtREAL -valued Function-like V32( dom c, ExtREAL ) V59() Element of bool [:(dom c),ExtREAL:]
ff . x is ext-real Element of ExtREAL
ff " {-infty} is Element of bool (dom c)
bool (dom c) is non empty set
ff " {+infty} is Element of bool (dom c)
x is Element of X
(c | E) . x is ext-real Element of ExtREAL
|.((c | E) . x).| is ext-real Element of ExtREAL
x is Element of S
x is Element of S
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
less_dom (((f | E) + (c | E)),(R_EAL x)) is Element of bool X
E /\ (less_dom (((f | E) + (c | E)),(R_EAL x))) is Element of bool X
((f " {-infty}) /\ (x \ (c " {+infty}))) \/ ((c " {-infty}) /\ (x \ (f " {+infty}))) is Element of bool X
(E /\ (less_dom (((f | E) + (c | E)),(R_EAL x)))) \/ (((f " {-infty}) /\ (x \ (c " {+infty}))) \/ ((c " {-infty}) /\ (x \ (f " {+infty})))) is Element of bool X
less_dom ((f + c),(R_EAL x)) is Element of bool X
x /\ (less_dom ((f + c),(R_EAL x))) is Element of bool X
(E /\ (less_dom (((f | E) + (c | E)),(R_EAL x)))) \/ ((f " {-infty}) /\ (x \ (c " {+infty}))) is Element of bool X
((E /\ (less_dom (((f | E) + (c | E)),(R_EAL x)))) \/ ((f " {-infty}) /\ (x \ (c " {+infty})))) \/ ((c " {-infty}) /\ (x \ (f " {+infty}))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is Element of S
B is Element of S
x is Element of S
x is Element of S
B /\ x is M13(X,S)
c | (B /\ x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(c | (B /\ x)) " {-infty} is Element of bool X
c " {-infty} is Element of bool X
(B /\ x) /\ (c " {-infty}) is Element of bool X
f | (B /\ x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (B /\ x)) is Element of bool X
(dom f) /\ (B /\ x) is Element of bool X
(c | (B /\ x)) " {+infty} is Element of bool X
c " {+infty} is Element of bool X
(B /\ x) /\ (c " {+infty}) is Element of bool X
dom (c | (B /\ x)) is Element of bool X
(dom c) /\ (B /\ x) is Element of bool X
(f | (B /\ x)) " {+infty} is Element of bool X
f " {+infty} is Element of bool X
(B /\ x) /\ (f " {+infty}) is Element of bool X
((f | (B /\ x)) " {+infty}) /\ ((c | (B /\ x)) " {-infty}) is Element of bool X
(B /\ x) /\ ((B /\ x) /\ (c " {-infty})) is Element of bool X
(f " {+infty}) /\ ((B /\ x) /\ ((B /\ x) /\ (c " {-infty}))) is Element of bool X
(B /\ x) /\ (B /\ x) is M13(X,S)
((B /\ x) /\ (B /\ x)) /\ (c " {-infty}) is Element of bool X
(f " {+infty}) /\ (((B /\ x) /\ (B /\ x)) /\ (c " {-infty})) is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
((f " {+infty}) /\ (c " {-infty})) /\ (B /\ x) is Element of bool X
(f | (B /\ x)) + (c | (B /\ x)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (B /\ x)) + (c | (B /\ x))) is Element of bool X
(dom (f | (B /\ x))) /\ (dom (c | (B /\ x))) is Element of bool X
(f | (B /\ x)) " {-infty} is Element of bool X
((f | (B /\ x)) " {-infty}) /\ ((c | (B /\ x)) " {+infty}) is Element of bool X
(((f | (B /\ x)) " {-infty}) /\ ((c | (B /\ x)) " {+infty})) \/ (((f | (B /\ x)) " {+infty}) /\ ((c | (B /\ x)) " {-infty})) is Element of bool X
((dom (f | (B /\ x))) /\ (dom (c | (B /\ x)))) \ ((((f | (B /\ x)) " {-infty}) /\ ((c | (B /\ x)) " {+infty})) \/ (((f | (B /\ x)) " {+infty}) /\ ((c | (B /\ x)) " {-infty}))) is Element of bool X
E is Element of S
(f + c) | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f + c) | E) is Element of bool X
(dom (f + c)) /\ E is Element of bool X
f " {-infty} is Element of bool X
(B /\ x) /\ (f " {-infty}) is Element of bool X
(B /\ x) /\ ((B /\ x) /\ (c " {+infty})) is Element of bool X
(f " {-infty}) /\ ((B /\ x) /\ ((B /\ x) /\ (c " {+infty}))) is Element of bool X
((B /\ x) /\ (B /\ x)) /\ (c " {+infty}) is Element of bool X
(f " {-infty}) /\ (((B /\ x) /\ (B /\ x)) /\ (c " {+infty})) is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) /\ (B /\ x) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
(B /\ x) /\ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
(dom f) /\ (dom c) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
x is Element of X
((f + c) | E) . x is ext-real Element of ExtREAL
(f + c) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
(f . x) + (c . x) is ext-real Element of ExtREAL
((f | (B /\ x)) + (c | (B /\ x))) . x is ext-real Element of ExtREAL
(f | (B /\ x)) . x is ext-real Element of ExtREAL
(c | (B /\ x)) . x is ext-real Element of ExtREAL
((f | (B /\ x)) . x) + ((c | (B /\ x)) . x) is ext-real Element of ExtREAL
(f . x) + ((c | (B /\ x)) . x) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
c /\ B is M13(X,S)
I1 is set
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
less_dom (f,(R_EAL x)) is Element of bool X
c /\ (less_dom (f,(R_EAL x))) is Element of bool X
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
less_dom (f,(R_EAL x)) is Element of bool X
(c /\ B) /\ (less_dom (f,(R_EAL x))) is Element of bool X
c /\ (less_dom (f,(R_EAL x))) is Element of bool X
B /\ (c /\ (less_dom (f,(R_EAL x)))) is Element of bool X
B /\ (less_dom (f,(R_EAL x))) is Element of bool X
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
less_dom (f,(R_EAL x)) is Element of bool X
(c /\ B) /\ (less_dom (f,(R_EAL x))) is Element of bool X
c /\ (less_dom (f,(R_EAL x))) is Element of bool X
B /\ (c /\ (less_dom (f,(R_EAL x)))) is Element of bool X
B /\ (less_dom (f,(R_EAL x))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
c is Element of S
B is complex real ext-real Element of REAL
B (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of S
c /\ x is M13(X,S)
dom (B (#) f) is Element of bool X
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
R_EAL (- 1) is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
R_EAL 1 is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (f,c) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (f,c)) is ext-real Element of ExtREAL
S is complex real ext-real set
R_EAL S is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
(X . f) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . f) + (- (R_EAL S)) is ext-real set
|.((X . f) - (R_EAL S)).| is ext-real Element of ExtREAL
S + S is complex real ext-real set
S + S is complex real ext-real set
R_EAL (S + S) is ext-real Element of ExtREAL
2 * S is complex real ext-real Element of REAL
R_EAL (2 * S) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (c,B) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (c,B)) is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (c,B) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (c,B)) is ext-real Element of ExtREAL
(X . (max (c,B))) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . (max (c,B))) + (- (R_EAL S)) is ext-real set
|.((X . (max (c,B))) - (R_EAL S)).| is ext-real Element of ExtREAL
(R_EAL 1) + (R_EAL S) is ext-real Element of ExtREAL
- S is complex real ext-real set
- S is complex real ext-real set
R_EAL (- S) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . c is ext-real Element of ExtREAL
(X . c) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . c) + (- (R_EAL S)) is ext-real set
|.((X . c) - (R_EAL S)).| is ext-real Element of ExtREAL
(- S) + S is complex real ext-real set
(- S) + S is complex real ext-real set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (B,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (B,x)) is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
R_EAL 1 is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
R_EAL (- 1) is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (f,c) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (f,c)) is ext-real Element of ExtREAL
S is complex real ext-real set
R_EAL S is ext-real Element of ExtREAL
- S is complex real ext-real set
- S is complex real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . c is ext-real Element of ExtREAL
(X . c) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . c) + (- (R_EAL S)) is ext-real set
|.((X . c) - (R_EAL S)).| is ext-real Element of ExtREAL
- (R_EAL S) is ext-real Element of ExtREAL
(- S) + S is complex real ext-real set
(- S) + S is complex real ext-real set
M is complex real ext-real Element of REAL
R_EAL M is ext-real Element of ExtREAL
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (B,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (B,x)) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (B,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (B,x)) is ext-real Element of ExtREAL
(X . (max (B,x))) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . (max (B,x))) + (- (R_EAL S)) is ext-real set
|.((X . (max (B,x))) - (R_EAL S)).| is ext-real Element of ExtREAL
- (R_EAL 1) is ext-real Element of ExtREAL
(- (R_EAL 1)) + (R_EAL S) is ext-real Element of ExtREAL
M is complex real ext-real Element of REAL
R_EAL M is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . c is ext-real Element of ExtREAL
(X . c) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . c) + (- (R_EAL S)) is ext-real set
|.((X . c) - (R_EAL S)).| is ext-real Element of ExtREAL
- (R_EAL M) is ext-real Element of ExtREAL
(- (R_EAL M)) + (R_EAL S) is ext-real Element of ExtREAL
- M is complex real ext-real Element of REAL
(- M) + S is complex real ext-real Element of REAL
R_EAL ((- M) + S) is ext-real Element of ExtREAL
2 * S is complex real ext-real Element of REAL
R_EAL (2 * S) is ext-real Element of ExtREAL
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (B,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (B,x)) is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S is complex real ext-real set
R_EAL S is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
M is ext-real Element of ExtREAL
f is complex real ext-real set
f is complex real ext-real set
c is complex real ext-real set
c is complex real ext-real set
B is complex set
x is complex set
B - x is complex set
|.(B - x).| is complex real ext-real non negative Element of REAL
|.(B - x).| / 2 is complex real ext-real non negative Element of REAL
R_EAL (|.(B - x).| / 2) is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (f1,E) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . (max (f1,E)) is ext-real Element of ExtREAL
(X . (max (f1,E))) - S is ext-real Element of ExtREAL
- S is ext-real set
(X . (max (f1,E))) + (- S) is ext-real set
|.((X . (max (f1,E))) - S).| is ext-real Element of ExtREAL
(X . (max (f1,E))) - M is ext-real Element of ExtREAL
- M is ext-real set
(X . (max (f1,E))) + (- M) is ext-real set
|.((X . (max (f1,E))) - M).| is ext-real Element of ExtREAL
DFPG is complex real ext-real Element of REAL
NFPG is complex real ext-real Element of REAL
DFPG - NFPG is complex real ext-real Element of REAL
|.(DFPG - NFPG).| is complex real ext-real non negative Element of REAL
|.(DFPG - NFPG).| / 2 is complex real ext-real non negative Element of REAL
R_EAL (|.(DFPG - NFPG).| / 2) is ext-real Element of ExtREAL
- (R_EAL (|.(DFPG - NFPG).| / 2)) is ext-real Element of ExtREAL
- (X . (max (f1,E))) is ext-real Element of ExtREAL
(- (X . (max (f1,E)))) + S is ext-real Element of ExtREAL
- ((X . (max (f1,E))) - S) is ext-real Element of ExtREAL
|.((- (X . (max (f1,E)))) + S).| is ext-real Element of ExtREAL
S - M is ext-real Element of ExtREAL
S + (- M) is ext-real set
|.(S - M).| is ext-real Element of ExtREAL
S + 0. is ext-real Element of ExtREAL
(S + 0.) - M is ext-real Element of ExtREAL
(S + 0.) + (- M) is ext-real set
|.((S + 0.) - M).| is ext-real Element of ExtREAL
(X . (max (f1,E))) + (- (X . (max (f1,E)))) is ext-real Element of ExtREAL
S + ((X . (max (f1,E))) + (- (X . (max (f1,E))))) is ext-real Element of ExtREAL
(S + ((X . (max (f1,E))) + (- (X . (max (f1,E)))))) - M is ext-real Element of ExtREAL
(S + ((X . (max (f1,E))) + (- (X . (max (f1,E)))))) + (- M) is ext-real set
|.((S + ((X . (max (f1,E))) + (- (X . (max (f1,E)))))) - M).| is ext-real Element of ExtREAL
((- (X . (max (f1,E)))) + S) + (X . (max (f1,E))) is ext-real Element of ExtREAL
(((- (X . (max (f1,E)))) + S) + (X . (max (f1,E)))) - M is ext-real Element of ExtREAL
(((- (X . (max (f1,E)))) + S) + (X . (max (f1,E)))) + (- M) is ext-real set
|.((((- (X . (max (f1,E)))) + S) + (X . (max (f1,E)))) - M).| is ext-real Element of ExtREAL
((- (X . (max (f1,E)))) + S) + ((X . (max (f1,E))) - M) is ext-real Element of ExtREAL
|.(((- (X . (max (f1,E)))) + S) + ((X . (max (f1,E))) - M)).| is ext-real Element of ExtREAL
|.((- (X . (max (f1,E)))) + S).| + |.((X . (max (f1,E))) - M).| is ext-real Element of ExtREAL
|.((X . (max (f1,E))) - S).| + |.((X . (max (f1,E))) - M).| is ext-real Element of ExtREAL
(R_EAL (|.(DFPG - NFPG).| / 2)) + (R_EAL (|.(DFPG - NFPG).| / 2)) is ext-real Element of ExtREAL
(|.(DFPG - NFPG).| / 2) + (|.(DFPG - NFPG).| / 2) is complex real ext-real non negative Element of REAL
x is complex real ext-real Element of REAL
|.x.| is complex real ext-real non negative Element of REAL
(R_EAL (|.(DFPG - NFPG).| / 2)) + |.((X . (max (f1,E))) - M).| is ext-real Element of ExtREAL
f is complex real ext-real set
c is complex real ext-real set
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X) is ext-real Element of ExtREAL
S is complex real ext-real set
f is complex real ext-real set
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . B is ext-real Element of ExtREAL
R_EAL S is ext-real Element of ExtREAL
(X . B) - (R_EAL S) is ext-real Element of ExtREAL
- (R_EAL S) is ext-real set
(X . B) + (- (R_EAL S)) is ext-real set
|.((X . B) - (R_EAL S)).| is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Sum X is ext-real Element of ExtREAL
len X is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S . (len X) is ext-real Element of ExtREAL
S . 0 is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . M is ext-real Element of ExtREAL
M + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
S . (M + 1) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg (len X) is finite V44( len X) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
X . (f + 1) is ext-real Element of ExtREAL
S . (f + 1) is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
(S . f) + (X . (f + 1)) is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X) is ext-real Element of ExtREAL
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . M is ext-real Element of ExtREAL
X . S is ext-real Element of ExtREAL
S is ext-real set
M is set
X . M is ext-real Element of ExtREAL
S is ext-real UpperBound of rng X
M is complex real ext-real set
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . c is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . S is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . S is ext-real Element of ExtREAL
f is complex real ext-real set
f is complex real ext-real set
c is ext-real set
R_EAL f is ext-real Element of ExtREAL
B is set
X . B is ext-real Element of ExtREAL
B is complex real ext-real set
x is complex real ext-real Element of REAL
R_EAL x is ext-real Element of ExtREAL
(sup (rng X)) - (R_EAL x) is ext-real Element of ExtREAL
- (R_EAL x) is ext-real set
(sup (rng X)) + (- (R_EAL x)) is ext-real set
I1 is ext-real set
a is set
X . a is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (f1,M) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f1 is ext-real Element of ExtREAL
X . x is ext-real Element of ExtREAL
(X . x) + (R_EAL x) is ext-real Element of ExtREAL
(sup (rng X)) - (X . x) is ext-real Element of ExtREAL
- (X . x) is ext-real set
(sup (rng X)) + (- (X . x)) is ext-real set
- ((sup (rng X)) - (X . x)) is ext-real Element of ExtREAL
- (R_EAL x) is ext-real Element of ExtREAL
(X . x) - (sup (rng X)) is ext-real Element of ExtREAL
- (sup (rng X)) is ext-real set
(X . x) + (- (sup (rng X))) is ext-real set
(sup (rng X)) + x is ext-real set
(R_EAL x) + (sup (rng X)) is ext-real Element of ExtREAL
- (sup (rng X)) is ext-real Element of ExtREAL
((R_EAL x) + (sup (rng X))) + (- (sup (rng X))) is ext-real Element of ExtREAL
(sup (rng X)) + (- (sup (rng X))) is ext-real Element of ExtREAL
(R_EAL x) + ((sup (rng X)) + (- (sup (rng X)))) is ext-real Element of ExtREAL
(R_EAL x) + 0 is ext-real set
R_EAL 0 is ext-real Element of ExtREAL
(sup (rng X)) + (R_EAL 0) is ext-real Element of ExtREAL
(sup (rng X)) + (R_EAL x) is ext-real Element of ExtREAL
|.((X . x) - (sup (rng X))).| is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
R_EAL c is ext-real Element of ExtREAL
f is complex real ext-real set
R_EAL f is ext-real Element of ExtREAL
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . c is ext-real Element of ExtREAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . B is ext-real Element of ExtREAL
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . M is ext-real Element of ExtREAL
f is complex real ext-real Element of REAL
max (0,f) is complex real ext-real Element of REAL
(max (0,f)) + 1 is complex real ext-real Element of REAL
R_EAL ((max (0,f)) + 1) is ext-real Element of ExtREAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . B is ext-real Element of ExtREAL
f + 0 is complex real ext-real Element of REAL
f is complex real ext-real set
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . S is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
rng S is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng S) is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S . M is ext-real Element of ExtREAL
X . M is ext-real Element of ExtREAL
M is ext-real set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
f is set
X . f is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
S is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . S is ext-real Element of ExtREAL
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
M is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . M is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
M is ext-real set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
f is set
X . f is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
S is ext-real Element of ExtREAL
M is ext-real set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
f is set
X . f is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
X . 1 is ext-real Element of ExtREAL
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
{-infty,+infty} is non empty finite ext-real-membered left_end right_end set
S is complex real ext-real set
max (S,1) is complex real ext-real set
M is complex real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
R_EAL M is ext-real Element of ExtREAL
S is complex real ext-real set
M is ext-real set
R_EAL S is ext-real Element of ExtREAL
f is set
X . f is ext-real Element of ExtREAL
S is complex real ext-real set
M is complex real ext-real set
R_EAL M is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X) is ext-real Element of ExtREAL
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
S is ext-real set
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
{-infty,+infty} is non empty finite ext-real-membered left_end right_end set
X . 1 is ext-real Element of ExtREAL
M is ext-real set
f is set
X . f is ext-real Element of ExtREAL
c is complex real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . x is ext-real Element of ExtREAL
M is complex real ext-real set
R_EAL M is ext-real Element of ExtREAL
(X . x) - (R_EAL M) is ext-real Element of ExtREAL
- (R_EAL M) is ext-real set
(X . x) + (- (R_EAL M)) is ext-real set
(X . x) - (X . x) is ext-real Element of ExtREAL
- (X . x) is ext-real set
(X . x) + (- (X . x)) is ext-real set
(X . x) - M is ext-real set
- M is complex real ext-real set
- M is complex real ext-real set
(X . x) + (- M) is ext-real set
|.((X . x) - (R_EAL M)).| is ext-real Element of ExtREAL
M is complex real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
M is complex real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
X . 1 is ext-real Element of ExtREAL
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
M is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(M) is ext-real Element of ExtREAL
rng M is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng M) is ext-real Element of ExtREAL
(X) is ext-real Element of ExtREAL
(S) is ext-real Element of ExtREAL
(X) + (S) is ext-real Element of ExtREAL
rng S is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng S) is ext-real Element of ExtREAL
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
(sup (rng S)) + (sup (rng X)) is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
B is complex real ext-real set
B / 2 is complex real ext-real Element of REAL
R_EAL (B / 2) is ext-real Element of ExtREAL
(sup (rng X)) - (R_EAL (B / 2)) is ext-real Element of ExtREAL
- (R_EAL (B / 2)) is ext-real set
(sup (rng X)) + (- (R_EAL (B / 2))) is ext-real set
x is ext-real set
I1 is set
X . I1 is ext-real Element of ExtREAL
(sup (rng S)) - (R_EAL (B / 2)) is ext-real Element of ExtREAL
(sup (rng S)) + (- (R_EAL (B / 2))) is ext-real set
f1 is ext-real set
E is set
S . E is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (a,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(B / 2) + (B / 2) is complex real ext-real Element of REAL
R_EAL B is ext-real Element of ExtREAL
(R_EAL (B / 2)) + (R_EAL (B / 2)) is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff is ext-real Element of ExtREAL
S . KB is ext-real Element of ExtREAL
c is complex real ext-real set
R_EAL c is ext-real Element of ExtREAL
(R_EAL c) - (S . KB) is ext-real Element of ExtREAL
- (S . KB) is ext-real set
(R_EAL c) + (- (S . KB)) is ext-real set
(R_EAL c) - ff is ext-real Element of ExtREAL
- ff is ext-real set
(R_EAL c) + (- ff) is ext-real set
x is ext-real Element of ExtREAL
X . KB is ext-real Element of ExtREAL
f is complex real ext-real set
R_EAL f is ext-real Element of ExtREAL
(R_EAL f) - (X . KB) is ext-real Element of ExtREAL
- (X . KB) is ext-real set
(R_EAL f) + (- (X . KB)) is ext-real set
(R_EAL f) - x is ext-real Element of ExtREAL
- x is ext-real set
(R_EAL f) + (- x) is ext-real set
((R_EAL f) - (X . KB)) + ((R_EAL c) - (S . KB)) is ext-real Element of ExtREAL
((R_EAL f) - x) + ((R_EAL c) - ff) is ext-real Element of ExtREAL
M . KB is ext-real Element of ExtREAL
f + c is complex real ext-real set
f + c is complex real ext-real set
R_EAL (f + c) is ext-real Element of ExtREAL
(M . KB) - (R_EAL (f + c)) is ext-real Element of ExtREAL
- (R_EAL (f + c)) is ext-real set
(M . KB) + (- (R_EAL (f + c))) is ext-real set
- ((M . KB) - (R_EAL (f + c))) is ext-real Element of ExtREAL
(R_EAL (f + c)) - (M . KB) is ext-real Element of ExtREAL
- (M . KB) is ext-real set
(R_EAL (f + c)) + (- (M . KB)) is ext-real set
(R_EAL (B / 2)) + ff is ext-real Element of ExtREAL
(R_EAL (B / 2)) + ((R_EAL c) - ff) is ext-real Element of ExtREAL
(R_EAL (B / 2)) + x is ext-real Element of ExtREAL
S . x is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S . n is ext-real Element of ExtREAL
(X . KB) + (S . KB) is ext-real Element of ExtREAL
X . n is ext-real Element of ExtREAL
(R_EAL f) + (R_EAL c) is ext-real Element of ExtREAL
|.((M . KB) - (R_EAL (f + c))).| is ext-real Element of ExtREAL
KAB is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
KAB - m is complex real ext-real Element of REAL
((R_EAL f) - (X . KB)) + (R_EAL c) is ext-real Element of ExtREAL
(((R_EAL f) - (X . KB)) + (R_EAL c)) - (S . KB) is ext-real Element of ExtREAL
(((R_EAL f) - (X . KB)) + (R_EAL c)) + (- (S . KB)) is ext-real set
(R_EAL c) + (R_EAL f) is ext-real Element of ExtREAL
((R_EAL c) + (R_EAL f)) - (X . KB) is ext-real Element of ExtREAL
((R_EAL c) + (R_EAL f)) + (- (X . KB)) is ext-real set
(((R_EAL c) + (R_EAL f)) - (X . KB)) - (S . KB) is ext-real Element of ExtREAL
(((R_EAL c) + (R_EAL f)) - (X . KB)) + (- (S . KB)) is ext-real set
c + f is complex real ext-real set
c + f is complex real ext-real set
R_EAL (c + f) is ext-real Element of ExtREAL
(R_EAL (c + f)) - ((X . KB) + (S . KB)) is ext-real Element of ExtREAL
- ((X . KB) + (S . KB)) is ext-real set
(R_EAL (c + f)) + (- ((X . KB) + (S . KB))) is ext-real set
(R_EAL (c + f)) - (M . KB) is ext-real Element of ExtREAL
(R_EAL (c + f)) + (- (M . KB)) is ext-real set
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
B is complex real ext-real set
B / 2 is complex real ext-real Element of REAL
R_EAL (B / 2) is ext-real Element of ExtREAL
(sup (rng X)) - (R_EAL (B / 2)) is ext-real Element of ExtREAL
- (R_EAL (B / 2)) is ext-real set
(sup (rng X)) + (- (R_EAL (B / 2))) is ext-real set
a is ext-real set
f1 is set
X . f1 is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
x is complex real ext-real Element of REAL
c - x is complex real ext-real Element of REAL
I1 is complex real ext-real Element of REAL
I1 - (c - x) is complex real ext-real Element of REAL
R_EAL B is ext-real Element of ExtREAL
(R_EAL B) - ((sup (rng X)) - (R_EAL (B / 2))) is ext-real Element of ExtREAL
- ((sup (rng X)) - (R_EAL (B / 2))) is ext-real set
(R_EAL B) + (- ((sup (rng X)) - (R_EAL (B / 2)))) is ext-real set
E is ext-real Element of ExtREAL
(I1 - (c - x)) + (c - x) is complex real ext-real Element of REAL
((R_EAL B) - ((sup (rng X)) - (R_EAL (B / 2)))) + ((sup (rng X)) - (R_EAL (B / 2))) is ext-real Element of ExtREAL
DFPG is set
S . DFPG is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (x,NFPG) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . x is ext-real Element of ExtREAL
X . (max (x,NFPG)) is ext-real Element of ExtREAL
S . NFPG is ext-real Element of ExtREAL
S . (max (x,NFPG)) is ext-real Element of ExtREAL
(X . (max (x,NFPG))) + (S . (max (x,NFPG))) is ext-real Element of ExtREAL
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . ff is ext-real Element of ExtREAL
X . ff is ext-real Element of ExtREAL
(X . ff) + (S . ff) is ext-real Element of ExtREAL
M . ff is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
f is complex real ext-real set
f / 2 is complex real ext-real Element of REAL
c is complex real ext-real Element of REAL
B is complex real ext-real Element of REAL
c - B is complex real ext-real Element of REAL
R_EAL (f / 2) is ext-real Element of ExtREAL
(sup (rng S)) - (R_EAL (f / 2)) is ext-real Element of ExtREAL
- (R_EAL (f / 2)) is ext-real set
(sup (rng S)) + (- (R_EAL (f / 2))) is ext-real set
x is complex real ext-real Element of REAL
x - (c - B) is complex real ext-real Element of REAL
R_EAL f is ext-real Element of ExtREAL
(R_EAL f) - ((sup (rng S)) - (R_EAL (f / 2))) is ext-real Element of ExtREAL
- ((sup (rng S)) - (R_EAL (f / 2))) is ext-real set
(R_EAL f) + (- ((sup (rng S)) - (R_EAL (f / 2)))) is ext-real set
I1 is ext-real Element of ExtREAL
(x - (c - B)) + (c - B) is complex real ext-real Element of REAL
((R_EAL f) - ((sup (rng S)) - (R_EAL (f / 2)))) + ((sup (rng S)) - (R_EAL (f / 2))) is ext-real Element of ExtREAL
a is ext-real set
f1 is set
S . f1 is ext-real Element of ExtREAL
E is set
X . E is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (x,DFPG) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . DFPG is ext-real Element of ExtREAL
X . (max (x,DFPG)) is ext-real Element of ExtREAL
S . x is ext-real Element of ExtREAL
S . (max (x,DFPG)) is ext-real Element of ExtREAL
(X . (max (x,DFPG))) + (S . (max (x,DFPG))) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . x is ext-real Element of ExtREAL
X . x is ext-real Element of ExtREAL
(X . x) + (S . x) is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
f / 2 is complex real ext-real Element of REAL
R_EAL (f / 2) is ext-real Element of ExtREAL
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . c is ext-real Element of ExtREAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . B is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (x,I1) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(f / 2) + (f / 2) is complex real ext-real Element of REAL
S . x is ext-real Element of ExtREAL
S . (max (x,I1)) is ext-real Element of ExtREAL
X . I1 is ext-real Element of ExtREAL
X . (max (x,I1)) is ext-real Element of ExtREAL
(X . (max (x,I1))) + (S . (max (x,I1))) is ext-real Element of ExtREAL
(R_EAL (f / 2)) + (R_EAL (f / 2)) is ext-real Element of ExtREAL
M . (max (x,I1)) is ext-real Element of ExtREAL
R_EAL f is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . f1 is ext-real Element of ExtREAL
X . f1 is ext-real Element of ExtREAL
(X . f1) + (S . f1) is ext-real Element of ExtREAL
M . f1 is ext-real Element of ExtREAL
S . 0 is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
(sup (rng X)) + (sup (rng S)) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . f is ext-real Element of ExtREAL
S . c is ext-real Element of ExtREAL
X . f is ext-real Element of ExtREAL
X . c is ext-real Element of ExtREAL
(X . f) + (S . f) is ext-real Element of ExtREAL
(X . c) + (S . c) is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
M . c is ext-real Element of ExtREAL
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng S is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng S) is ext-real Element of ExtREAL
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
M is complex real ext-real Element of REAL
R_EAL M is ext-real Element of ExtREAL
(R_EAL M) * (sup (rng X)) is ext-real Element of ExtREAL
f is ext-real UpperBound of rng S
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
S . 1 is ext-real Element of ExtREAL
c is ext-real Element of ExtREAL
X . 1 is ext-real Element of ExtREAL
M * (X . 1) is ext-real set
B is ext-real set
x is set
X . x is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S . I1 is ext-real Element of ExtREAL
X . I1 is ext-real Element of ExtREAL
(R_EAL M) * (X . I1) is ext-real Element of ExtREAL
((R_EAL M) * (X . I1)) / (R_EAL M) is ext-real Element of ExtREAL
(R_EAL M) " is ext-real set
((R_EAL M) * (X . I1)) * ((R_EAL M) ") is ext-real set
c is ext-real Element of ExtREAL
c / (R_EAL M) is ext-real Element of ExtREAL
c * ((R_EAL M) ") is ext-real set
S . 1 is ext-real Element of ExtREAL
X . 1 is ext-real Element of ExtREAL
(R_EAL M) * (X . 1) is ext-real Element of ExtREAL
B is complex real ext-real Element of REAL
B / M is complex real ext-real Element of REAL
x is complex real ext-real Element of REAL
x * M is complex real ext-real Element of REAL
f is ext-real set
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
c is set
S . c is ext-real Element of ExtREAL
dom X is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
X . B is ext-real Element of ExtREAL
(R_EAL M) * (X . B) is ext-real Element of ExtREAL
M * (sup (rng X)) is ext-real set
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
S . f is ext-real Element of ExtREAL
X . f is ext-real Element of ExtREAL
M * (X . f) is ext-real set
(S) is ext-real Element of ExtREAL
M * (sup (rng X)) is ext-real set
f is set
X . f is ext-real Element of ExtREAL
(R_EAL M) * (X . f) is ext-real Element of ExtREAL
-infty * (R_EAL M) is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S . c is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
S . f is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
f is complex real ext-real set
f / M is complex real ext-real Element of REAL
R_EAL (f / M) is ext-real Element of ExtREAL
(R_EAL (f / M)) * (R_EAL M) is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
c / M is complex real ext-real Element of REAL
(c / M) * M is complex real ext-real Element of REAL
M / M is complex real ext-real Element of REAL
c / (M / M) is complex real ext-real Element of REAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . B is ext-real Element of ExtREAL
(R_EAL M) * (X . B) is ext-real Element of ExtREAL
S . B is ext-real Element of ExtREAL
R_EAL f is ext-real Element of ExtREAL
M * (sup (rng X)) is ext-real set
M * (sup (rng X)) is ext-real set
M * (sup (rng X)) is ext-real set
M * (sup (rng X)) is ext-real set
M * (sup (rng X)) is ext-real set
f is set
X . f is ext-real Element of ExtREAL
(R_EAL M) * (X . f) is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
S . c is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
S . f is ext-real Element of ExtREAL
dom S is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
X is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
S is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(S) is ext-real Element of ExtREAL
rng S is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng S) is ext-real Element of ExtREAL
(X) is ext-real Element of ExtREAL
M is complex real ext-real Element of REAL
R_EAL M is ext-real Element of ExtREAL
(R_EAL M) * (X) is ext-real Element of ExtREAL
rng X is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng X) is ext-real Element of ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
X . f is ext-real Element of ExtREAL
(R_EAL M) * (X . f) is ext-real Element of ExtREAL
X . c is ext-real Element of ExtREAL
(R_EAL M) * (X . c) is ext-real Element of ExtREAL
S . f is ext-real Element of ExtREAL
S . c 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 set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
M is Element of X
f is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f . c is ext-real Element of ExtREAL
S . c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
(S . c) . M is ext-real Element of ExtREAL
f is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
c is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f . B is ext-real Element of ExtREAL
S . B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
(S . B) . M is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
X is set
S is set
PFuncs (X,S) is non empty functional PartFunc-set of X,S
[:NAT,(PFuncs (X,S)):] is non empty set
bool [:NAT,(PFuncs (X,S)):] is non empty set
M is Relation-like NAT -defined PFuncs (X,S) -valued Function-like V32( NAT , PFuncs (X,S)) Element of bool [:NAT,(PFuncs (X,S)):]
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
M . f is set
[:X,S:] is set
bool [:X,S:] is non empty set
dom M is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
M . f is Relation-like X -defined S -valued Function-like Element of bool [:X,S:]
rng M is Element of bool (PFuncs (X,S))
bool (PFuncs (X,S)) is non empty set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
M is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom M is Element of bool X
PFuncs (X,ExtREAL) is non empty functional PartFunc-set of X, ExtREAL
f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ f is complex real ext-real Element of REAL
(2 |^ f) * f is complex real ext-real Element of REAL
2 |^ f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ f) * f is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
c is set
M . c is ext-real Element of ExtREAL
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B - 1 is complex real ext-real integer rational Element of REAL
(B - 1) / (2 |^ f) is complex real ext-real rational Element of REAL
B / (2 |^ f) is complex real ext-real non negative rational Element of REAL
x is complex real ext-real rational Element of REAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 - 1 is complex real ext-real integer rational Element of REAL
(I1 - 1) / (2 |^ f) is complex real ext-real rational Element of REAL
I1 / (2 |^ f) is complex real ext-real non negative rational Element of REAL
I1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
B + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 - 1 is complex real ext-real integer rational Element of REAL
(I1 - 1) / (2 |^ f) is complex real ext-real rational Element of REAL
I1 / (2 |^ f) is complex real ext-real non negative rational Element of REAL
B is complex real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) / (2 |^ f) is complex real ext-real rational Element of REAL
x / (2 |^ f) is complex real ext-real non negative rational Element of REAL
c is Relation-like Function-like set
dom c is set
B is set
rng c is set
x is set
c . x is set
R_EAL f is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
R_EAL f is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 - 1 is complex real ext-real integer rational Element of REAL
(I1 - 1) / (2 |^ f) is complex real ext-real rational Element of REAL
I1 / (2 |^ f) is complex real ext-real non negative rational Element of REAL
R_EAL f is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
[:(dom M),ExtREAL:] is V59() set
bool [:(dom M),ExtREAL:] is non empty set
B is Relation-like dom M -defined ExtREAL -valued Function-like V59() Element of bool [:(dom M),ExtREAL:]
x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of PFuncs (X,ExtREAL)
dom I1 is Element of bool X
a is Element of X
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 - 1 is complex real ext-real integer rational Element of REAL
(f1 - 1) / (2 |^ f) is complex real ext-real Element of REAL
M . a is ext-real Element of ExtREAL
f1 / (2 |^ f) is complex real ext-real Element of REAL
I1 . a is ext-real Element of ExtREAL
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
f is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
c is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,f,c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,f,c) is Element of bool X
B is Element of S
B is Element of S
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
E is Element of X
(X,ExtREAL,c,x) . E is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) . E is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL f1 is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL a is ext-real Element of ExtREAL
(X,ExtREAL,c,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,f1) . E is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL f1 is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
2 |^ f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ f1) * f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) / (2 |^ f1) is complex real ext-real rational Element of REAL
x / (2 |^ f1) is complex real ext-real non negative rational Element of REAL
(X,ExtREAL,c,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,f1) . E is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG - 1 is complex real ext-real integer rational Element of REAL
(DFPG - 1) / (2 |^ f1) is complex real ext-real rational Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL a is ext-real Element of ExtREAL
DFPG / (2 |^ f1) is complex real ext-real non negative rational Element of REAL
(2 |^ f1) * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(NFPG * a) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,a) . E is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL a is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a + NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ a) * a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB - 1 is complex real ext-real integer rational Element of REAL
(KB - 1) / (2 |^ a) is complex real ext-real rational Element of REAL
KB / (2 |^ a) is complex real ext-real non negative rational Element of REAL
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,a) . E is ext-real Element of ExtREAL
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KAB - 1 is complex real ext-real integer rational Element of REAL
(KAB - 1) / (2 |^ a) is complex real ext-real rational Element of REAL
a + x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ (a + x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG / (2 |^ (a + x)) is complex real ext-real non negative rational Element of REAL
R_EAL (DFPG / (2 |^ (a + x))) is ext-real Element of ExtREAL
R_EAL ((KAB - 1) / (2 |^ a)) is ext-real Element of ExtREAL
(2 |^ a) * (2 |^ x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG / ((2 |^ a) * (2 |^ x)) is complex real ext-real non negative rational Element of REAL
DFPG / (2 |^ x) is complex real ext-real non negative rational Element of REAL
(DFPG / (2 |^ x)) / (2 |^ a) is complex real ext-real non negative rational Element of REAL
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
ff * (KAB - 1) is complex real ext-real integer rational Element of REAL
(ff * (KAB - 1)) + 1 is complex real ext-real integer rational Element of REAL
(ff * (KAB - 1)) / (2 |^ (a + x)) is complex real ext-real rational Element of REAL
(DFPG - 1) / (2 |^ (a + x)) is complex real ext-real rational Element of REAL
(ff * (KAB - 1)) / ((2 |^ a) * (2 |^ x)) is complex real ext-real rational Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL a is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL f1 is ext-real Element of ExtREAL
M . E is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
a * I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(a * I1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
R_EAL I1 is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL I1)) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL I1))) is Element of bool X
f1 is Element of X
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,I1) is Element of bool X
M . f1 is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) . f1 is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
|.((X,ExtREAL,c,I1) . f1).| is ext-real Element of ExtREAL
M . f1 is ext-real Element of ExtREAL
E is complex real ext-real Element of REAL
(2 |^ I1) * E is complex real ext-real Element of REAL
[\((2 |^ I1) * E)/] is complex real ext-real integer rational set
[\((2 |^ I1) * E)/] + 1 is complex real ext-real integer rational Element of REAL
((2 |^ I1) * E) - 1 is complex real ext-real Element of REAL
(2 |^ I1) * I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG - 1 is complex real ext-real integer rational Element of REAL
(DFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
DFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
(X,ExtREAL,c,I1) . f1 is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
|.((X,ExtREAL,c,I1) . f1).| is ext-real Element of ExtREAL
M . f1 is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
Seg ((a * I1) + 1) is non empty finite V44((a * I1) + 1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (a * I1) + 1 ) } is set
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E - 1 is complex real ext-real integer rational Element of REAL
(E - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((E - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1))))) is Element of bool X
E / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (E / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (E / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (E / (2 |^ I1))))) is Element of bool X
DFPG is Element of S
NFPG is Element of S
f1 - 1 is complex real ext-real integer rational Element of REAL
(f1 - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((f1 - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1))))) is Element of bool X
f1 / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (f1 / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (f1 / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (f1 / (2 |^ I1))))) is Element of bool X
DFPG is Element of S
NFPG is Element of S
f1 - 1 is complex real ext-real integer rational Element of REAL
(f1 - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((f1 - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1))))) is Element of bool X
f1 / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (f1 / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (f1 / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((f1 - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (f1 / (2 |^ I1))))) is Element of bool X
f1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(2 |^ I1) * I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f1 . E is set
E - 1 is complex real ext-real integer rational Element of REAL
(E - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((E - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1))))) is Element of bool X
E / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (E / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (E / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((E - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (E / (2 |^ I1))))) is Element of bool X
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((2 |^ I1) * I1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
E is set
x is set
f1 . E is set
f1 . x is set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f1 . x is set
f1 . E is set
DFPG - 1 is complex real ext-real integer rational Element of REAL
(DFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((DFPG - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1))))) is Element of bool X
DFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (DFPG / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (DFPG / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (DFPG / (2 |^ I1))))) is Element of bool X
(f1 . E) /\ (f1 . x) is set
x is set
x is set
M . x is ext-real Element of ExtREAL
NFPG - 1 is complex real ext-real integer rational Element of REAL
(NFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((NFPG - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1))))) is Element of bool X
NFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (NFPG / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (NFPG / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (NFPG / (2 |^ I1))))) is Element of bool X
x is set
x is set
M . x is ext-real Element of ExtREAL
DFPG + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f1 . x is set
NFPG - 1 is complex real ext-real integer rational Element of REAL
(NFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((NFPG - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1))))) is Element of bool X
NFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (NFPG / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (NFPG / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((NFPG - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (NFPG / (2 |^ I1))))) is Element of bool X
f1 . E is set
DFPG - 1 is complex real ext-real integer rational Element of REAL
(DFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((DFPG - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1))))) is Element of bool X
DFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (DFPG / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (DFPG / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (DFPG / (2 |^ I1))))) is Element of bool X
(f1 . E) /\ (f1 . x) is set
x is set
x is set
M . x is ext-real Element of ExtREAL
NFPG + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
x is set
x is set
M . x is ext-real Element of ExtREAL
f1 . E is set
f1 . x is set
f1 . E is set
f1 . x is set
E is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom E is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is set
DFPG is Element of X
NFPG is Element of X
(X,ExtREAL,c,I1) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) . NFPG is ext-real Element of ExtREAL
M . DFPG is ext-real Element of ExtREAL
M . NFPG is ext-real Element of ExtREAL
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((x - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1))))) is Element of bool X
x / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (x / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (x / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (x / (2 |^ I1))))) is Element of bool X
M . DFPG is ext-real Element of ExtREAL
M . NFPG is ext-real Element of ExtREAL
rng E is finite Element of bool S
bool S is non empty set
union (rng E) is set
x is set
M . x is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG - 1 is complex real ext-real integer rational Element of REAL
(DFPG - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
DFPG / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL ((DFPG - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1))))) is Element of bool X
R_EAL (DFPG / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (DFPG / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((DFPG - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (DFPG / (2 |^ I1))))) is Element of bool X
E . DFPG is set
M . x is ext-real Element of ExtREAL
E . ((a * I1) + 1) is set
M . x is ext-real Element of ExtREAL
DFPG is set
x is set
DFPG is set
NFPG is set
E . NFPG is set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
R_EAL ((x - 1) / (2 |^ I1)) is ext-real Element of ExtREAL
great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1)))) is Element of bool X
B /\ (great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1))))) is Element of bool X
x / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
R_EAL (x / (2 |^ I1)) is ext-real Element of ExtREAL
less_dom (M,(R_EAL (x / (2 |^ I1)))) is Element of bool X
(B /\ (great_eq_dom (M,(R_EAL ((x - 1) / (2 |^ I1)))))) /\ (less_dom (M,(R_EAL (x / (2 |^ I1))))) is Element of bool X
x is Element of X
M . x is ext-real Element of ExtREAL
|.(M . x).| is ext-real Element of ExtREAL
- (M . x) is ext-real Element of ExtREAL
(X,c,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
I1 is complex real ext-real set
[/I1\] is complex real ext-real integer rational set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,x) . f1 is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL E is ext-real Element of ExtREAL
(X,ExtREAL,c,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,E) . x is ext-real Element of ExtREAL
((X,c,x)) is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
|.(M . x).| is ext-real Element of ExtREAL
(X,c,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
I1 is complex real ext-real Element of REAL
[/I1\] is complex real ext-real integer rational set
[/I1\] + 1 is complex real ext-real integer rational Element of REAL
f1 is complex real ext-real set
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
2 |^ x is complex real ext-real Element of REAL
1 / (2 |^ x) is complex real ext-real Element of REAL
(X,c,x) . x is ext-real Element of ExtREAL
((X,c,x) . x) - (M . x) is ext-real Element of ExtREAL
- (M . x) is ext-real set
((X,c,x) . x) + (- (M . x)) is ext-real set
|.(((X,c,x) . x) - (M . x)).| is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ DFPG) * DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG - 1 is complex real ext-real integer rational Element of REAL
(NFPG - 1) / (2 |^ DFPG) is complex real ext-real rational Element of REAL
NFPG / (2 |^ DFPG) is complex real ext-real non negative rational Element of REAL
(X,c,x) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,c,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,DFPG) . x is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x - 1 is complex real ext-real integer rational Element of REAL
(x - 1) / (2 |^ DFPG) is complex real ext-real rational Element of REAL
x / (2 |^ DFPG) is complex real ext-real non negative rational Element of REAL
R_EAL (x / (2 |^ DFPG)) is ext-real Element of ExtREAL
R_EAL ((x - 1) / (2 |^ DFPG)) is ext-real Element of ExtREAL
(R_EAL (x / (2 |^ DFPG))) - (R_EAL ((x - 1) / (2 |^ DFPG))) is ext-real Element of ExtREAL
- (R_EAL ((x - 1) / (2 |^ DFPG))) is ext-real set
(R_EAL (x / (2 |^ DFPG))) + (- (R_EAL ((x - 1) / (2 |^ DFPG)))) is ext-real set
(x / (2 |^ DFPG)) - ((x - 1) / (2 |^ DFPG)) is complex real ext-real rational Element of REAL
- ((x - 1) / (2 |^ DFPG)) is complex real ext-real rational Element of REAL
(x / (2 |^ DFPG)) + (- ((x - 1) / (2 |^ DFPG))) is complex real ext-real rational Element of REAL
- (x - 1) is complex real ext-real integer rational Element of REAL
(- (x - 1)) / (2 |^ DFPG) is complex real ext-real rational Element of REAL
(x / (2 |^ DFPG)) + ((- (x - 1)) / (2 |^ DFPG)) is complex real ext-real rational Element of REAL
x + (- (x - 1)) is complex real ext-real integer rational Element of REAL
(x + (- (x - 1))) / (2 |^ DFPG) is complex real ext-real rational Element of REAL
1 / (2 |^ DFPG) is complex real ext-real non negative rational Element of REAL
(M . x) - ((X,c,x) . DFPG) is ext-real Element of ExtREAL
- ((X,c,x) . DFPG) is ext-real set
(M . x) + (- ((X,c,x) . DFPG)) is ext-real set
((X,c,x) . DFPG) - (M . x) is ext-real Element of ExtREAL
((X,c,x) . DFPG) + (- (M . x)) is ext-real set
- (((X,c,x) . DFPG) - (M . x)) is ext-real Element of ExtREAL
|.(((X,c,x) . DFPG) - (M . x)).| is ext-real Element of ExtREAL
|.((M . x) - ((X,c,x) . DFPG)).| is ext-real Element of ExtREAL
R_EAL (1 / (2 |^ DFPG)) is ext-real Element of ExtREAL
- (R_EAL (1 / (2 |^ DFPG))) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
1 / (2 |^ x) is complex real ext-real non negative rational Element of REAL
max (E,x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
2 |^ NFPG is complex real ext-real Element of REAL
1 / (2 |^ NFPG) is complex real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x + x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
2 |^ x is complex real ext-real Element of REAL
(2 |^ x) * (2 |^ x) is complex real ext-real Element of REAL
(2 |^ NFPG) " is complex real ext-real Element of REAL
(2 |^ x) " is complex real ext-real non negative rational Element of REAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
2 |^ NFPG is complex real ext-real Element of REAL
1 / (2 |^ NFPG) is complex real ext-real Element of REAL
R_EAL (1 / (2 |^ NFPG)) is ext-real Element of ExtREAL
(X,c,x) . NFPG is ext-real Element of ExtREAL
((X,c,x) . NFPG) - (M . x) is ext-real Element of ExtREAL
- (M . x) is ext-real set
((X,c,x) . NFPG) + (- (M . x)) is ext-real set
|.(((X,c,x) . NFPG) - (M . x)).| is ext-real Element of ExtREAL
I1 is complex real ext-real Element of REAL
R_EAL I1 is ext-real Element of ExtREAL
((X,c,x)) is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
|.(M . x).| is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
a is set
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,I1) is Element of bool X
M . a is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) . a is ext-real Element of ExtREAL
M . a is ext-real Element of ExtREAL
2 |^ I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(2 |^ I1) * I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 - 1 is complex real ext-real integer rational Element of REAL
(f1 - 1) / (2 |^ I1) is complex real ext-real rational Element of REAL
f1 / (2 |^ I1) is complex real ext-real non negative rational Element of REAL
(X,ExtREAL,c,I1) . a is ext-real Element of ExtREAL
M . a is ext-real Element of ExtREAL
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 V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
integral (X,S,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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
dom f is Element of bool X
dom c is Element of bool X
(dom f) /\ (dom c) is Element of bool X
(X,S,M,(f + c)) is ext-real Element of ExtREAL
f | (dom (f + c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (dom (f + c)))) is ext-real Element of ExtREAL
c | (dom (f + c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | (dom (f + c)))) is ext-real Element of ExtREAL
(X,S,M,(f | (dom (f + c)))) + (X,S,M,(c | (dom (f + c)))) is ext-real Element of ExtREAL
dom (f | (dom (f + c))) is Element of bool X
dom (c | (dom (f + c))) is Element of bool X
B is set
(f | (dom (f + c))) . B is ext-real Element of ExtREAL
x is set
(c | (dom (f + c))) . x is ext-real Element of ExtREAL
rng c is ext-real-membered Element of bool ExtREAL
c " {-infty} is Element of bool X
rng f is ext-real-membered Element of bool ExtREAL
f " {-infty} is Element of bool X
c " {+infty} is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
f " {+infty} is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
(dom f) /\ (dom (f + c)) is Element of bool X
(dom f) /\ (dom f) is Element of bool X
((dom f) /\ (dom f)) /\ (dom c) is Element of bool X
(dom c) /\ (dom (f + c)) is Element of bool X
(dom c) /\ (dom c) is Element of bool X
((dom c) /\ (dom c)) /\ (dom f) is Element of bool X
(c | (dom (f + c))) " {-infty} is Element of bool X
(dom (f + c)) /\ (c " {-infty}) is Element of bool X
(f | (dom (f + c))) " {-infty} is Element of bool X
(dom (f + c)) /\ (f " {-infty}) is Element of bool X
(dom (f | (dom (f + c)))) /\ (dom (c | (dom (f + c)))) is Element of bool X
(c | (dom (f + c))) " {+infty} is Element of bool X
((f | (dom (f + c))) " {-infty}) /\ ((c | (dom (f + c))) " {+infty}) is Element of bool X
(f | (dom (f + c))) " {+infty} is Element of bool X
((f | (dom (f + c))) " {+infty}) /\ ((c | (dom (f + c))) " {-infty}) is Element of bool X
(((f | (dom (f + c))) " {-infty}) /\ ((c | (dom (f + c))) " {+infty})) \/ (((f | (dom (f + c))) " {+infty}) /\ ((c | (dom (f + c))) " {-infty})) is Element of bool X
((dom (f | (dom (f + c)))) /\ (dom (c | (dom (f + c))))) \ ((((f | (dom (f + c))) " {-infty}) /\ ((c | (dom (f + c))) " {+infty})) \/ (((f | (dom (f + c))) " {+infty}) /\ ((c | (dom (f + c))) " {-infty}))) is Element of bool X
(f | (dom (f + c))) + (c | (dom (f + c))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (dom (f + c))) + (c | (dom (f + c)))) is Element of bool X
B is Element of X
((f | (dom (f + c))) + (c | (dom (f + c)))) . B is ext-real Element of ExtREAL
(f + c) . B is ext-real Element of ExtREAL
(f | (dom (f + c))) . B is ext-real Element of ExtREAL
(c | (dom (f + c))) . B is ext-real Element of ExtREAL
((f | (dom (f + c))) . B) + ((c | (dom (f + c))) . B) is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
(f . B) + ((c | (dom (f + c))) . B) is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
(f . B) + (c . B) is ext-real Element of ExtREAL
integral (X,S,M,((f | (dom (f + c))) + (c | (dom (f + c))))) is ext-real Element of ExtREAL
integral (X,S,M,(f | (dom (f + c)))) is ext-real Element of ExtREAL
integral (X,S,M,(c | (dom (f + c)))) is ext-real Element of ExtREAL
(integral (X,S,M,(f | (dom (f + c))))) + (integral (X,S,M,(c | (dom (f + c))))) is ext-real Element of ExtREAL
integral (X,S,M,(f + c)) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
c (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) f)) is ext-real Element of ExtREAL
R_EAL c is ext-real Element of ExtREAL
(R_EAL c) * (X,S,M,f) is ext-real Element of ExtREAL
dom f is Element of bool X
B is set
f . B is ext-real Element of ExtREAL
dom (c (#) f) is Element of bool X
x is set
(c (#) f) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
(R_EAL c) * (f . x) is ext-real Element of ExtREAL
c * (X,S,M,f) is ext-real set
integral (X,S,M,f) is ext-real Element of ExtREAL
integral (X,S,M,(c (#) 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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
B is Element of S
c \/ B is M13(X,S)
f | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
dom (f | (c \/ B)) is Element of bool X
f1 is set
(f | (c \/ B)) . f1 is ext-real Element of ExtREAL
f1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
E is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
E . 1 is ext-real Element of ExtREAL
dom E is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
f1 . DFPG is set
(f1 . DFPG) /\ c is Element of bool X
DFPG is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom DFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . NFPG is set
f1 . NFPG is set
(f1 . NFPG) /\ B is Element of bool X
Seg (len DFPG) is finite V44( len DFPG) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len DFPG ) } is set
(f | (c \/ B)) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (c \/ B)) | B) is Element of bool X
(dom (f | (c \/ B))) /\ B is Element of bool X
dom f is Element of bool X
(dom f) /\ (c \/ B) is Element of bool X
((dom f) /\ (c \/ B)) /\ B is Element of bool X
(c \/ B) /\ B is M13(X,S)
(dom f) /\ ((c \/ B) /\ B) is Element of bool X
(dom f) /\ B is Element of bool X
dom (f | B) is Element of bool X
x is set
((f | (c \/ B)) | B) . x is ext-real Element of ExtREAL
(f | B) . x is ext-real Element of ExtREAL
(f | (c \/ B)) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
NFPG is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
(f | (c \/ B)) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (c \/ B)) | c) is Element of bool X
(dom (f | (c \/ B))) /\ c is Element of bool X
((dom f) /\ (c \/ B)) /\ c is Element of bool X
(c \/ B) /\ c is M13(X,S)
(dom f) /\ ((c \/ B) /\ c) is Element of bool X
(dom f) /\ c is Element of bool X
dom (f | c) is Element of bool X
ff is set
((f | (c \/ B)) | c) . ff is ext-real Element of ExtREAL
(f | c) . ff is ext-real Element of ExtREAL
(f | (c \/ B)) . ff is ext-real Element of ExtREAL
f . ff is ext-real Element of ExtREAL
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
M * f1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
ff is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom ff is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ff) is finite V44( len ff) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len ff ) } is set
integral (X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
Sum ff is ext-real Element of ExtREAL
((dom f) /\ c) \/ ((dom f) /\ B) is Element of bool X
(dom (f | c)) \/ ((dom f) /\ B) is Element of bool X
KB is set
(f | (c \/ B)) . KB is ext-real Element of ExtREAL
f . KB is ext-real Element of ExtREAL
(f | B) . KB is ext-real Element of ExtREAL
((dom f) /\ B) \/ ((dom f) /\ c) is Element of bool X
(dom (f | B)) \/ ((dom f) /\ c) is Element of bool X
KB is set
(f | (c \/ B)) . KB is ext-real Element of ExtREAL
f . KB is ext-real Element of ExtREAL
(f | c) . KB is ext-real Element of ExtREAL
KB is set
(f | B) . KB is ext-real Element of ExtREAL
M * NFPG is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom KB is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . KAB is ext-real Element of ExtREAL
Seg (len KB) is finite V44( len KB) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len KB ) } is set
E . KAB is ext-real Element of ExtREAL
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
E . KAB is ext-real Element of ExtREAL
E . KAB is ext-real Element of ExtREAL
E . KAB is ext-real Element of ExtREAL
Seg (len NFPG) is finite V44( len NFPG) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len NFPG ) } is set
dom NFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(M * NFPG) . KAB is ext-real Element of ExtREAL
NFPG . KAB is set
M . (NFPG . KAB) is ext-real Element of ExtREAL
rng NFPG is finite Element of bool S
bool S is non empty set
n is Element of S
M . n is ext-real Element of ExtREAL
(E . KAB) * ((M * NFPG) . KAB) is ext-real Element of ExtREAL
KAB is set
KB . KAB is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB . n is ext-real Element of ExtREAL
KB " {-infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB is set
KB . KAB is ext-real Element of ExtREAL
Seg (len NFPG) is finite V44( len NFPG) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len NFPG ) } is set
dom NFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
integral (X,S,M,(f | B)) is ext-real Element of ExtREAL
Sum KB is ext-real Element of ExtREAL
KAB is set
(f | c) . KAB is ext-real Element of ExtREAL
M * x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KAB is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom KAB is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len f1)) /\ (Seg (len f1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(dom KAB) /\ (Seg (len NFPG)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(dom KAB) /\ (dom KB) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
ff . n is ext-real Element of ExtREAL
KAB . n is ext-real Element of ExtREAL
KB . n is ext-real Element of ExtREAL
(KAB . n) + (KB . n) is ext-real Element of ExtREAL
NFPG . n is set
f1 . n is set
(f1 . n) /\ B is Element of bool X
m is set
rng f1 is finite Element of bool S
bool S is non empty set
union (rng f1) is set
(f1 . n) /\ (c \/ B) is Element of bool X
(f1 . n) /\ c is Element of bool X
((f1 . n) /\ c) \/ ((f1 . n) /\ B) is Element of bool X
x . n is set
(x . n) \/ (NFPG . n) is set
rng NFPG is finite Element of bool S
m is Element of S
M . m is ext-real Element of ExtREAL
(M * NFPG) . n is ext-real Element of ExtREAL
n is set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
rng x is finite Element of bool S
n is Element of S
M . n is ext-real Element of ExtREAL
(M * x) . n is ext-real Element of ExtREAL
(M * f1) . n is ext-real Element of ExtREAL
M . (f1 . n) is ext-real Element of ExtREAL
(M . n) + (M . m) is ext-real Element of ExtREAL
M . (NFPG . n) is ext-real Element of ExtREAL
((M * x) . n) + (M . (NFPG . n)) is ext-real Element of ExtREAL
((M * x) . n) + ((M * NFPG) . n) is ext-real Element of ExtREAL
E . n is ext-real Element of ExtREAL
(E . n) * ((M * f1) . n) is ext-real Element of ExtREAL
(E . n) * ((M * x) . n) is ext-real Element of ExtREAL
(E . n) * ((M * NFPG) . n) is ext-real Element of ExtREAL
((E . n) * ((M * x) . n)) + ((E . n) * ((M * NFPG) . n)) is ext-real Element of ExtREAL
(KAB . n) + ((E . n) * ((M * NFPG) . n)) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . n is ext-real Element of ExtREAL
Seg (len KAB) is finite V44( len KAB) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len KAB ) } is set
E . n is ext-real Element of ExtREAL
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
E . n is ext-real Element of ExtREAL
E . n is ext-real Element of ExtREAL
E . n is ext-real Element of ExtREAL
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(M * x) . n is ext-real Element of ExtREAL
x . n is set
M . (x . n) is ext-real Element of ExtREAL
rng x is finite Element of bool S
bool S is non empty set
m is Element of S
M . m is ext-real Element of ExtREAL
(E . n) * ((M * x) . n) is ext-real Element of ExtREAL
n is set
KAB . n is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KAB . m is ext-real Element of ExtREAL
KAB " {-infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
n is set
KAB . n is ext-real Element of ExtREAL
KB " {+infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(KAB " {-infty}) /\ (KB " {+infty}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB " {+infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(KAB " {+infty}) /\ (KB " {-infty}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
((KAB " {-infty}) /\ (KB " {+infty})) \/ ((KAB " {+infty}) /\ (KB " {-infty})) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
((dom KAB) /\ (dom KB)) \ (((KAB " {-infty}) /\ (KB " {+infty})) \/ ((KAB " {+infty}) /\ (KB " {-infty}))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB + KB is Relation-like NAT -defined ExtREAL -valued Function-like V59() Element of bool [:NAT,ExtREAL:]
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
integral (X,S,M,(f | c)) is ext-real Element of ExtREAL
Sum KAB is ext-real Element of ExtREAL
Seg (len KB) is finite V44( len KB) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len KB ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
dom f is Element of bool X
c is set
f . c is ext-real Element of ExtREAL
integral (X,S,M,f) is ext-real Element of ExtREAL
c is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
B is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * c is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum x is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . I1 is ext-real Element of ExtREAL
c . I1 is set
M . (c . I1) is ext-real Element of ExtREAL
(M * c) . I1 is ext-real Element of ExtREAL
B . I1 is ext-real Element of ExtREAL
(B . I1) * ((M * c) . I1) is ext-real Element of ExtREAL
c . I1 is set
a is set
rng c is finite Element of bool S
bool S is non empty set
f1 is Element of S
M . f1 is ext-real Element of ExtREAL
(M * c) . I1 is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
B . I1 is ext-real Element of ExtREAL
(B . I1) * ((M * c) . I1) is ext-real Element of ExtREAL
c . I1 is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f - c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - c) is Element of bool X
integral (X,S,M,f) is ext-real Element of ExtREAL
integral (X,S,M,(f - c)) is ext-real Element of ExtREAL
integral (X,S,M,c) is ext-real Element of ExtREAL
(integral (X,S,M,(f - c))) + (integral (X,S,M,c)) is ext-real Element of ExtREAL
B is set
c . B is ext-real Element of ExtREAL
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * B is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum I1 is ext-real Element of ExtREAL
(dom f) /\ (dom c) is Element of bool X
a is set
f . a is ext-real Element of ExtREAL
a is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
f1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
E is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom E is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * a is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum E is ext-real Element of ExtREAL
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(len f1) * (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
NFPG is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom NFPG is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg ((len f1) * (len x)) is finite V44((len f1) * (len x)) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= (len f1) * (len x) ) } is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is set
rng NFPG is finite set
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
ff -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(ff -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((ff -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(ff -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((ff -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
a . (((ff -' 1) div (len x)) + 1) is set
rng a is finite Element of bool S
bool S is non empty set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
B . (((ff -' 1) mod (len x)) + 1) is set
rng B is finite Element of bool S
NFPG . ff is set
(a . (((ff -' 1) div (len x)) + 1)) /\ (B . (((ff -' 1) mod (len x)) + 1)) is set
ff is set
NFPG . ff is set
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . ff is set
x . KB is set
KB -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(KB -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KB -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(KB -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KB -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
ff -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(ff -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((ff -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(ff -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((ff -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
a . (((ff -' 1) div (len x)) + 1) is set
B . (((ff -' 1) mod (len x)) + 1) is set
(a . (((ff -' 1) div (len x)) + 1)) /\ (B . (((ff -' 1) mod (len x)) + 1)) is set
(x . ff) /\ (x . KB) is set
a . (((KB -' 1) div (len x)) + 1) is set
B . (((KB -' 1) mod (len x)) + 1) is set
(a . (((KB -' 1) div (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1)) is set
((a . (((ff -' 1) div (len x)) + 1)) /\ (B . (((ff -' 1) mod (len x)) + 1))) /\ ((a . (((KB -' 1) div (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1))) is set
(B . (((ff -' 1) mod (len x)) + 1)) /\ ((a . (((KB -' 1) div (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1))) is set
(a . (((ff -' 1) div (len x)) + 1)) /\ ((B . (((ff -' 1) mod (len x)) + 1)) /\ ((a . (((KB -' 1) div (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1)))) is set
(B . (((ff -' 1) mod (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1)) is set
(a . (((KB -' 1) div (len x)) + 1)) /\ ((B . (((ff -' 1) mod (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1))) is set
(a . (((ff -' 1) div (len x)) + 1)) /\ ((a . (((KB -' 1) div (len x)) + 1)) /\ ((B . (((ff -' 1) mod (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1)))) is set
(a . (((ff -' 1) div (len x)) + 1)) /\ (a . (((KB -' 1) div (len x)) + 1)) is set
((a . (((ff -' 1) div (len x)) + 1)) /\ (a . (((KB -' 1) div (len x)) + 1))) /\ ((B . (((ff -' 1) mod (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1))) is set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
(KB -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((KB -' 1) div (len x)) + 1) - 1 is complex real ext-real integer rational Element of REAL
((((KB -' 1) div (len x)) + 1) - 1) * (len x) is complex real ext-real integer rational Element of REAL
(((KB -' 1) mod (len x)) + 1) - 1 is complex real ext-real integer rational Element of REAL
(((((KB -' 1) div (len x)) + 1) - 1) * (len x)) + ((((KB -' 1) mod (len x)) + 1) - 1) is complex real ext-real integer rational Element of REAL
((((((KB -' 1) div (len x)) + 1) - 1) * (len x)) + ((((KB -' 1) mod (len x)) + 1) - 1)) + 1 is complex real ext-real integer rational Element of REAL
KB - 1 is complex real ext-real integer rational Element of REAL
(KB - 1) + 1 is complex real ext-real integer rational Element of REAL
(((((KB -' 1) div (len x)) + 1) - 1) * (len x)) + (((KB -' 1) mod (len x)) + 1) is complex real ext-real integer rational Element of REAL
(ff -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((ff -' 1) div (len x)) + 1) - 1 is complex real ext-real integer rational Element of REAL
((((ff -' 1) div (len x)) + 1) - 1) * (len x) is complex real ext-real integer rational Element of REAL
(((ff -' 1) mod (len x)) + 1) - 1 is complex real ext-real integer rational Element of REAL
(((((ff -' 1) div (len x)) + 1) - 1) * (len x)) + ((((ff -' 1) mod (len x)) + 1) - 1) is complex real ext-real integer rational Element of REAL
((((((ff -' 1) div (len x)) + 1) - 1) * (len x)) + ((((ff -' 1) mod (len x)) + 1) - 1)) + 1 is complex real ext-real integer rational Element of REAL
ff - 1 is complex real ext-real integer rational Element of REAL
(ff - 1) + 1 is complex real ext-real integer rational Element of REAL
(((((ff -' 1) div (len x)) + 1) - 1) * (len x)) + (((ff -' 1) mod (len x)) + 1) is complex real ext-real integer rational Element of REAL
{} /\ ((B . (((ff -' 1) mod (len x)) + 1)) /\ (B . (((KB -' 1) mod (len x)) + 1))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded set
((a . (((ff -' 1) div (len x)) + 1)) /\ (a . (((KB -' 1) div (len x)) + 1))) /\ {} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded set
rng a is finite Element of bool S
bool S is non empty set
union (rng a) is set
ff is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
len ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len ff) is finite V44( len ff) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len ff ) } is set
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(KB -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KB -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
a . (((KB -' 1) div (len x)) + 1) is set
f1 . (((KB -' 1) div (len x)) + 1) is ext-real Element of ExtREAL
KB is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom KB is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
rng B is finite Element of bool S
union (rng B) is set
rng ff is finite Element of bool S
union (rng ff) is set
KAB is set
n is set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . m is set
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
1 + n is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
m is set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
z is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
1 + z is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
n * (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(n * (len x)) + x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m - 1 is complex real ext-real integer rational Element of REAL
(m - 1) * (len x) is complex real ext-real integer rational Element of REAL
((m - 1) * (len x)) + x is complex real ext-real integer rational Element of REAL
(len f1) - 1 is complex real ext-real integer rational Element of REAL
((len f1) - 1) * (len x) is complex real ext-real integer rational Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) - 1) * (len x)) + x is complex real ext-real integer rational Element of REAL
0 + x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x - 1 is complex real ext-real integer rational Element of REAL
(n * (len x)) + z is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) - 1) * (len x)) + (len x) is complex real ext-real integer rational Element of REAL
dom ff is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
ff . x is set
(x -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(x -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(a . m) /\ (B . x) is set
KAB is set
n is set
dom ff is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . m is set
m -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(m -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((m -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
(m -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((m -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
a . (((m -' 1) div (len x)) + 1) is set
B . (((m -' 1) mod (len x)) + 1) is set
(a . (((m -' 1) div (len x)) + 1)) /\ (B . (((m -' 1) mod (len x)) + 1)) is set
dom ff is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . KAB is set
n is Element of X
m is Element of X
(f - c) . n is ext-real Element of ExtREAL
(f - c) . m is ext-real Element of ExtREAL
KAB -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(KAB -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KAB -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(KAB -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((KAB -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
a . (((KAB -' 1) div (len x)) + 1) is set
B . (((KAB -' 1) mod (len x)) + 1) is set
(a . (((KAB -' 1) div (len x)) + 1)) /\ (B . (((KAB -' 1) mod (len x)) + 1)) is set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
f . m is ext-real Element of ExtREAL
f1 . (((KAB -' 1) div (len x)) + 1) is ext-real Element of ExtREAL
f . n is ext-real Element of ExtREAL
c . n is ext-real Element of ExtREAL
(f . n) - (c . n) is ext-real Element of ExtREAL
- (c . n) is ext-real set
(f . n) + (- (c . n)) is ext-real set
x . (((KAB -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
(f1 . (((KAB -' 1) div (len x)) + 1)) - (x . (((KAB -' 1) mod (len x)) + 1)) is ext-real Element of ExtREAL
- (x . (((KAB -' 1) mod (len x)) + 1)) is ext-real set
(f1 . (((KAB -' 1) div (len x)) + 1)) + (- (x . (((KAB -' 1) mod (len x)) + 1))) is ext-real set
c . m is ext-real Element of ExtREAL
(f . m) - (c . m) is ext-real Element of ExtREAL
- (c . m) is ext-real set
(f . m) + (- (c . m)) is ext-real set
M * ff is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
KAB is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom KAB is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . n is set
KB . n is ext-real Element of ExtREAL
n -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(n -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((n -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
n is set
f . n is ext-real Element of ExtREAL
a . (((n -' 1) div (len x)) + 1) is set
(n -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((n -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
B . (((n -' 1) mod (len x)) + 1) is set
(a . (((n -' 1) div (len x)) + 1)) /\ (B . (((n -' 1) mod (len x)) + 1)) is set
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
f1 . (((n -' 1) div (len x)) + 1) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(n -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((n -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
B . (((n -' 1) mod (len x)) + 1) is set
x . (((n -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
R_EAL 0 is ext-real Element of ExtREAL
n is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom n is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
m is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom m is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . n is set
n . n is ext-real Element of ExtREAL
n -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(n -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((n -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
x is set
c . x is ext-real Element of ExtREAL
(n -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((n -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
a . (((n -' 1) div (len x)) + 1) is set
B . (((n -' 1) mod (len x)) + 1) is set
(a . (((n -' 1) div (len x)) + 1)) /\ (B . (((n -' 1) mod (len x)) + 1)) is set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
x . (((n -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . n is set
m . n is ext-real Element of ExtREAL
m is set
(f - c) . m is ext-real Element of ExtREAL
f . m is ext-real Element of ExtREAL
c . m is ext-real Element of ExtREAL
(f . m) - (c . m) is ext-real Element of ExtREAL
- (c . m) is ext-real set
(f . m) + (- (c . m)) is ext-real set
KB . n is ext-real Element of ExtREAL
n . n is ext-real Element of ExtREAL
(KB . n) - (n . n) is ext-real Element of ExtREAL
- (n . n) is ext-real set
(KB . n) + (- (n . n)) is ext-real set
(KB . n) - (c . m) is ext-real Element of ExtREAL
(KB . n) + (- (c . m)) is ext-real set
n is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom n is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
m is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
len m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom m is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
z is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n . z is ext-real Element of ExtREAL
(M * ff) . z is ext-real Element of ExtREAL
ff . z is set
M . (ff . z) is ext-real Element of ExtREAL
x is Element of S
M . x is ext-real Element of ExtREAL
x is Element of S
M . x is ext-real Element of ExtREAL
N0 is set
c . N0 is ext-real Element of ExtREAL
f . N0 is ext-real Element of ExtREAL
KB . z is ext-real Element of ExtREAL
n . z is ext-real Element of ExtREAL
(KB . z) - (n . z) is ext-real Element of ExtREAL
- (n . z) is ext-real set
(KB . z) + (- (n . z)) is ext-real set
m . z is ext-real Element of ExtREAL
(m . z) * ((M * ff) . z) is ext-real Element of ExtREAL
m . z is ext-real Element of ExtREAL
(m . z) * ((M * ff) . z) is ext-real Element of ExtREAL
rng n is finite ext-real-membered Element of bool ExtREAL
x is set
n . x is ext-real Element of ExtREAL
n " {-infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
m " {+infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(n " {-infty}) /\ (m " {+infty}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{} /\ (m " {+infty}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m . x is ext-real Element of ExtREAL
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(x -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
n . x is ext-real Element of ExtREAL
(M * ff) . x is ext-real Element of ExtREAL
(n . x) * ((M * ff) . x) is ext-real Element of ExtREAL
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
B . (((x -' 1) mod (len x)) + 1) is set
ff . x is set
N0 is Element of S
x is Element of S
M . {} is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
m is set
c . m is ext-real Element of ExtREAL
x . (((x -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
B . (((x -' 1) mod (len x)) + 1) is set
ff . x is set
(x -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
a . (((x -' 1) div (len x)) + 1) is set
(a . (((x -' 1) div (len x)) + 1)) /\ (B . (((x -' 1) mod (len x)) + 1)) is set
M . (ff . x) is ext-real Element of ExtREAL
B . (((x -' 1) mod (len x)) + 1) is set
rng m is finite ext-real-membered Element of bool ExtREAL
x is set
m . x is ext-real Element of ExtREAL
n " {+infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
m " {-infty} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(n " {+infty}) /\ (m " {-infty}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
(n " {+infty}) /\ {} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
n + m is Relation-like NAT -defined ExtREAL -valued Function-like V59() Element of bool [:NAT,ExtREAL:]
dom (n + m) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
(dom n) /\ (dom m) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{} \/ {} is finite V41() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded set
((dom n) /\ (dom m)) \ ({} \/ {}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . x is ext-real Element of ExtREAL
(n + m) . x is ext-real Element of ExtREAL
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(x -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) div (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
(x -' 1) mod (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((x -' 1) mod (len x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
((len f1) * (len x)) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) -' 1) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((len f1) * (len x)) div (len x) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(((len f1) * (len x)) div (len x)) - 1 is complex real ext-real integer rational Element of REAL
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
m . x is ext-real Element of ExtREAL
n . x is ext-real Element of ExtREAL
(m . x) + (n . x) is ext-real Element of ExtREAL
(M * ff) . x is ext-real Element of ExtREAL
((m . x) + (n . x)) * ((M * ff) . x) is ext-real Element of ExtREAL
(m . x) * ((M * ff) . x) is ext-real Element of ExtREAL
(n . x) * ((M * ff) . x) is ext-real Element of ExtREAL
((m . x) * ((M * ff) . x)) + ((n . x) * ((M * ff) . x)) is ext-real Element of ExtREAL
ff . x is set
a . (((x -' 1) div (len x)) + 1) is set
B . (((x -' 1) mod (len x)) + 1) is set
(a . (((x -' 1) div (len x)) + 1)) /\ (B . (((x -' 1) mod (len x)) + 1)) is set
N0 is set
x . (((x -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
c . N0 is ext-real Element of ExtREAL
f1 . (((x -' 1) div (len x)) + 1) is ext-real Element of ExtREAL
f . N0 is ext-real Element of ExtREAL
KB . x is ext-real Element of ExtREAL
(KB . x) - (n . x) is ext-real Element of ExtREAL
- (n . x) is ext-real set
(KB . x) + (- (n . x)) is ext-real set
ff . x is set
M . (ff . x) is ext-real Element of ExtREAL
ff . x is set
KB . x is ext-real Element of ExtREAL
a . (((x -' 1) div (len x)) + 1) is set
N0 is set
f1 . (((x -' 1) div (len x)) + 1) is ext-real Element of ExtREAL
f . N0 is ext-real Element of ExtREAL
a . (((x -' 1) div (len x)) + 1) is set
a . (((x -' 1) div (len x)) + 1) is set
B . (((x -' 1) mod (len x)) + 1) is set
N0 is set
x . (((x -' 1) mod (len x)) + 1) is ext-real Element of ExtREAL
c . N0 is ext-real Element of ExtREAL
B . (((x -' 1) mod (len x)) + 1) is set
B . (((x -' 1) mod (len x)) + 1) is set
(n . x) - (n . x) is ext-real Element of ExtREAL
- (n . x) is ext-real set
(n . x) + (- (n . x)) is ext-real set
- 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() Element of REAL
(KB . x) - (n . x) is ext-real Element of ExtREAL
(KB . x) + (- (n . x)) is ext-real set
(KB . x) - ((n . x) - (n . x)) is ext-real Element of ExtREAL
- ((n . x) - (n . x)) is ext-real set
(KB . x) + (- ((n . x) - (n . x))) is ext-real set
(KB . x) + (- 0) is ext-real set
n . x is ext-real Element of ExtREAL
(n . x) + ((n . x) * ((M * ff) . x)) is ext-real Element of ExtREAL
m . x is ext-real Element of ExtREAL
(n . x) + (m . x) is ext-real Element of ExtREAL
x is Element of X
c . x is ext-real Element of ExtREAL
|.(c . x).| is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
|.(f . x).| is ext-real Element of ExtREAL
|.(f . x).| + |.(c . x).| is ext-real Element of ExtREAL
(f - c) . x is ext-real Element of ExtREAL
|.((f - c) . x).| is ext-real Element of ExtREAL
(f . x) - (c . x) is ext-real Element of ExtREAL
- (c . x) is ext-real set
(f . x) + (- (c . x)) is ext-real set
|.((f . x) - (c . x)).| is ext-real Element of ExtREAL
Sum KAB is ext-real Element of ExtREAL
Seg (len KAB) is finite V44( len KAB) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len KAB ) } is set
Sum m is ext-real Element of ExtREAL
x is set
(f - c) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
(f . x) - (c . x) is ext-real Element of ExtREAL
- (c . x) is ext-real set
(f . x) + (- (c . x)) is ext-real set
Sum n 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - c) is Element of bool X
dom f is Element of bool X
dom c is Element of bool X
(dom f) /\ (dom c) is Element of bool X
f | (dom (f - c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (dom (f - c)))) is ext-real Element of ExtREAL
(X,S,M,(f - c)) is ext-real Element of ExtREAL
c | (dom (f - c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | (dom (f - c)))) is ext-real Element of ExtREAL
(X,S,M,(f - c)) + (X,S,M,(c | (dom (f - c)))) is ext-real Element of ExtREAL
(- 1) (#) c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + (- c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
rng c is ext-real-membered Element of bool ExtREAL
c " {-infty} is Element of bool X
rng f is ext-real-membered Element of bool ExtREAL
f " {+infty} is Element of bool X
c " {+infty} is Element of bool X
(f " {+infty}) /\ (c " {+infty}) is Element of bool X
f " {-infty} is Element of bool X
(f " {-infty}) /\ (c " {-infty}) is Element of bool X
((f " {+infty}) /\ (c " {+infty})) \/ ((f " {-infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {+infty}) /\ (c " {+infty})) \/ ((f " {-infty}) /\ (c " {-infty}))) is Element of bool X
dom (f | (dom (f - c))) is Element of bool X
(dom f) /\ (dom (f - c)) is Element of bool X
(dom f) /\ (dom f) is Element of bool X
((dom f) /\ (dom f)) /\ (dom c) is Element of bool X
B is set
(c | (dom (f - c))) . B is ext-real Element of ExtREAL
(f | (dom (f - c))) . B is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
dom (c | (dom (f - c))) is Element of bool X
(dom c) /\ (dom (f - c)) is Element of bool X
(dom c) /\ (dom c) is Element of bool X
((dom c) /\ (dom c)) /\ (dom f) is Element of bool X
(c | (dom (f - c))) " {-infty} is Element of bool X
(dom (f - c)) /\ (c " {-infty}) is Element of bool X
(f | (dom (f - c))) " {+infty} is Element of bool X
(dom (f - c)) /\ (f " {+infty}) is Element of bool X
(dom (f | (dom (f - c)))) /\ (dom (c | (dom (f - c)))) is Element of bool X
(c | (dom (f - c))) " {+infty} is Element of bool X
((f | (dom (f - c))) " {+infty}) /\ ((c | (dom (f - c))) " {+infty}) is Element of bool X
(f | (dom (f - c))) " {-infty} is Element of bool X
((f | (dom (f - c))) " {-infty}) /\ ((c | (dom (f - c))) " {-infty}) is Element of bool X
(((f | (dom (f - c))) " {+infty}) /\ ((c | (dom (f - c))) " {+infty})) \/ (((f | (dom (f - c))) " {-infty}) /\ ((c | (dom (f - c))) " {-infty})) is Element of bool X
((dom (f | (dom (f - c)))) /\ (dom (c | (dom (f - c))))) \ ((((f | (dom (f - c))) " {+infty}) /\ ((c | (dom (f - c))) " {+infty})) \/ (((f | (dom (f - c))) " {-infty}) /\ ((c | (dom (f - c))) " {-infty}))) is Element of bool X
(f | (dom (f - c))) - (c | (dom (f - c))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (dom (f - c))) - (c | (dom (f - c)))) is Element of bool X
B is Element of X
((f | (dom (f - c))) - (c | (dom (f - c)))) . B is ext-real Element of ExtREAL
(f - c) . B is ext-real Element of ExtREAL
(f | (dom (f - c))) . B is ext-real Element of ExtREAL
(c | (dom (f - c))) . B is ext-real Element of ExtREAL
((f | (dom (f - c))) . B) - ((c | (dom (f - c))) . B) is ext-real Element of ExtREAL
- ((c | (dom (f - c))) . B) is ext-real set
((f | (dom (f - c))) . B) + (- ((c | (dom (f - c))) . B)) is ext-real set
f . B is ext-real Element of ExtREAL
(f . B) - ((c | (dom (f - c))) . B) is ext-real Element of ExtREAL
(f . B) + (- ((c | (dom (f - c))) . B)) is ext-real set
c . B is ext-real Element of ExtREAL
(f . B) - (c . B) is ext-real Element of ExtREAL
- (c . B) is ext-real set
(f . B) + (- (c . B)) is ext-real set
integral (X,S,M,(f | (dom (f - c)))) is ext-real Element of ExtREAL
integral (X,S,M,((f | (dom (f - c))) - (c | (dom (f - c))))) is ext-real Element of ExtREAL
integral (X,S,M,(c | (dom (f - c)))) is ext-real Element of ExtREAL
(integral (X,S,M,((f | (dom (f - c))) - (c | (dom (f - c)))))) + (integral (X,S,M,(c | (dom (f - c))))) is ext-real Element of ExtREAL
integral (X,S,M,(f - c)) is ext-real Element of ExtREAL
(integral (X,S,M,(f - c))) + (integral (X,S,M,(c | (dom (f - c))))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - c) is Element of bool X
c | (dom (f - c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | (dom (f - c)))) is ext-real Element of ExtREAL
f | (dom (f - c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (dom (f - c)))) is ext-real Element of ExtREAL
(- 1) (#) c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + (- c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f - c)) is ext-real Element of ExtREAL
(X,S,M,(f - c)) + (X,S,M,(c | (dom (f - c)))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
M . (dom f) is ext-real Element of ExtREAL
c is ext-real Element of ExtREAL
c * (M . (dom f)) is ext-real Element of ExtREAL
B is set
f . B is ext-real Element of ExtREAL
B is Element of S
M . B is ext-real Element of ExtREAL
B is Element of S
M . B is ext-real Element of ExtREAL
c * (M . B) is ext-real Element of ExtREAL
<*(c * (M . B))*> is Relation-like NAT -defined ExtREAL -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
[1,(c * (M . B))] is set
{1,(c * (M . B))} is non empty finite ext-real-membered left_end right_end set
{{1,(c * (M . B))},{1}} is non empty finite V41() set
{[1,(c * (M . B))]} is non empty finite set
<*c*> is Relation-like NAT -defined ExtREAL -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
[1,c] is set
{1,c} is non empty finite ext-real-membered left_end right_end set
{{1,c},{1}} is non empty finite V41() set
{[1,c]} is non empty finite set
<*(dom f)*> is Relation-like NAT -defined bool X -valued non empty Function-like finite V44(1) FinSequence-like FinSubsequence-like FinSequence of bool X
[1,(dom f)] is set
{1,(dom f)} is non empty finite set
{{1,(dom f)},{1}} is non empty finite V41() set
{[1,(dom f)]} is non empty finite set
rng <*(dom f)*> is finite Element of bool (bool X)
{B} is non empty finite set
E is set
E is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom E is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is set
E . DFPG is set
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
I1 . DFPG is ext-real Element of ExtREAL
NFPG is set
f . NFPG is ext-real Element of ExtREAL
f1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . DFPG is ext-real Element of ExtREAL
M * x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . DFPG is ext-real Element of ExtREAL
I1 . DFPG is ext-real Element of ExtREAL
(M * x) . DFPG is ext-real Element of ExtREAL
(I1 . DFPG) * ((M * x) . DFPG) is ext-real Element of ExtREAL
x . DFPG is set
M . (x . DFPG) is ext-real Element of ExtREAL
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
integral (X,S,M,f) is ext-real Element of ExtREAL
Sum f1 is ext-real Element of ExtREAL
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
R_EAL 1 is ext-real Element of ExtREAL
(R_EAL 1) * (c * (M . B)) is ext-real Element of ExtREAL
R_EAL 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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
eq_dom (f,(R_EAL 0)) is Element of bool X
f | (eq_dom (f,(R_EAL 0))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (eq_dom (f,(R_EAL 0))))) is ext-real Element of ExtREAL
dom f is Element of bool X
(dom f) /\ (eq_dom (f,(R_EAL 0))) is Element of bool X
f | ((dom f) /\ (eq_dom (f,(R_EAL 0)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is set
dom (f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) is Element of bool X
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
len x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
I1 is Element of S
less_dom (f,(R_EAL 0)) is Element of bool X
I1 /\ (less_dom (f,(R_EAL 0))) is Element of bool X
I1 \ (I1 /\ (less_dom (f,(R_EAL 0)))) is Element of bool X
great_eq_dom (f,0.) is Element of bool X
I1 /\ (great_eq_dom (f,0.)) is Element of bool X
f1 is Element of S
great_eq_dom (f,(R_EAL 0)) is Element of bool X
I1 /\ (great_eq_dom (f,(R_EAL 0))) is Element of bool X
less_eq_dom (f,(R_EAL 0)) is Element of bool X
(I1 /\ (great_eq_dom (f,(R_EAL 0)))) /\ (less_eq_dom (f,(R_EAL 0))) is Element of bool X
I1 /\ (eq_dom (f,(R_EAL 0))) is Element of bool X
Seg (len x) is finite V44( len x) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len x ) } is set
E is Element of S
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . x is set
x . x is set
(x . x) /\ E is Element of bool X
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
union (rng x) is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
x . DFPG is set
E /\ (x . DFPG) is Element of bool X
E /\ (dom f) is Element of bool X
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
NFPG is Element of X
x is Element of X
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . NFPG is ext-real Element of ExtREAL
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . x is ext-real Element of ExtREAL
x . DFPG is set
(x . DFPG) /\ E is Element of bool X
f . NFPG is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG is Element of X
x . DFPG is set
x is Element of X
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . NFPG is ext-real Element of ExtREAL
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . x is ext-real Element of ExtREAL
x is set
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
integral (X,S,M,(f | ((dom f) /\ (eq_dom (f,(R_EAL 0)))))) is ext-real Element of ExtREAL
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
I1 . 1 is ext-real Element of ExtREAL
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
a is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum a is ext-real Element of ExtREAL
f1 is set
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . f1 is ext-real Element of ExtREAL
(dom f) /\ ((dom f) /\ (eq_dom (f,(R_EAL 0)))) is Element of bool X
f . f1 is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . f1 is ext-real Element of ExtREAL
x . f1 is set
E is set
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
(f | ((dom f) /\ (eq_dom (f,(R_EAL 0))))) . E is ext-real Element of ExtREAL
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a . f1 is ext-real Element of ExtREAL
I1 . f1 is ext-real Element of ExtREAL
(M * x) . f1 is ext-real Element of ExtREAL
(I1 . f1) * ((M * x) . f1) is ext-real Element of ExtREAL
x . f1 is set
M . (x . f1) is ext-real Element of ExtREAL
(M * x) . f1 is ext-real Element of ExtREAL
I1 . f1 is ext-real Element of ExtREAL
(I1 . f1) * ((M * x) . f1) is ext-real Element of ExtREAL
I1 . f1 is ext-real Element of ExtREAL
x . f1 is set
len a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
f1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
f1 . (len a) is ext-real Element of ExtREAL
f1 . 0 is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . E is ext-real Element of ExtREAL
E + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f1 . (E + 1) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg (len a) is finite V44( len a) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len a ) } is set
a . (x + 1) is ext-real Element of ExtREAL
f1 . (x + 1) is ext-real Element of ExtREAL
f1 . x is ext-real Element of ExtREAL
(f1 . x) + (a . (x + 1)) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c | f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | f)) is ext-real Element of ExtREAL
dom c is Element of bool X
(dom c) /\ f is Element of bool X
c | ((dom c) /\ f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c | ((dom c) /\ f)) is Element of bool X
I1 is set
(c | ((dom c) /\ f)) . I1 is ext-real Element of ExtREAL
c . I1 is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng I1 is finite Element of bool S
bool S is non empty set
union (rng I1) is set
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
a is Element of S
a /\ f is M13(X,S)
len I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
dom E is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg (len I1) is finite V44( len I1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len I1 ) } is set
f1 is Element of S
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is set
I1 . x is set
(I1 . x) /\ f1 is Element of bool X
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng x is finite Element of bool S
union (rng x) is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
I1 . DFPG is set
f1 /\ (I1 . DFPG) is Element of bool X
f1 /\ (dom c) is Element of bool X
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is set
NFPG is Element of X
x is Element of X
(c | ((dom c) /\ f)) . NFPG is ext-real Element of ExtREAL
(c | ((dom c) /\ f)) . x is ext-real Element of ExtREAL
I1 . DFPG is set
(I1 . DFPG) /\ f1 is Element of bool X
c . NFPG is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG is Element of X
x . DFPG is set
x is Element of X
(c | ((dom c) /\ f)) . NFPG is ext-real Element of ExtREAL
(c | ((dom c) /\ f)) . x is ext-real Element of ExtREAL
(dom c) /\ ((dom c) /\ f) is Element of bool X
(dom c) /\ (dom c) is Element of bool X
((dom c) /\ (dom c)) /\ f is Element of bool X
dom (c | f) is Element of bool X
I1 is set
(c | ((dom c) /\ f)) . I1 is ext-real Element of ExtREAL
(c | f) . I1 is ext-real Element of ExtREAL
c . I1 is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng I1 is finite Element of bool S
bool S is non empty set
union (rng I1) is set
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
integral (X,S,M,(c | ((dom c) /\ f))) is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
a is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
a . 1 is ext-real Element of ExtREAL
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
f1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom f1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum f1 is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . x is set
M . (I1 . x) is ext-real Element of ExtREAL
rng I1 is finite Element of bool S
bool S is non empty set
union (rng I1) is set
NFPG is set
DFPG is Element of S
E is measure_zero of M
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . E is ext-real Element of ExtREAL
I1 . E is set
M . (I1 . E) is ext-real Element of ExtREAL
(M * I1) . E is ext-real Element of ExtREAL
a . E is ext-real Element of ExtREAL
(a . E) * ((M * I1) . E) is ext-real Element of ExtREAL
len f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
E . (len f1) is ext-real Element of ExtREAL
E . 0 is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is ext-real Element of ExtREAL
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
E . (x + 1) is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
Seg (len f1) is finite V44( len f1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len f1 ) } is set
f1 . (DFPG + 1) is ext-real Element of ExtREAL
E . (DFPG + 1) is ext-real Element of ExtREAL
E . DFPG is ext-real Element of ExtREAL
(E . DFPG) + (f1 . (DFPG + 1)) 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 V59() set
bool [:X,ExtREAL:] is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
B is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(B) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,x) is Element of bool X
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
B . x is ext-real Element of ExtREAL
x is set
f . x is ext-real Element of ExtREAL
x is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
I1 . 1 is ext-real Element of ExtREAL
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
len I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len I1) is finite V44( len I1) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len I1 ) } is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is ext-real Element of ExtREAL
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
f1 is complex real ext-real Element of REAL
a is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V58() V59() V60() FinSequence of REAL
dom a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
Seg 0 is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty proper complex real ext-real non positive non negative integer functional finite V41() V44( 0 ) FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= 0 ) } is set
rng x is finite Element of bool S
bool S is non empty set
f1 is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
E is set
x . E is set
union (rng x) is set
f1 is set
E is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
x . 1 is set
{(x . 1)} is non empty finite set
union {(x . 1)} is set
f1 is set
f . f1 is ext-real Element of ExtREAL
a . 1 is complex real ext-real Element of ExtREAL
a . 2 is complex real ext-real Element of ExtREAL
I1 . 2 is ext-real Element of ExtREAL
{(a . 1)} is non empty finite complex-membered ext-real-membered real-membered left_end right_end bounded_below bounded_above real-bounded set
rng a is finite complex-membered ext-real-membered real-membered bounded_below bounded_above real-bounded Element of bool REAL
(rng a) \ {(a . 1)} is finite complex-membered ext-real-membered real-membered bounded_below bounded_above real-bounded Element of bool REAL
f1 is non empty finite complex-membered ext-real-membered real-membered left_end right_end bounded_below bounded_above real-bounded set
inf f1 is complex real ext-real set
R_EAL (inf f1) is ext-real Element of ExtREAL
sup f1 is complex real ext-real set
DFPG is set
a . DFPG is complex real ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
I1 . NFPG is ext-real Element of ExtREAL
a . NFPG is complex real ext-real Element of ExtREAL
R_EAL (a . NFPG) is ext-real Element of ExtREAL
len a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
Seg (len a) is finite V44( len a) complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT : ( 1 <= b₁ & b₁ <= len a ) } is set
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
R_EAL (sup f1) is ext-real Element of ExtREAL
ff is set
f . ff is ext-real Element of ExtREAL
rng x is finite Element of bool S
bool S is non empty set
union (rng x) is set
KB is set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
KAB is set
x . KAB is set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
I1 . n is ext-real Element of ExtREAL
a . n is complex real ext-real Element of ExtREAL
rng f is ext-real-membered Element of bool ExtREAL
f " {-infty} is Element of bool X
f " {+infty} is Element of bool X
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - (X,ExtREAL,c,ff)) is Element of bool X
dom (X,ExtREAL,c,ff) is Element of bool X
(dom (X,ExtREAL,c,ff)) /\ (dom f) is Element of bool X
(X,ExtREAL,c,ff) " {+infty} is Element of bool X
((X,ExtREAL,c,ff) " {+infty}) /\ (f " {+infty}) is Element of bool X
(X,ExtREAL,c,ff) " {-infty} is Element of bool X
((X,ExtREAL,c,ff) " {-infty}) /\ (f " {-infty}) is Element of bool X
(((X,ExtREAL,c,ff) " {+infty}) /\ (f " {+infty})) \/ (((X,ExtREAL,c,ff) " {-infty}) /\ (f " {-infty})) is Element of bool X
((dom (X,ExtREAL,c,ff)) /\ (dom f)) \ ((((X,ExtREAL,c,ff) " {+infty}) /\ (f " {+infty})) \/ (((X,ExtREAL,c,ff) " {-infty}) /\ (f " {-infty}))) is Element of bool X
K403(X) is non empty Element of bool (bool X)
[:NAT,K403(X):] is non empty set
bool [:NAT,K403(X):] is non empty set
M . (dom f) is ext-real Element of ExtREAL
ff is ext-real Element of ExtREAL
KB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . KAB is Element of K403(X)
(X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,KAB)),ff) is Element of bool X
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . KAB is Element of K403(X)
n is set
(X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,KAB)),ff) is Element of bool X
dom (f - (X,ExtREAL,c,KAB)) is Element of bool X
Union KB is Element of bool X
rng KB is set
union (rng KB) is set
KAB is set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB . n is Element of K403(X)
KAB is set
n is Element of X
(X,c,n) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
m is complex real ext-real Element of REAL
- m is complex real ext-real Element of REAL
R_EAL (- m) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,n) . KAB is ext-real Element of ExtREAL
(X,c,n) . n is ext-real Element of ExtREAL
((X,c,n)) is ext-real Element of ExtREAL
m is complex real ext-real Element of REAL
m / 2 is complex real ext-real Element of REAL
R_EAL (m / 2) is ext-real Element of ExtREAL
n is complex real ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f . KAB is ext-real Element of ExtREAL
(f . KAB) - (R_EAL (m / 2)) is ext-real Element of ExtREAL
- (R_EAL (m / 2)) is ext-real set
(f . KAB) + (- (R_EAL (m / 2))) is ext-real set
((X,c,n)) - 0. is ext-real Element of ExtREAL
- 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
- 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
((X,c,n)) + (- 0.) is ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m + x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,(m + x)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,(m + x)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - (X,ExtREAL,c,(m + x))) is Element of bool X
x is complex real ext-real set
R_EAL x is ext-real Element of ExtREAL
N0 is complex real ext-real set
N0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
A is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
max (m,A) is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,c,n) . (max (m,A)) is ext-real Element of ExtREAL
((X,c,n) . (max (m,A))) - ((X,c,n)) is ext-real Element of ExtREAL
- ((X,c,n)) is ext-real set
((X,c,n) . (max (m,A))) + (- ((X,c,n))) is ext-real set
|.(((X,c,n) . (max (m,A))) - ((X,c,n))).| is ext-real Element of ExtREAL
(X,ExtREAL,c,(m + x)) . n is ext-real Element of ExtREAL
(X,ExtREAL,c,(max (m,A))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,(max (m,A))) . n is ext-real Element of ExtREAL
(X,c,n) . (m + x) is ext-real Element of ExtREAL
(R_EAL x) + ((X,c,n)) is ext-real Element of ExtREAL
betae is complex real ext-real set
((X,c,n)) + x is ext-real set
((X,c,n)) - ((X,c,n) . (m + x)) is ext-real Element of ExtREAL
- ((X,c,n) . (m + x)) is ext-real set
((X,c,n)) + (- ((X,c,n) . (m + x))) is ext-real set
((X,c,n) . (m + x)) - ((X,c,n)) is ext-real Element of ExtREAL
((X,c,n) . (m + x)) + (- ((X,c,n))) is ext-real set
|.(((X,c,n) . (m + x)) - ((X,c,n))).| is ext-real Element of ExtREAL
|.(((X,c,n)) - ((X,c,n) . (m + x))).| is ext-real Element of ExtREAL
((X,c,n)) - ((X,ExtREAL,c,(m + x)) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,(m + x)) . n) is ext-real set
((X,c,n)) + (- ((X,ExtREAL,c,(m + x)) . n)) is ext-real set
x is complex real ext-real set
(X,ExtREAL,c,(m + x)) . KAB is ext-real Element of ExtREAL
x is complex real ext-real set
((X,ExtREAL,c,(m + x)) . KAB) + (R_EAL (m / 2)) is ext-real Element of ExtREAL
(((X,ExtREAL,c,(m + x)) . KAB) + (R_EAL (m / 2))) + (m / 2) is ext-real set
((X,c,n)) + (m / 2) is ext-real set
((X,c,n)) + (R_EAL (m / 2)) is ext-real Element of ExtREAL
((X,ExtREAL,c,(m + x)) . n) + (R_EAL (m / 2)) is ext-real Element of ExtREAL
(((X,ExtREAL,c,(m + x)) . n) + (R_EAL (m / 2))) + (R_EAL (m / 2)) is ext-real Element of ExtREAL
(R_EAL (m / 2)) + (R_EAL (m / 2)) is ext-real Element of ExtREAL
((X,ExtREAL,c,(m + x)) . n) + ((R_EAL (m / 2)) + (R_EAL (m / 2))) is ext-real Element of ExtREAL
(m / 2) + (m / 2) is complex real ext-real Element of REAL
R_EAL ((m / 2) + (m / 2)) is ext-real Element of ExtREAL
((X,ExtREAL,c,(m + x)) . n) + (R_EAL ((m / 2) + (m / 2))) is ext-real Element of ExtREAL
(f . KAB) - ((X,ExtREAL,c,(m + x)) . n) is ext-real Element of ExtREAL
(f . KAB) + (- ((X,ExtREAL,c,(m + x)) . n)) is ext-real set
(f - (X,ExtREAL,c,(m + x))) . n is ext-real Element of ExtREAL
less_dom ((f - (X,ExtREAL,c,(m + x))),ff) is Element of bool X
KB . (m + x) is Element of K403(X)
inferior_setsequence KB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
(inferior_setsequence KB) . m is Element of K403(X)
dom (inferior_setsequence KB) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
f . n is ext-real Element of ExtREAL
(f . n) - ff is ext-real Element of ExtREAL
- ff is ext-real set
(f . n) + (- ff) is ext-real set
R_EAL 1 is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,n) . n is ext-real Element of ExtREAL
((X,c,n) . n) + ff is ext-real Element of ExtREAL
(f . n) - ((X,c,n) . n) is ext-real Element of ExtREAL
- ((X,c,n) . n) is ext-real set
(f . n) + (- ((X,c,n) . n)) is ext-real set
(X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,n) . n is ext-real Element of ExtREAL
(f . n) - ((X,ExtREAL,c,n) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,n) . n) is ext-real set
(f . n) + (- ((X,ExtREAL,c,n) . n)) is ext-real set
(f . n) - ff is ext-real Element of ExtREAL
- ff is ext-real set
(f . n) + (- ff) is ext-real set
f . KAB is ext-real Element of ExtREAL
(f . KAB) - ff is ext-real Element of ExtREAL
(f . KAB) + (- ff) is ext-real set
n is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
n - m is complex real ext-real Element of REAL
m is complex real ext-real Element of REAL
m + 1 is complex real ext-real Element of REAL
R_EAL (m + 1) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
R_EAL m is ext-real Element of ExtREAL
z is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,n) . z is ext-real Element of ExtREAL
((X,c,n) . z) + ff is ext-real Element of ExtREAL
(f . n) - ((X,c,n) . z) is ext-real Element of ExtREAL
- ((X,c,n) . z) is ext-real set
(f . n) + (- ((X,c,n) . z)) is ext-real set
(X,ExtREAL,c,z) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,z) . n is ext-real Element of ExtREAL
(f . n) - ((X,ExtREAL,c,z) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,z) . n) is ext-real set
(f . n) + (- ((X,ExtREAL,c,z) . n)) is ext-real set
(f . n) - ff is ext-real Element of ExtREAL
- ff is ext-real set
(f . n) + (- ff) is ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - (X,ExtREAL,c,m)) is Element of bool X
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,m) . n is ext-real Element of ExtREAL
(f . n) - ((X,ExtREAL,c,m) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,m) . n) is ext-real set
(f . n) + (- ((X,ExtREAL,c,m) . n)) is ext-real set
(f - (X,ExtREAL,c,m)) . n is ext-real Element of ExtREAL
less_dom ((f - (X,ExtREAL,c,m)),ff) is Element of bool X
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
n + m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,(n + m)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,(n + m)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,(n + m))),ff) is Element of bool X
KB . (n + m) is Element of K403(X)
inferior_setsequence KB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
(inferior_setsequence KB) . n is Element of K403(X)
dom (inferior_setsequence KB) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
inferior_setsequence KB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
dom (inferior_setsequence KB) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
inferior_setsequence KB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
dom (inferior_setsequence KB) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(inferior_setsequence KB) . m is Element of K403(X)
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(inferior_setsequence KB) . m is Element of K403(X)
rng (inferior_setsequence KB) is Element of bool K403(X)
bool K403(X) is non empty set
Union (inferior_setsequence KB) is Element of bool X
rng (inferior_setsequence KB) is set
union (rng (inferior_setsequence KB)) is set
lim_inf KB is Element of bool X
lim_sup KB is Element of bool X
KAB is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . n is Element of K403(X)
KB . m is Element of K403(X)
n is set
(X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,n)),ff) is Element of bool X
dom (f - (X,ExtREAL,c,n)) is Element of bool X
(f - (X,ExtREAL,c,n)) . n is ext-real Element of ExtREAL
f . n is ext-real Element of ExtREAL
(X,ExtREAL,c,n) . n is ext-real Element of ExtREAL
(f . n) - ((X,ExtREAL,c,n) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,n) . n) is ext-real set
(f . n) + (- ((X,ExtREAL,c,n) . n)) is ext-real set
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,m) . n is ext-real Element of ExtREAL
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f - (X,ExtREAL,c,m)) is Element of bool X
(f - (X,ExtREAL,c,m)) . n is ext-real Element of ExtREAL
(f . n) - ((X,ExtREAL,c,m) . n) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,m) . n) is ext-real set
(f . n) + (- ((X,ExtREAL,c,m) . n)) is ext-real set
less_dom ((f - (X,ExtREAL,c,m)),ff) is Element of bool X
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB . n is Element of K403(X)
KB . m is Element of K403(X)
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . n is ext-real Element of ExtREAL
KB . n is Element of K403(X)
M . (KB . n) is ext-real Element of ExtREAL
n is set
m is set
n is set
(inferior_setsequence KB) . n is set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
m + 0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB . (m + 0) is Element of K403(X)
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,m)),ff) is Element of bool X
dom (f - (X,ExtREAL,c,m)) is Element of bool X
rng KAB is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng KAB) is ext-real Element of ExtREAL
m is set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
n is complex real ext-real Element of REAL
R_EAL n is ext-real Element of ExtREAL
less_dom ((f - (X,ExtREAL,c,n)),(R_EAL n)) is Element of bool X
dom (f - (X,ExtREAL,c,n)) is Element of bool X
m is set
dom (X,ExtREAL,c,n) is Element of bool X
m is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng m is finite Element of bool S
bool S is non empty set
union (rng m) is set
dom m is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is Element of S
x /\ (less_dom ((f - (X,ExtREAL,c,n)),(R_EAL n))) is Element of bool X
(dom (f - (X,ExtREAL,c,n))) /\ (less_dom ((f - (X,ExtREAL,c,n)),(R_EAL n))) is Element of bool X
KB . m is set
dom KB is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
[:NAT,S:] is non empty set
bool [:NAT,S:] is non empty set
n is Relation-like NAT -defined S -valued Function-like V32( NAT ,S) Element of bool [:NAT,S:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
n . m is Element of S
m + 1 is epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below Element of NAT
n . (m + 1) is Element of S
rng n is non empty Element of bool (bool X)
rng KB is Element of bool K403(X)
dom M is Element of bool S
union (rng KB) is set
M . (union (rng KB)) is ext-real Element of ExtREAL
dom KAB is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
m is set
KB . m is set
m is set
KAB . m is ext-real Element of ExtREAL
KB . m is set
M . (KB . m) is ext-real Element of ExtREAL
M * KB is Relation-like NAT -defined ExtREAL -valued Function-like V59() Element of bool [:NAT,ExtREAL:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
KB . m is Element of K403(X)
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . n is Element of K403(X)
ff is set
f . ff is ext-real Element of ExtREAL
M . {} is ext-real Element of ExtREAL
KAB is set
f . KAB is ext-real Element of ExtREAL
KB is Element of S
M . KB is ext-real Element of ExtREAL
(R_EAL (sup f1)) * (M . KB) is ext-real Element of ExtREAL
KAB is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom KAB is Element of bool X
KAB - f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (KAB - f) is Element of bool X
n is set
f . n is ext-real Element of ExtREAL
KAB . n is ext-real Element of ExtREAL
(dom KAB) /\ (dom f) is Element of bool X
KAB " {+infty} is Element of bool X
f " {+infty} is Element of bool X
(KAB " {+infty}) /\ (f " {+infty}) is Element of bool X
KAB " {-infty} is Element of bool X
f " {-infty} is Element of bool X
(KAB " {-infty}) /\ (f " {-infty}) is Element of bool X
((KAB " {+infty}) /\ (f " {+infty})) \/ ((KAB " {-infty}) /\ (f " {-infty})) is Element of bool X
((dom KAB) /\ (dom f)) \ (((KAB " {+infty}) /\ (f " {+infty})) \/ ((KAB " {-infty}) /\ (f " {-infty}))) is Element of bool X
n is set
KAB . n is ext-real Element of ExtREAL
(dom KAB) /\ (dom f) is Element of bool X
f | (dom (KAB - f)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
KAB | (dom (KAB - f)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (dom (KAB - f)))) is ext-real Element of ExtREAL
(X,S,M,(KAB | (dom (KAB - f)))) is ext-real Element of ExtREAL
x is complex real ext-real Element of REAL
ff is complex real ext-real Element of REAL
x * ff is complex real ext-real Element of REAL
(R_EAL (sup f1)) + (M . (dom f)) is ext-real Element of ExtREAL
KAB is ext-real Element of ExtREAL
KAB * ((R_EAL (sup f1)) + (M . (dom f))) is ext-real Element of ExtREAL
(X,S,M,f) - (KAB * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real Element of ExtREAL
- (KAB * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real set
(X,S,M,f) + (- (KAB * ((R_EAL (sup f1)) + (M . (dom f))))) is ext-real set
n is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
m is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng m is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng m) is ext-real Element of ExtREAL
(sup (rng m)) - KAB is ext-real Element of ExtREAL
- KAB is ext-real set
(sup (rng m)) + (- KAB) is ext-real set
n is ext-real set
dom m is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
m is set
m . m is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
n . x is Element of K403(X)
z is Element of S
M . z is ext-real Element of ExtREAL
KB \ z is M13(X,S)
M . (KB \ z) is ext-real Element of ExtREAL
(M . KB) - (M . z) is ext-real Element of ExtREAL
- (M . z) is ext-real set
(M . KB) + (- (M . z)) is ext-real set
M . (n . x) is ext-real Element of ExtREAL
(M . (dom f)) - KAB is ext-real Element of ExtREAL
(M . (dom f)) + (- KAB) is ext-real set
(M . (n . x)) + KAB is ext-real Element of ExtREAL
(M . (dom f)) - (M . (n . x)) is ext-real Element of ExtREAL
- (M . (n . x)) is ext-real set
(M . (dom f)) + (- (M . (n . x))) is ext-real set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n . x is Element of K403(X)
N0 is Element of S
KB \ N0 is M13(X,S)
M . (KB \ N0) is ext-real Element of ExtREAL
(dom f) \ (n . x) is Element of bool X
M . ((dom f) \ (n . x)) is ext-real Element of ExtREAL
N0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,N0) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,N0) is Element of bool X
m is set
(X,ExtREAL,c,N0) . m is ext-real Element of ExtREAL
n . N0 is Element of K403(X)
X \ (n . N0) is Element of bool X
KB /\ (X \ (n . N0)) is Element of bool X
KB /\ X is Element of bool X
(KB /\ X) \ (n . N0) is Element of bool X
KB \ (n . N0) is Element of bool X
A is Element of S
m is Element of S
(R_EAL (sup f1)) * KAB is ext-real Element of ExtREAL
ee is complex real ext-real Element of REAL
x * ee is complex real ext-real Element of REAL
GP is set
f . GP is ext-real Element of ExtREAL
M . m is ext-real Element of ExtREAL
KB \/ (n . N0) is Element of bool X
(KB \ (n . N0)) \/ (n . N0) is Element of bool X
A \/ m is M13(X,S)
(A \/ m) /\ (dom f) is Element of bool X
f | (A \/ m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | A is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | A)) is ext-real Element of ExtREAL
f | m is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | m)) is ext-real Element of ExtREAL
(X,S,M,(f | A)) + (X,S,M,(f | m)) is ext-real Element of ExtREAL
M . A is ext-real Element of ExtREAL
(R_EAL (sup f1)) * +infty is ext-real Element of ExtREAL
(R_EAL (sup f1)) * (M . A) is ext-real Element of ExtREAL
(A \/ m) /\ (dom (X,ExtREAL,c,N0)) is Element of bool X
(X,ExtREAL,c,N0) | (A \/ m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
GP is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom GP is Element of bool X
(X,S,M,GP) is ext-real Element of ExtREAL
dom (f | A) is Element of bool X
GP - (f | A) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (GP - (f | A)) is Element of bool X
XSMGP is set
(f | A) . XSMGP is ext-real Element of ExtREAL
GP . XSMGP is ext-real Element of ExtREAL
(dom GP) /\ (dom (f | A)) is Element of bool X
GP " {+infty} is Element of bool X
(f | A) " {+infty} is Element of bool X
(GP " {+infty}) /\ ((f | A) " {+infty}) is Element of bool X
GP " {-infty} is Element of bool X
(f | A) " {-infty} is Element of bool X
(GP " {-infty}) /\ ((f | A) " {-infty}) is Element of bool X
((GP " {+infty}) /\ ((f | A) " {+infty})) \/ ((GP " {-infty}) /\ ((f | A) " {-infty})) is Element of bool X
((dom GP) /\ (dom (f | A))) \ (((GP " {+infty}) /\ ((f | A) " {+infty})) \/ ((GP " {-infty}) /\ ((f | A) " {-infty}))) is Element of bool X
(dom f) /\ A is Element of bool X
f . XSMGP is ext-real Element of ExtREAL
XSMGP is set
GP . XSMGP is ext-real Element of ExtREAL
(X,S,M,f) - (X,S,M,GP) is ext-real Element of ExtREAL
- (X,S,M,GP) is ext-real set
(X,S,M,f) + (- (X,S,M,GP)) is ext-real set
x is complex real ext-real Element of REAL
XSMGP is complex real ext-real Element of REAL
x - XSMGP is complex real ext-real Element of REAL
(dom GP) /\ (dom (f | A)) is Element of bool X
GP | (dom (GP - (f | A))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | A) | (dom (GP - (f | A))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) - (X,S,M,(f | A)) is ext-real Element of ExtREAL
- (X,S,M,(f | A)) is ext-real set
(X,S,M,f) + (- (X,S,M,(f | A))) is ext-real set
XSMgA is complex real ext-real Element of REAL
x - XSMgA is complex real ext-real Element of REAL
betae is complex real ext-real Element of REAL
x - betae is complex real ext-real Element of REAL
XSMgB is complex real ext-real Element of REAL
EP is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom EP is Element of bool X
(X,S,M,EP) is ext-real Element of ExtREAL
KAB * (M . m) is ext-real Element of ExtREAL
XSMEP is set
EP . XSMEP is ext-real Element of ExtREAL
KAB * +infty is ext-real Element of ExtREAL
(X,ExtREAL,c,N0) | A is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,N0) | A)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,c,N0)) is ext-real Element of ExtREAL
(X,ExtREAL,c,N0) | m is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,N0) | m)) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,N0) | A)) + (X,S,M,((X,ExtREAL,c,N0) | m)) is ext-real Element of ExtREAL
MB is complex real ext-real Element of REAL
MG is complex real ext-real Element of REAL
ee * MB is complex real ext-real Element of REAL
ee * MG is complex real ext-real Element of REAL
(x - betae) - (ee * MG) is complex real ext-real Element of REAL
(x - betae) - (ee * MB) is complex real ext-real Element of REAL
XSMEP is complex real ext-real Element of REAL
XSMgB - XSMEP is complex real ext-real Element of REAL
ee * x is complex real ext-real Element of REAL
x + MG is complex real ext-real Element of REAL
ee * (x + MG) is complex real ext-real Element of REAL
x - (ee * (x + MG)) is complex real ext-real Element of REAL
dom ((X,ExtREAL,c,N0) | m) is Element of bool X
(dom (X,ExtREAL,c,N0)) /\ m is Element of bool X
((X,ExtREAL,c,N0) | m) + EP is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (((X,ExtREAL,c,N0) | m) + EP) is Element of bool X
(dom ((X,ExtREAL,c,N0) | m)) /\ (dom EP) is Element of bool X
dom (f | m) is Element of bool X
(((X,ExtREAL,c,N0) | m) + EP) - (f | m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((((X,ExtREAL,c,N0) | m) + EP) - (f | m)) is Element of bool X
f - (X,ExtREAL,c,N0) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is set
(f | m) . x is ext-real Element of ExtREAL
(((X,ExtREAL,c,N0) | m) + EP) . x is ext-real Element of ExtREAL
(dom (((X,ExtREAL,c,N0) | m) + EP)) /\ (dom (f | m)) is Element of bool X
(((X,ExtREAL,c,N0) | m) + EP) " {+infty} is Element of bool X
(f | m) " {+infty} is Element of bool X
((((X,ExtREAL,c,N0) | m) + EP) " {+infty}) /\ ((f | m) " {+infty}) is Element of bool X
(((X,ExtREAL,c,N0) | m) + EP) " {-infty} is Element of bool X
(f | m) " {-infty} is Element of bool X
((((X,ExtREAL,c,N0) | m) + EP) " {-infty}) /\ ((f | m) " {-infty}) is Element of bool X
(((((X,ExtREAL,c,N0) | m) + EP) " {+infty}) /\ ((f | m) " {+infty})) \/ (((((X,ExtREAL,c,N0) | m) + EP) " {-infty}) /\ ((f | m) " {-infty})) is Element of bool X
((dom (((X,ExtREAL,c,N0) | m) + EP)) /\ (dom (f | m))) \ ((((((X,ExtREAL,c,N0) | m) + EP) " {+infty}) /\ ((f | m) " {+infty})) \/ (((((X,ExtREAL,c,N0) | m) + EP) " {-infty}) /\ ((f | m) " {-infty}))) is Element of bool X
((X,ExtREAL,c,N0) | m) . x is ext-real Element of ExtREAL
EP . x is ext-real Element of ExtREAL
(((X,ExtREAL,c,N0) | m) . x) + (EP . x) is ext-real Element of ExtREAL
(X,ExtREAL,c,N0) . x is ext-real Element of ExtREAL
((X,ExtREAL,c,N0) . x) + (EP . x) is ext-real Element of ExtREAL
((X,ExtREAL,c,N0) . x) + KAB is ext-real Element of ExtREAL
less_dom ((f - (X,ExtREAL,c,N0)),KAB) is Element of bool X
(f - (X,ExtREAL,c,N0)) . x is ext-real Element of ExtREAL
dom (f - (X,ExtREAL,c,N0)) is Element of bool X
f . x is ext-real Element of ExtREAL
(f . x) - ((X,ExtREAL,c,N0) . x) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,N0) . x) is ext-real set
(f . x) + (- ((X,ExtREAL,c,N0) . x)) is ext-real set
(dom (((X,ExtREAL,c,N0) | m) + EP)) /\ (dom (f | m)) is Element of bool X
(f | m) | (dom ((((X,ExtREAL,c,N0) | m) + EP) - (f | m))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(((X,ExtREAL,c,N0) | m) + EP) | (dom ((((X,ExtREAL,c,N0) | m) + EP) - (f | m))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(((X,ExtREAL,c,N0) | m) + EP)) is ext-real Element of ExtREAL
((X,ExtREAL,c,N0) | m) | m is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(((X,ExtREAL,c,N0) | m) | m)) is ext-real Element of ExtREAL
EP | m is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(EP | m)) is ext-real Element of ExtREAL
(X,S,M,(((X,ExtREAL,c,N0) | m) | m)) + (X,S,M,(EP | m)) is ext-real Element of ExtREAL
(X,S,M,(((X,ExtREAL,c,N0) | m) | m)) + (X,S,M,EP) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,N0) | m)) + (X,S,M,EP) is ext-real Element of ExtREAL
(X,S,M,(f | m)) - (X,S,M,EP) is ext-real Element of ExtREAL
- (X,S,M,EP) is ext-real set
(X,S,M,(f | m)) + (- (X,S,M,EP)) is ext-real set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
KAB is ext-real Element of ExtREAL
KAB * ((R_EAL (sup f1)) + (M . (dom f))) is ext-real Element of ExtREAL
(X,S,M,f) - (KAB * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real Element of ExtREAL
- (KAB * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real set
(X,S,M,f) + (- (KAB * ((R_EAL (sup f1)) + (M . (dom f))))) is ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,m)) is ext-real Element of ExtREAL
B . m is ext-real Element of ExtREAL
m is ext-real Element of ExtREAL
n is complex real ext-real Element of REAL
KAB is complex real ext-real Element of REAL
KAB / 2 is complex real ext-real Element of REAL
R_EAL (KAB / 2) is ext-real Element of ExtREAL
(R_EAL (KAB / 2)) * ((R_EAL (sup f1)) + (M . (dom f))) is ext-real Element of ExtREAL
n is complex real ext-real Element of REAL
x + n is complex real ext-real Element of REAL
n / (x + n) is complex real ext-real Element of REAL
KAB is complex real ext-real Element of REAL
KAB / 2 is complex real ext-real Element of REAL
min ((n / (x + n)),(KAB / 2)) is complex real ext-real Element of REAL
R_EAL (x + n) is ext-real Element of ExtREAL
R_EAL x is ext-real Element of ExtREAL
R_EAL n is ext-real Element of ExtREAL
(R_EAL x) + (R_EAL n) is ext-real Element of ExtREAL
(min ((n / (x + n)),(KAB / 2))) * (x + n) is complex real ext-real Element of REAL
R_EAL (min ((n / (x + n)),(KAB / 2))) is ext-real Element of ExtREAL
(R_EAL (min ((n / (x + n)),(KAB / 2)))) * ((R_EAL (sup f1)) + (M . (dom f))) is ext-real Element of ExtREAL
(n / (x + n)) * (x + n) is complex real ext-real Element of REAL
KAB is ext-real Element of ExtREAL
(X,S,M,f) - KAB is ext-real Element of ExtREAL
- KAB is ext-real set
(X,S,M,f) + (- KAB) is ext-real set
n is ext-real Element of ExtREAL
n * ((R_EAL (sup f1)) + (M . (dom f))) is ext-real Element of ExtREAL
(X,S,M,f) - (n * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real Element of ExtREAL
- (n * ((R_EAL (sup f1)) + (M . (dom f)))) is ext-real set
(X,S,M,f) + (- (n * ((R_EAL (sup f1)) + (M . (dom f))))) is ext-real set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . n is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . n is ext-real Element of ExtREAL
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . KAB is ext-real Element of ExtREAL
(X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,KAB)) is ext-real Element of ExtREAL
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . KAB is ext-real Element of ExtREAL
B . n is ext-real Element of ExtREAL
(X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,KAB) is Element of bool X
(X,ExtREAL,c,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,n) is Element of bool X
(X,ExtREAL,c,n) - (X,ExtREAL,c,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,n) - (X,ExtREAL,c,KAB)) is Element of bool X
m is set
(X,ExtREAL,c,KAB) . m is ext-real Element of ExtREAL
(X,ExtREAL,c,n) . m is ext-real Element of ExtREAL
(dom (X,ExtREAL,c,n)) /\ (dom (X,ExtREAL,c,KAB)) is Element of bool X
(X,ExtREAL,c,n) " {+infty} is Element of bool X
(X,ExtREAL,c,KAB) " {+infty} is Element of bool X
((X,ExtREAL,c,n) " {+infty}) /\ ((X,ExtREAL,c,KAB) " {+infty}) is Element of bool X
(X,ExtREAL,c,n) " {-infty} is Element of bool X
(X,ExtREAL,c,KAB) " {-infty} is Element of bool X
((X,ExtREAL,c,n) " {-infty}) /\ ((X,ExtREAL,c,KAB) " {-infty}) is Element of bool X
(((X,ExtREAL,c,n) " {+infty}) /\ ((X,ExtREAL,c,KAB) " {+infty})) \/ (((X,ExtREAL,c,n) " {-infty}) /\ ((X,ExtREAL,c,KAB) " {-infty})) is Element of bool X
((dom (X,ExtREAL,c,n)) /\ (dom (X,ExtREAL,c,KAB))) \ ((((X,ExtREAL,c,n) " {+infty}) /\ ((X,ExtREAL,c,KAB) " {+infty})) \/ (((X,ExtREAL,c,n) " {-infty}) /\ ((X,ExtREAL,c,KAB) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,c,n)) /\ (dom (X,ExtREAL,c,KAB)) is Element of bool X
(X,ExtREAL,c,n) | (dom ((X,ExtREAL,c,n) - (X,ExtREAL,c,KAB))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,KAB) | (dom ((X,ExtREAL,c,n) - (X,ExtREAL,c,KAB))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,KAB) | (dom ((X,ExtREAL,c,n) - (X,ExtREAL,c,KAB))))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,n) | (dom ((X,ExtREAL,c,n) - (X,ExtREAL,c,KAB))))) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,c,n)) is ext-real Element of ExtREAL
KAB is complex real ext-real set
KAB is complex real ext-real set
n is ext-real set
rng B is non empty ext-real-membered V93() Element of bool ExtREAL
dom B is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
R_EAL KAB is ext-real Element of ExtREAL
m is set
B . m is ext-real Element of ExtREAL
sup (rng B) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . n is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
B . m is ext-real Element of ExtREAL
B . 1 is ext-real Element of ExtREAL
m is complex real ext-real set
R_EAL m is ext-real Element of ExtREAL
(sup (rng B)) + (R_EAL m) is ext-real Element of ExtREAL
(R_EAL m) + (sup (rng B)) is ext-real Element of ExtREAL
- (sup (rng B)) is ext-real Element of ExtREAL
((R_EAL m) + (sup (rng B))) + (- (sup (rng B))) is ext-real Element of ExtREAL
(sup (rng B)) + (- (sup (rng B))) is ext-real Element of ExtREAL
(R_EAL m) + ((sup (rng B)) + (- (sup (rng B)))) is ext-real Element of ExtREAL
(R_EAL m) + 0 is ext-real set
(sup (rng B)) - (R_EAL m) is ext-real Element of ExtREAL
- (R_EAL m) is ext-real set
(sup (rng B)) + (- (R_EAL m)) is ext-real set
n is ext-real set
m is set
B . m is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
z is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . z is ext-real Element of ExtREAL
(B . z) - (sup (rng B)) is ext-real Element of ExtREAL
- (sup (rng B)) is ext-real set
(B . z) + (- (sup (rng B))) is ext-real set
|.((B . z) - (sup (rng B))).| is ext-real Element of ExtREAL
B . x is ext-real Element of ExtREAL
(B . z) + (R_EAL m) is ext-real Element of ExtREAL
(sup (rng B)) - (B . z) is ext-real Element of ExtREAL
- (B . z) is ext-real set
(sup (rng B)) + (- (B . z)) is ext-real set
- ((sup (rng B)) - (B . z)) is ext-real Element of ExtREAL
- (R_EAL m) is ext-real Element of ExtREAL
(sup (rng B)) + 0. is ext-real Element of ExtREAL
n is complex real ext-real Element of REAL
R_EAL n is ext-real Element of ExtREAL
m is complex real ext-real set
R_EAL m is ext-real Element of ExtREAL
(X,S,M,f) - (R_EAL m) is ext-real Element of ExtREAL
- (R_EAL m) is ext-real set
(X,S,M,f) + (- (R_EAL m)) is ext-real set
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . n is ext-real Element of ExtREAL
(B) + m is ext-real set
KAB is complex real ext-real set
R_EAL KAB is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . n is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . m is ext-real Element of ExtREAL
KAB is complex real ext-real set
KB is ext-real Element of ExtREAL
(R_EAL (inf f1)) - KB is ext-real Element of ExtREAL
- KB is ext-real set
(R_EAL (inf f1)) + (- KB) is ext-real set
KAB is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
n is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng n is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng n) is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . m is Element of K403(X)
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,m) is Element of bool X
m is set
(X,ExtREAL,c,m) . m is ext-real Element of ExtREAL
X \ (KAB . m) is Element of bool X
ff is Element of S
ff /\ (X \ (KAB . m)) is Element of bool X
ff /\ X is Element of bool X
(ff /\ X) \ (KAB . m) is Element of bool X
ff \ (KAB . m) is Element of bool X
ff \/ (KAB . m) is Element of bool X
(ff \ (KAB . m)) \/ (KAB . m) is Element of bool X
m is Element of S
n is Element of S
m \/ n is M13(X,S)
(m \/ n) /\ (dom (X,ExtREAL,c,m)) is Element of bool X
(X,ExtREAL,c,m) | (m \/ n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom x is Element of bool X
z is set
x . z is ext-real Element of ExtREAL
(X,ExtREAL,c,m) | n is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,m) | n) is Element of bool X
(dom (X,ExtREAL,c,m)) /\ n is Element of bool X
((X,ExtREAL,c,m) | n) - x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (((X,ExtREAL,c,m) | n) - x) is Element of bool X
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is set
x . x is ext-real Element of ExtREAL
((X,ExtREAL,c,m) | n) . x is ext-real Element of ExtREAL
(dom ((X,ExtREAL,c,m) | n)) /\ (dom x) is Element of bool X
((X,ExtREAL,c,m) | n) " {+infty} is Element of bool X
x " {+infty} is Element of bool X
(((X,ExtREAL,c,m) | n) " {+infty}) /\ (x " {+infty}) is Element of bool X
((X,ExtREAL,c,m) | n) " {-infty} is Element of bool X
x " {-infty} is Element of bool X
(((X,ExtREAL,c,m) | n) " {-infty}) /\ (x " {-infty}) is Element of bool X
((((X,ExtREAL,c,m) | n) " {+infty}) /\ (x " {+infty})) \/ ((((X,ExtREAL,c,m) | n) " {-infty}) /\ (x " {-infty})) is Element of bool X
((dom ((X,ExtREAL,c,m) | n)) /\ (dom x)) \ (((((X,ExtREAL,c,m) | n) " {+infty}) /\ (x " {+infty})) \/ ((((X,ExtREAL,c,m) | n) " {-infty}) /\ (x " {-infty}))) is Element of bool X
(X,ExtREAL,c,m) . x is ext-real Element of ExtREAL
less_dom ((f - (X,ExtREAL,c,m)),KB) is Element of bool X
dom (f - (X,ExtREAL,c,m)) is Element of bool X
(f - (X,ExtREAL,c,m)) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
(f . x) - ((X,ExtREAL,c,m) . x) is ext-real Element of ExtREAL
- ((X,ExtREAL,c,m) . x) is ext-real set
(f . x) + (- ((X,ExtREAL,c,m) . x)) is ext-real set
((X,ExtREAL,c,m) . x) + KB is ext-real Element of ExtREAL
(f . x) - KB is ext-real Element of ExtREAL
(f . x) + (- KB) is ext-real set
(X,ExtREAL,c,m) | m is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,m) | m)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,c,m)) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,m) | n)) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,m) | m)) + (X,S,M,((X,ExtREAL,c,m) | n)) is ext-real Element of ExtREAL
(dom ((X,ExtREAL,c,m) | n)) /\ (dom x) is Element of bool X
x | (dom (((X,ExtREAL,c,m) | n) - x)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((X,ExtREAL,c,m) | n) | (dom (((X,ExtREAL,c,m) | n) - x)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(x | (dom (((X,ExtREAL,c,m) | n) - x)))) is ext-real Element of ExtREAL
(X,S,M,(((X,ExtREAL,c,m) | n) | (dom (((X,ExtREAL,c,m) | n) - x)))) is ext-real Element of ExtREAL
(X,S,M,x) is ext-real Element of ExtREAL
M . n is ext-real Element of ExtREAL
((R_EAL (inf f1)) - KB) * (M . n) is ext-real Element of ExtREAL
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,m)),KB) is Element of bool X
M . (less_dom ((f - (X,ExtREAL,c,m)),KB)) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - KB) * (M . (less_dom ((f - (X,ExtREAL,c,m)),KB))) is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,m)),KB) is Element of bool X
M . (less_dom ((f - (X,ExtREAL,c,m)),KB)) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - KB) * (M . (less_dom ((f - (X,ExtREAL,c,m)),KB))) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,c,m)) is ext-real Element of ExtREAL
R_EAL 2 is ext-real Element of ExtREAL
(R_EAL (inf f1)) / (R_EAL 2) is ext-real Element of ExtREAL
(R_EAL 2) " is ext-real set
(R_EAL (inf f1)) * ((R_EAL 2) ") is ext-real set
n is complex real ext-real set
KB is complex real ext-real Element of REAL
KB / 2 is complex real ext-real Element of REAL
KB - (KB / 2) is complex real ext-real Element of REAL
n is complex real ext-real Element of REAL
n / (KB - (KB / 2)) is complex real ext-real Element of REAL
(KB - (KB / 2)) * (n / (KB - (KB / 2))) is complex real ext-real Element of REAL
R_EAL (KB - (KB / 2)) is ext-real Element of ExtREAL
R_EAL (n / (KB - (KB / 2))) is ext-real Element of ExtREAL
(R_EAL (KB - (KB / 2))) * (R_EAL (n / (KB - (KB / 2)))) is ext-real Element of ExtREAL
m is Relation-like NAT -defined K403(X) -valued Function-like V32( NAT ,K403(X)) Element of bool [:NAT,K403(X):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng x is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng x) is ext-real Element of ExtREAL
n / (KB - (KB / 2)) is complex real ext-real Element of REAL
R_EAL n is ext-real Element of ExtREAL
(R_EAL n) / (R_EAL (KB - (KB / 2))) is ext-real Element of ExtREAL
(R_EAL (KB - (KB / 2))) " is ext-real set
(R_EAL n) * ((R_EAL (KB - (KB / 2))) ") is ext-real set
R_EAL n is ext-real Element of ExtREAL
(R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2)) is ext-real Element of ExtREAL
- ((R_EAL (inf f1)) / (R_EAL 2)) is ext-real set
(R_EAL (inf f1)) + (- ((R_EAL (inf f1)) / (R_EAL 2))) is ext-real set
(R_EAL n) / ((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) " is ext-real set
(R_EAL n) * (((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) ") is ext-real set
z is ext-real set
z is ext-real set
z is ext-real Element of ExtREAL
(KB / 2) - (KB / 2) is complex real ext-real Element of REAL
dom x is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
x is set
x . x is ext-real Element of ExtREAL
N0 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) * ((R_EAL n) / ((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2)))) is ext-real Element of ExtREAL
m . N0 is Element of K403(X)
M . (m . N0) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) * (M . (m . N0)) is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . m is ext-real Element of ExtREAL
m . m is Element of K403(X)
M . (m . m) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) * (M . (m . m)) is ext-real Element of ExtREAL
(X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f - (X,ExtREAL,c,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
less_dom ((f - (X,ExtREAL,c,m)),((R_EAL (inf f1)) / (R_EAL 2))) is Element of bool X
M . (less_dom ((f - (X,ExtREAL,c,m)),((R_EAL (inf f1)) / (R_EAL 2)))) is ext-real Element of ExtREAL
((R_EAL (inf f1)) - ((R_EAL (inf f1)) / (R_EAL 2))) * (M . (less_dom ((f - (X,ExtREAL,c,m)),((R_EAL (inf f1)) / (R_EAL 2))))) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,c,m)) is ext-real Element of ExtREAL
KB is complex real ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
eq_dom (f,(R_EAL 0)) is Element of bool X
x is Element of S
great_eq_dom (f,(R_EAL 0)) is Element of bool X
x /\ (great_eq_dom (f,(R_EAL 0))) is Element of bool X
a is set
x \/ (eq_dom (f,(R_EAL 0))) is Element of bool X
x \ (eq_dom (f,(R_EAL 0))) is Element of bool X
(x \ (eq_dom (f,(R_EAL 0)))) \/ (eq_dom (f,(R_EAL 0))) is Element of bool X
(dom f) \ (eq_dom (f,(R_EAL 0))) is Element of bool X
I1 is Element of S
less_eq_dom (f,(R_EAL 0)) is Element of bool X
I1 /\ (less_eq_dom (f,(R_EAL 0))) is Element of bool X
x /\ (eq_dom (f,(R_EAL 0))) is Element of bool X
X \ (eq_dom (f,(R_EAL 0))) is Element of bool X
x /\ (X \ (eq_dom (f,(R_EAL 0)))) is Element of bool X
x /\ X is Element of bool X
(x /\ X) \ (eq_dom (f,(R_EAL 0))) is Element of bool X
E is Element of S
f1 is Element of S
f | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | f1) is Element of bool X
(dom f) /\ f1 is Element of bool X
x is set
(f | f1) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
x is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
DFPG is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,c,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,NFPG) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,NFPG) | f1)) is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,x,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,x,NFPG)) is ext-real Element of ExtREAL
(X,ExtREAL,c,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,NFPG) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,NFPG) | f1)) is ext-real Element of ExtREAL
NFPG is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng NFPG is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng NFPG) is ext-real Element of ExtREAL
(NFPG) is ext-real Element of ExtREAL
x is set
f . x is ext-real Element of ExtREAL
E \/ f1 is M13(X,S)
(E \/ f1) /\ (dom f) is Element of bool X
f | (E \/ f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | E)) is ext-real Element of ExtREAL
(X,S,M,(f | f1)) is ext-real Element of ExtREAL
(X,S,M,(f | E)) + (X,S,M,(f | f1)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,x) | f1) is Element of bool X
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,x) is Element of bool X
dom (X,ExtREAL,c,x) is Element of bool X
f1 \/ E is M13(X,S)
(dom (X,ExtREAL,c,x)) /\ f1 is Element of bool X
x is Element of X
(X,x,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(f | f1) . x is ext-real Element of ExtREAL
((X,x,x)) is ext-real Element of ExtREAL
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,c,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,ff) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,ff) | f1) is Element of bool X
(X,x,x) . ff is ext-real Element of ExtREAL
(X,ExtREAL,x,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,ff) . x is ext-real Element of ExtREAL
((X,ExtREAL,c,ff) | f1) . x is ext-real Element of ExtREAL
(X,ExtREAL,c,ff) . x is ext-real Element of ExtREAL
(X,c,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,c,x) . ff is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
((X,c,x)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,ff) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
KB is Element of X
((X,ExtREAL,c,x) | f1) . KB is ext-real Element of ExtREAL
((X,ExtREAL,c,ff) | f1) . KB is ext-real Element of ExtREAL
(X,ExtREAL,x,x) . KB is ext-real Element of ExtREAL
(X,ExtREAL,x,ff) . KB is ext-real Element of ExtREAL
dom ((X,ExtREAL,c,x) | f1) is Element of bool X
(X,ExtREAL,c,x) . KB is ext-real Element of ExtREAL
(X,ExtREAL,c,ff) . KB is ext-real Element of ExtREAL
dom ((X,ExtREAL,c,ff) | f1) is Element of bool X
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB is Element of X
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) . KB is ext-real Element of ExtREAL
(X,ExtREAL,x,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,ff) . KB is ext-real Element of ExtREAL
(DFPG) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . x is ext-real Element of ExtREAL
DFPG . ff is ext-real Element of ExtREAL
(X,ExtREAL,x,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,ff) is Element of bool X
(X,S,M,(X,ExtREAL,x,ff)) is ext-real Element of ExtREAL
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,x,x)) is ext-real Element of ExtREAL
dom (X,ExtREAL,x,x) is Element of bool X
(X,ExtREAL,x,ff) - (X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,ff) - (X,ExtREAL,x,x)) is Element of bool X
KB is set
(X,ExtREAL,x,x) . KB is ext-real Element of ExtREAL
(X,ExtREAL,x,ff) . KB is ext-real Element of ExtREAL
(dom (X,ExtREAL,x,ff)) /\ (dom (X,ExtREAL,x,x)) is Element of bool X
(X,ExtREAL,x,ff) " {+infty} is Element of bool X
(X,ExtREAL,x,x) " {+infty} is Element of bool X
((X,ExtREAL,x,ff) " {+infty}) /\ ((X,ExtREAL,x,x) " {+infty}) is Element of bool X
(X,ExtREAL,x,ff) " {-infty} is Element of bool X
(X,ExtREAL,x,x) " {-infty} is Element of bool X
((X,ExtREAL,x,ff) " {-infty}) /\ ((X,ExtREAL,x,x) " {-infty}) is Element of bool X
(((X,ExtREAL,x,ff) " {+infty}) /\ ((X,ExtREAL,x,x) " {+infty})) \/ (((X,ExtREAL,x,ff) " {-infty}) /\ ((X,ExtREAL,x,x) " {-infty})) is Element of bool X
((dom (X,ExtREAL,x,ff)) /\ (dom (X,ExtREAL,x,x))) \ ((((X,ExtREAL,x,ff) " {+infty}) /\ ((X,ExtREAL,x,x) " {+infty})) \/ (((X,ExtREAL,x,ff) " {-infty}) /\ ((X,ExtREAL,x,x) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,x,ff)) /\ (dom (X,ExtREAL,x,x)) is Element of bool X
(X,ExtREAL,x,ff) | (dom ((X,ExtREAL,x,ff) - (X,ExtREAL,x,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) | (dom ((X,ExtREAL,x,ff) - (X,ExtREAL,x,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
rng DFPG is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng DFPG) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG . x is ext-real Element of ExtREAL
NFPG . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,x) is Element of bool X
f1 \/ E is M13(X,S)
(E \/ f1) /\ (dom (X,ExtREAL,c,x)) is Element of bool X
ff is set
(X,ExtREAL,c,x) . ff is ext-real Element of ExtREAL
(X,ExtREAL,c,x) | (E \/ f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
(X,ExtREAL,c,x) | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,x) | E)) is ext-real Element of ExtREAL
(X,ExtREAL,c,x) | f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,x) | f1)) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,x) | E)) + (X,S,M,((X,ExtREAL,c,x) | f1)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG . x is ext-real Element of ExtREAL
NFPG . ff is ext-real Element of ExtREAL
(X,ExtREAL,c,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,ff) is Element of bool X
(X,S,M,(X,ExtREAL,c,ff)) is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
dom (X,ExtREAL,c,x) is Element of bool X
(X,ExtREAL,c,ff) - (X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,ff) - (X,ExtREAL,c,x)) is Element of bool X
KB is set
(X,ExtREAL,c,x) . KB is ext-real Element of ExtREAL
(X,ExtREAL,c,ff) . KB is ext-real Element of ExtREAL
(dom (X,ExtREAL,c,ff)) /\ (dom (X,ExtREAL,c,x)) is Element of bool X
(X,ExtREAL,c,ff) " {+infty} is Element of bool X
(X,ExtREAL,c,x) " {+infty} is Element of bool X
((X,ExtREAL,c,ff) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty}) is Element of bool X
(X,ExtREAL,c,ff) " {-infty} is Element of bool X
(X,ExtREAL,c,x) " {-infty} is Element of bool X
((X,ExtREAL,c,ff) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty}) is Element of bool X
(((X,ExtREAL,c,ff) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty})) \/ (((X,ExtREAL,c,ff) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty})) is Element of bool X
((dom (X,ExtREAL,c,ff)) /\ (dom (X,ExtREAL,c,x))) \ ((((X,ExtREAL,c,ff) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty})) \/ (((X,ExtREAL,c,ff) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,c,ff)) /\ (dom (X,ExtREAL,c,x)) is Element of bool X
(X,ExtREAL,c,ff) | (dom ((X,ExtREAL,c,ff) - (X,ExtREAL,c,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) | (dom ((X,ExtREAL,c,ff) - (X,ExtREAL,c,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) 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 set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Element of S
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
B is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
I1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
(I1) is ext-real Element of ExtREAL
rng I1 is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng I1) is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . a is ext-real Element of ExtREAL
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f1 is Element of bool X
E is Element of X
(X,B,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
f1 . E is ext-real Element of ExtREAL
((X,B,E)) is ext-real Element of ExtREAL
(X,c,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,c,E) . a is ext-real Element of ExtREAL
rng (X,c,E) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,c,E)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,E) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,c,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,DFPG) . E is ext-real Element of ExtREAL
(X,c,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) . E is ext-real Element of ExtREAL
((X,c,E)) is ext-real Element of ExtREAL
(X,S,M,f1) is ext-real Element of ExtREAL
E is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng E is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng E) is ext-real Element of ExtREAL
(E) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,B,x)) is ext-real Element of ExtREAL
I1 . x is ext-real Element of ExtREAL
rng x is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng x) is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is ext-real Element of ExtREAL
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
(X,ExtREAL,B,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f1 is Element of bool X
E is Element of X
(X,c,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
f1 . E is ext-real Element of ExtREAL
((X,c,E)) is ext-real Element of ExtREAL
(X,B,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,B,E) . a is ext-real Element of ExtREAL
rng (X,B,E) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,B,E)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,B,E) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) . E is ext-real Element of ExtREAL
(X,B,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . E is ext-real Element of ExtREAL
((X,B,E)) is ext-real Element of ExtREAL
(X,S,M,f1) is ext-real Element of ExtREAL
E is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng E is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng E) is ext-real Element of ExtREAL
(E) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
E . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
x . x is ext-real Element of ExtREAL
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 V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
c is Element of S
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
(X,ExtREAL,c,0) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of X
(X,c,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
B . x is ext-real Element of ExtREAL
((X,c,x)) is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,x) . a is ext-real Element of ExtREAL
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,a) . x is ext-real Element of ExtREAL
(X,c,x) . I1 is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,I1) . x is ext-real Element of ExtREAL
(X,c,x) . 0 is ext-real Element of ExtREAL
rng (X,c,x) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,c,x)) is ext-real Element of ExtREAL
dom B is Element of bool X
(X,S,M,B) is ext-real Element of ExtREAL
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng x is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng x) is ext-real Element of ExtREAL
(x) is ext-real Element of ExtREAL
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
c is ext-real Element of ExtREAL
B is ext-real Element of ExtREAL
x is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(I1) is ext-real Element of ExtREAL
x is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(I1) is ext-real Element of ExtREAL
a is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(f1) is ext-real Element of ExtREAL
a is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(f1) is ext-real Element of ExtREAL
E is Element of X
(X,x,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,a,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,x,E)) is ext-real Element of ExtREAL
((X,a,E)) is ext-real Element of ExtREAL
f . E is ext-real Element of ExtREAL
E is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
(X,S,M,f) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
dom f is Element of bool X
B is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
I1 is Element of X
(X,ExtREAL,c,B) . I1 is ext-real Element of ExtREAL
(X,ExtREAL,c,x) . I1 is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
B is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) is ext-real Element of ExtREAL
(B) is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
rng I1 is finite Element of bool S
bool S is non empty set
union (rng I1) is set
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
x is Element of S
I1 is Element of X
(X,c,I1) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,c,I1)) is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,c,I1) . a is ext-real Element of ExtREAL
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,a) . I1 is ext-real Element of ExtREAL
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,c,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,a) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f + c)) is ext-real Element of ExtREAL
(X,S,M,f) is ext-real Element of ExtREAL
(X,S,M,c) is ext-real Element of ExtREAL
(X,S,M,f) + (X,S,M,c) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
B is Element of S
B is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
I1 is Element of S
I1 is Element of S
(dom f) /\ (dom c) is Element of bool X
dom (f + c) is Element of bool X
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . a is ext-real Element of ExtREAL
x . f1 is ext-real Element of ExtREAL
(X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,f1) is Element of bool X
(X,ExtREAL,B,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,a) is Element of bool X
E is set
(X,ExtREAL,B,f1) - (X,ExtREAL,B,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,f1) - (X,ExtREAL,B,a)) is Element of bool X
(dom (X,ExtREAL,B,f1)) /\ (dom (X,ExtREAL,B,a)) is Element of bool X
(X,ExtREAL,B,f1) " {+infty} is Element of bool X
(X,ExtREAL,B,a) " {+infty} is Element of bool X
((X,ExtREAL,B,f1) " {+infty}) /\ ((X,ExtREAL,B,a) " {+infty}) is Element of bool X
(X,ExtREAL,B,f1) " {-infty} is Element of bool X
(X,ExtREAL,B,a) " {-infty} is Element of bool X
((X,ExtREAL,B,f1) " {-infty}) /\ ((X,ExtREAL,B,a) " {-infty}) is Element of bool X
(((X,ExtREAL,B,f1) " {+infty}) /\ ((X,ExtREAL,B,a) " {+infty})) \/ (((X,ExtREAL,B,f1) " {-infty}) /\ ((X,ExtREAL,B,a) " {-infty})) is Element of bool X
((dom (X,ExtREAL,B,f1)) /\ (dom (X,ExtREAL,B,a))) \ ((((X,ExtREAL,B,f1) " {+infty}) /\ ((X,ExtREAL,B,a) " {+infty})) \/ (((X,ExtREAL,B,f1) " {-infty}) /\ ((X,ExtREAL,B,a) " {-infty}))) is Element of bool X
(X,ExtREAL,B,a) . E is ext-real Element of ExtREAL
(X,ExtREAL,B,f1) . E is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,f1)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,a)) is ext-real Element of ExtREAL
(X,ExtREAL,B,f1) | (dom ((X,ExtREAL,B,f1) - (X,ExtREAL,B,a))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,a) | (dom ((X,ExtREAL,B,f1) - (X,ExtREAL,B,a))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
a is Element of S
a is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(f1) is ext-real Element of ExtREAL
E is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,E,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,E,x) is Element of bool X
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,x) is Element of bool X
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) + (X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,x) is Element of bool X
(dom (X,ExtREAL,B,x)) /\ (dom (X,ExtREAL,a,x)) is Element of bool X
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,E,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) + (X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,DFPG) + (X,ExtREAL,a,DFPG)) is Element of bool X
dom (X,ExtREAL,E,DFPG) is Element of bool X
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) + (X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,x) + (X,ExtREAL,a,x)) is Element of bool X
dom (X,ExtREAL,E,x) is Element of bool X
NFPG is Element of X
(X,ExtREAL,E,x) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,E,DFPG) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,a,x) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,a,DFPG) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,DFPG) + (X,ExtREAL,a,DFPG)) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,DFPG) . NFPG) + ((X,ExtREAL,a,DFPG) . NFPG) is ext-real Element of ExtREAL
((X,ExtREAL,B,x) + (X,ExtREAL,a,x)) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,x) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,x) . NFPG) + ((X,ExtREAL,a,x) . NFPG) is ext-real Element of ExtREAL
x is set
dom f1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f1 . DFPG is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,DFPG)) is ext-real Element of ExtREAL
f1 . x is ext-real Element of ExtREAL
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,E,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,E,DFPG)) is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . DFPG is ext-real Element of ExtREAL
x . DFPG is ext-real Element of ExtREAL
f1 . DFPG is ext-real Element of ExtREAL
(x . DFPG) + (f1 . DFPG) is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) + (X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,DFPG) is Element of bool X
dom (X,ExtREAL,a,DFPG) is Element of bool X
(X,S,M,(X,ExtREAL,E,DFPG)) is ext-real Element of ExtREAL
dom (X,ExtREAL,E,DFPG) is Element of bool X
(dom (X,ExtREAL,B,DFPG)) /\ (dom (X,ExtREAL,a,DFPG)) is Element of bool X
(X,ExtREAL,B,DFPG) | (dom (X,ExtREAL,B,DFPG)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,B,DFPG) | (dom (X,ExtREAL,B,DFPG)))) is ext-real Element of ExtREAL
(X,ExtREAL,a,DFPG) | (dom (X,ExtREAL,a,DFPG)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,a,DFPG) | (dom (X,ExtREAL,a,DFPG)))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,B,DFPG) | (dom (X,ExtREAL,B,DFPG)))) + (X,S,M,((X,ExtREAL,a,DFPG) | (dom (X,ExtREAL,a,DFPG)))) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,DFPG)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,DFPG)) + (X,S,M,((X,ExtREAL,a,DFPG) | (dom (X,ExtREAL,a,DFPG)))) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,DFPG)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,DFPG)) + (X,S,M,(X,ExtREAL,a,DFPG)) is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . DFPG is ext-real Element of ExtREAL
f1 . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,a,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,NFPG) is Element of bool X
(X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,DFPG) is Element of bool X
x is set
(X,ExtREAL,a,NFPG) - (X,ExtREAL,a,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,a,NFPG) - (X,ExtREAL,a,DFPG)) is Element of bool X
(dom (X,ExtREAL,a,NFPG)) /\ (dom (X,ExtREAL,a,DFPG)) is Element of bool X
(X,ExtREAL,a,NFPG) " {+infty} is Element of bool X
(X,ExtREAL,a,DFPG) " {+infty} is Element of bool X
((X,ExtREAL,a,NFPG) " {+infty}) /\ ((X,ExtREAL,a,DFPG) " {+infty}) is Element of bool X
(X,ExtREAL,a,NFPG) " {-infty} is Element of bool X
(X,ExtREAL,a,DFPG) " {-infty} is Element of bool X
((X,ExtREAL,a,NFPG) " {-infty}) /\ ((X,ExtREAL,a,DFPG) " {-infty}) is Element of bool X
(((X,ExtREAL,a,NFPG) " {+infty}) /\ ((X,ExtREAL,a,DFPG) " {+infty})) \/ (((X,ExtREAL,a,NFPG) " {-infty}) /\ ((X,ExtREAL,a,DFPG) " {-infty})) is Element of bool X
((dom (X,ExtREAL,a,NFPG)) /\ (dom (X,ExtREAL,a,DFPG))) \ ((((X,ExtREAL,a,NFPG) " {+infty}) /\ ((X,ExtREAL,a,DFPG) " {+infty})) \/ (((X,ExtREAL,a,NFPG) " {-infty}) /\ ((X,ExtREAL,a,DFPG) " {-infty}))) is Element of bool X
(X,ExtREAL,a,DFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,a,NFPG) . x is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,NFPG)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,DFPG)) is ext-real Element of ExtREAL
(X,ExtREAL,a,NFPG) | (dom ((X,ExtREAL,a,NFPG) - (X,ExtREAL,a,DFPG))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,DFPG) | (dom ((X,ExtREAL,a,NFPG) - (X,ExtREAL,a,DFPG))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
DFPG is set
dom x is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x . NFPG is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,NFPG)) is ext-real Element of ExtREAL
x . DFPG is ext-real Element of ExtREAL
(x) is ext-real Element of ExtREAL
(x) + (f1) is ext-real Element of ExtREAL
DFPG is Element of X
(X,E,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,E,DFPG)) is ext-real Element of ExtREAL
(f + c) . DFPG is ext-real Element of ExtREAL
NFPG is set
(X,B,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
dom (X,B,DFPG) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,B,DFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . DFPG is ext-real Element of ExtREAL
(X,B,DFPG) . NFPG is ext-real Element of ExtREAL
(X,a,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
dom (X,a,DFPG) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,a,DFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,x) . DFPG is ext-real Element of ExtREAL
(X,a,DFPG) . NFPG is ext-real Element of ExtREAL
((X,B,DFPG)) is ext-real Element of ExtREAL
((X,a,DFPG)) is ext-real Element of ExtREAL
((X,B,DFPG)) + ((X,a,DFPG)) is ext-real Element of ExtREAL
f . DFPG is ext-real Element of ExtREAL
(f . DFPG) + ((X,a,DFPG)) is ext-real Element of ExtREAL
c . DFPG is ext-real Element of ExtREAL
(f . DFPG) + (c . DFPG) is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,a,DFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,x) . DFPG is ext-real Element of ExtREAL
(X,a,DFPG) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,a,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,NFPG) . DFPG is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,B,DFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . DFPG is ext-real Element of ExtREAL
(X,B,DFPG) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,NFPG) . DFPG is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,E,DFPG) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,E,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,NFPG) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,NFPG) + (X,ExtREAL,a,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((X,ExtREAL,B,NFPG) + (X,ExtREAL,a,NFPG)) . DFPG is ext-real Element of ExtREAL
dom ((X,ExtREAL,B,NFPG) + (X,ExtREAL,a,NFPG)) is Element of bool X
dom (X,ExtREAL,E,NFPG) is Element of bool X
(X,ExtREAL,B,NFPG) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,a,NFPG) . DFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,NFPG) . DFPG) + ((X,ExtREAL,a,NFPG) . DFPG) is ext-real Element of ExtREAL
(X,B,DFPG) . NFPG is ext-real Element of ExtREAL
((X,B,DFPG) . NFPG) + ((X,ExtREAL,a,NFPG) . DFPG) is ext-real Element of ExtREAL
(X,a,DFPG) . NFPG is ext-real Element of ExtREAL
((X,B,DFPG) . NFPG) + ((X,a,DFPG) . NFPG) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
(X,S,M,(f + c)) is ext-real Element of ExtREAL
(dom f) /\ (dom c) is Element of bool X
c | ((dom f) /\ (dom c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c | ((dom f) /\ (dom c))) is Element of bool X
(dom c) /\ ((dom f) /\ (dom c)) is Element of bool X
(dom c) /\ (dom c) is Element of bool X
((dom c) /\ (dom c)) /\ (dom f) is Element of bool X
x is Element of S
x is Element of S
I1 is Element of S
I1 is Element of S
I1 /\ x is M13(X,S)
a is M13(X,S)
f | a is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | a)) is ext-real Element of ExtREAL
c | a is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | a)) is ext-real Element of ExtREAL
(X,S,M,(f | a)) + (X,S,M,(c | a)) is ext-real Element of ExtREAL
(dom c) /\ a is Element of bool X
(dom f) /\ a is Element of bool X
f | ((dom f) /\ (dom c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | ((dom f) /\ (dom c))) is Element of bool X
(dom f) /\ ((dom f) /\ (dom c)) is Element of bool X
(dom f) /\ (dom f) is Element of bool X
((dom f) /\ (dom f)) /\ (dom c) is Element of bool X
(f | ((dom f) /\ (dom c))) + (c | ((dom f) /\ (dom c))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | ((dom f) /\ (dom c))) + (c | ((dom f) /\ (dom c)))) is Element of bool X
a /\ a is M13(X,S)
(dom (f | ((dom f) /\ (dom c)))) /\ (dom (c | ((dom f) /\ (dom c)))) is Element of bool X
E is set
((f | ((dom f) /\ (dom c))) + (c | ((dom f) /\ (dom c)))) . E is ext-real Element of ExtREAL
(f + c) . E is ext-real Element of ExtREAL
(f | ((dom f) /\ (dom c))) . E is ext-real Element of ExtREAL
(c | ((dom f) /\ (dom c))) . E is ext-real Element of ExtREAL
((f | ((dom f) /\ (dom c))) . E) + ((c | ((dom f) /\ (dom c))) . E) is ext-real Element of ExtREAL
f . E is ext-real Element of ExtREAL
(f . E) + ((c | ((dom f) /\ (dom c))) . E) is ext-real Element of ExtREAL
c . E is ext-real Element of ExtREAL
(f . E) + (c . E) is ext-real Element of ExtREAL
(X,S,M,((f | ((dom f) /\ (dom c))) + (c | ((dom f) /\ (dom c))))) is ext-real Element of ExtREAL
(X,S,M,(f | ((dom f) /\ (dom c)))) is ext-real Element of ExtREAL
(X,S,M,(c | ((dom f) /\ (dom c)))) is ext-real Element of ExtREAL
(X,S,M,(f | ((dom f) /\ (dom c)))) + (X,S,M,(c | ((dom f) /\ (dom c)))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
c is Element of S
c is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
B is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(B) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is ext-real Element of ExtREAL
B . I1 is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,I1) is Element of bool X
(X,S,M,(X,ExtREAL,c,I1)) is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,c,x) is Element of bool X
(X,ExtREAL,c,I1) - (X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,c,I1) - (X,ExtREAL,c,x)) is Element of bool X
a is set
(X,ExtREAL,c,x) . a is ext-real Element of ExtREAL
(X,ExtREAL,c,I1) . a is ext-real Element of ExtREAL
(dom (X,ExtREAL,c,I1)) /\ (dom (X,ExtREAL,c,x)) is Element of bool X
(X,ExtREAL,c,I1) " {+infty} is Element of bool X
(X,ExtREAL,c,x) " {+infty} is Element of bool X
((X,ExtREAL,c,I1) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty}) is Element of bool X
(X,ExtREAL,c,I1) " {-infty} is Element of bool X
(X,ExtREAL,c,x) " {-infty} is Element of bool X
((X,ExtREAL,c,I1) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty}) is Element of bool X
(((X,ExtREAL,c,I1) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty})) \/ (((X,ExtREAL,c,I1) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty})) is Element of bool X
((dom (X,ExtREAL,c,I1)) /\ (dom (X,ExtREAL,c,x))) \ ((((X,ExtREAL,c,I1) " {+infty}) /\ ((X,ExtREAL,c,x) " {+infty})) \/ (((X,ExtREAL,c,I1) " {-infty}) /\ ((X,ExtREAL,c,x) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,c,I1)) /\ (dom (X,ExtREAL,c,x)) is Element of bool X
(X,ExtREAL,c,x) | (dom ((X,ExtREAL,c,I1) - (X,ExtREAL,c,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,c,I1) | (dom ((X,ExtREAL,c,I1) - (X,ExtREAL,c,x))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,c,x) | (dom ((X,ExtREAL,c,I1) - (X,ExtREAL,c,x))))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,c,I1) | (dom ((X,ExtREAL,c,I1) - (X,ExtREAL,c,x))))) is ext-real Element of ExtREAL
rng B is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng B) is ext-real Element of ExtREAL
B . 0 is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
B . x is ext-real Element of ExtREAL
(X,ExtREAL,c,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,c,x)) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
B is Element of S
B is Element of S
B /\ c is M13(X,S)
dom (f | c) is Element of bool X
(dom f) /\ (B /\ c) is Element of bool X
f | (B /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (B /\ c)) is Element of bool X
I1 is set
(f | c) . I1 is ext-real Element of ExtREAL
(f | (B /\ c)) . I1 is ext-real Element of ExtREAL
f . I1 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
c \/ B is M13(X,S)
f | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
(X,S,M,f) is ext-real Element of ExtREAL
x is Element of S
x is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(I1) is ext-real Element of ExtREAL
a is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f1 is Element of S
f1 is Element of S
E is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
f1 /\ B is M13(X,S)
dom (f | B) is Element of bool X
(dom f) /\ (f1 /\ B) is Element of bool X
f | (f1 /\ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (f1 /\ B)) is Element of bool X
DFPG is set
(f | (f1 /\ B)) . DFPG is ext-real Element of ExtREAL
(f | B) . DFPG is ext-real Element of ExtREAL
f . DFPG is ext-real Element of ExtREAL
f1 /\ c is M13(X,S)
dom (f | c) is Element of bool X
(dom f) /\ (f1 /\ c) is Element of bool X
f | (f1 /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (f1 /\ c)) is Element of bool X
NFPG is set
(f | (f1 /\ c)) . NFPG is ext-real Element of ExtREAL
(f | c) . NFPG is ext-real Element of ExtREAL
f . NFPG is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,a,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,NFPG) is Element of bool X
dom (X,ExtREAL,E,NFPG) is Element of bool X
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,E,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,NFPG) is Element of bool X
(dom (X,ExtREAL,x,NFPG)) /\ B is Element of bool X
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(dom (X,ExtREAL,x,NFPG)) /\ c is Element of bool X
NFPG is Element of X
(X,a,NFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,a,NFPG)) is ext-real Element of ExtREAL
(f | c) . NFPG is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,a,NFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,a,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,x) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((X,ExtREAL,x,x) | c) . NFPG is ext-real Element of ExtREAL
dom ((X,ExtREAL,x,x) | c) is Element of bool X
dom (X,ExtREAL,a,x) is Element of bool X
(X,ExtREAL,x,x) . NFPG is ext-real Element of ExtREAL
(X,x,NFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,x,NFPG) . x is ext-real Element of ExtREAL
(dom f) /\ c is Element of bool X
((X,x,NFPG)) is ext-real Element of ExtREAL
f . NFPG is ext-real Element of ExtREAL
NFPG is Element of X
(X,E,NFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,E,NFPG)) is ext-real Element of ExtREAL
(f | B) . NFPG is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,x) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,x) | B) is Element of bool X
(X,ExtREAL,E,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,E,x) is Element of bool X
(X,E,NFPG) . x is ext-real Element of ExtREAL
(X,ExtREAL,E,x) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,x,x) | B) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,x,x) . NFPG is ext-real Element of ExtREAL
(X,x,NFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,x,NFPG) . x is ext-real Element of ExtREAL
(dom f) /\ B is Element of bool X
((X,x,NFPG)) is ext-real Element of ExtREAL
f . NFPG is ext-real Element of ExtREAL
f1 /\ (c \/ B) is M13(X,S)
(dom f) /\ (f1 /\ (c \/ B)) is Element of bool X
dom (f | (c \/ B)) is Element of bool X
f | (f1 /\ (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (f1 /\ (c \/ B))) is Element of bool X
x is set
(f | (c \/ B)) . x is ext-real Element of ExtREAL
(f | (f1 /\ (c \/ B))) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
x is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,a,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,KB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,KB) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,KB) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,E,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,KB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
m is Element of X
(X,ExtREAL,E,ff) . m is ext-real Element of ExtREAL
(X,ExtREAL,E,KB) . m is ext-real Element of ExtREAL
(dom f) /\ B is Element of bool X
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,n) | B) is Element of bool X
(X,ExtREAL,E,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,E,n) is Element of bool X
(X,ExtREAL,E,n) . m is ext-real Element of ExtREAL
((X,ExtREAL,x,n) | B) . m is ext-real Element of ExtREAL
(X,ExtREAL,x,n) . m is ext-real Element of ExtREAL
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,KAB) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,KAB) | B) is Element of bool X
(X,ExtREAL,E,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,E,KAB) is Element of bool X
(X,ExtREAL,E,KAB) . m is ext-real Element of ExtREAL
((X,ExtREAL,x,KAB) | B) . m is ext-real Element of ExtREAL
(X,ExtREAL,x,KAB) . m is ext-real Element of ExtREAL
ff is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
KB is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . KAB is ext-real Element of ExtREAL
(X,ExtREAL,E,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,E,KAB)) is ext-real Element of ExtREAL
KAB is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . KAB is ext-real Element of ExtREAL
(X,ExtREAL,a,KAB) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,a,KAB)) is ext-real Element of ExtREAL
dom ff is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
dom KB is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
KAB is set
ff . KAB is ext-real Element of ExtREAL
KB . KAB is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,a,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
ff . n is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,n)) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,E,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
KB . n is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,E,n)) is ext-real Element of ExtREAL
KAB is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . n is ext-real Element of ExtREAL
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,x,n)) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . n is ext-real Element of ExtREAL
ff . n is ext-real Element of ExtREAL
KB . n is ext-real Element of ExtREAL
(ff . n) + (KB . n) is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,a,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
KAB . m is ext-real Element of ExtREAL
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,x,m)) is ext-real Element of ExtREAL
(X,ExtREAL,x,m) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,x,m) | (c \/ B))) is ext-real Element of ExtREAL
ff . m is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,m)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,E,m)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,a,m)) + (X,S,M,(X,ExtREAL,E,m)) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,a,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of X
(X,ExtREAL,a,n) . x is ext-real Element of ExtREAL
(X,ExtREAL,a,m) . x is ext-real Element of ExtREAL
(dom f) /\ c is Element of bool X
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,m) | c) is Element of bool X
(X,ExtREAL,a,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,m) is Element of bool X
(X,ExtREAL,a,m) . x is ext-real Element of ExtREAL
((X,ExtREAL,x,m) | c) . x is ext-real Element of ExtREAL
(X,ExtREAL,x,m) . x is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,n) | c) is Element of bool X
(X,ExtREAL,a,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,n) is Element of bool X
(X,ExtREAL,a,n) . x is ext-real Element of ExtREAL
((X,ExtREAL,x,n) | c) . x is ext-real Element of ExtREAL
(X,ExtREAL,x,n) . x is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . n is ext-real Element of ExtREAL
ff . m is ext-real Element of ExtREAL
KB . n is ext-real Element of ExtREAL
KB . m is ext-real Element of ExtREAL
(X,ExtREAL,a,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,m) is Element of bool X
(X,S,M,(X,ExtREAL,a,m)) is ext-real Element of ExtREAL
(X,ExtREAL,a,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,a,n) is Element of bool X
(X,ExtREAL,a,m) - (X,ExtREAL,a,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,a,m) - (X,ExtREAL,a,n)) is Element of bool X
n is set
(X,ExtREAL,a,n) . n is ext-real Element of ExtREAL
(X,ExtREAL,a,m) . n is ext-real Element of ExtREAL
(dom (X,ExtREAL,a,m)) /\ (dom (X,ExtREAL,a,n)) is Element of bool X
(X,ExtREAL,a,m) " {+infty} is Element of bool X
(X,ExtREAL,a,n) " {+infty} is Element of bool X
((X,ExtREAL,a,m) " {+infty}) /\ ((X,ExtREAL,a,n) " {+infty}) is Element of bool X
(X,ExtREAL,a,m) " {-infty} is Element of bool X
(X,ExtREAL,a,n) " {-infty} is Element of bool X
((X,ExtREAL,a,m) " {-infty}) /\ ((X,ExtREAL,a,n) " {-infty}) is Element of bool X
(((X,ExtREAL,a,m) " {+infty}) /\ ((X,ExtREAL,a,n) " {+infty})) \/ (((X,ExtREAL,a,m) " {-infty}) /\ ((X,ExtREAL,a,n) " {-infty})) is Element of bool X
((dom (X,ExtREAL,a,m)) /\ (dom (X,ExtREAL,a,n))) \ ((((X,ExtREAL,a,m) " {+infty}) /\ ((X,ExtREAL,a,n) " {+infty})) \/ (((X,ExtREAL,a,m) " {-infty}) /\ ((X,ExtREAL,a,n) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,a,m)) /\ (dom (X,ExtREAL,a,n)) is Element of bool X
(X,ExtREAL,a,m) | (dom ((X,ExtREAL,a,m) - (X,ExtREAL,a,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,a,n) | (dom ((X,ExtREAL,a,m) - (X,ExtREAL,a,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,a,n) | (dom ((X,ExtREAL,a,m) - (X,ExtREAL,a,n))))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,a,m) | (dom ((X,ExtREAL,a,m) - (X,ExtREAL,a,n))))) is ext-real Element of ExtREAL
(X,ExtREAL,E,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,E,m)) is ext-real Element of ExtREAL
dom (X,ExtREAL,E,m) is Element of bool X
dom (X,ExtREAL,E,n) is Element of bool X
(X,ExtREAL,E,m) - (X,ExtREAL,E,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,E,m) - (X,ExtREAL,E,n)) is Element of bool X
n is set
(X,ExtREAL,E,n) . n is ext-real Element of ExtREAL
(X,ExtREAL,E,m) . n is ext-real Element of ExtREAL
(dom (X,ExtREAL,E,m)) /\ (dom (X,ExtREAL,E,n)) is Element of bool X
(X,ExtREAL,E,m) " {+infty} is Element of bool X
(X,ExtREAL,E,n) " {+infty} is Element of bool X
((X,ExtREAL,E,m) " {+infty}) /\ ((X,ExtREAL,E,n) " {+infty}) is Element of bool X
(X,ExtREAL,E,m) " {-infty} is Element of bool X
(X,ExtREAL,E,n) " {-infty} is Element of bool X
((X,ExtREAL,E,m) " {-infty}) /\ ((X,ExtREAL,E,n) " {-infty}) is Element of bool X
(((X,ExtREAL,E,m) " {+infty}) /\ ((X,ExtREAL,E,n) " {+infty})) \/ (((X,ExtREAL,E,m) " {-infty}) /\ ((X,ExtREAL,E,n) " {-infty})) is Element of bool X
((dom (X,ExtREAL,E,m)) /\ (dom (X,ExtREAL,E,n))) \ ((((X,ExtREAL,E,m) " {+infty}) /\ ((X,ExtREAL,E,n) " {+infty})) \/ (((X,ExtREAL,E,m) " {-infty}) /\ ((X,ExtREAL,E,n) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,E,m)) /\ (dom (X,ExtREAL,E,n)) is Element of bool X
(X,ExtREAL,E,m) | (dom ((X,ExtREAL,E,m) - (X,ExtREAL,E,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,E,n) | (dom ((X,ExtREAL,E,m) - (X,ExtREAL,E,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,E,n) | (dom ((X,ExtREAL,E,m) - (X,ExtREAL,E,n))))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,E,m) | (dom ((X,ExtREAL,E,m) - (X,ExtREAL,E,n))))) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
ff . n is ext-real Element of ExtREAL
ff . m is ext-real Element of ExtREAL
(ff) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,n) is Element of bool X
dom ((X,ExtREAL,x,n) | (c \/ B)) is Element of bool X
dom (X,ExtREAL,x,n) is Element of bool X
(dom (X,ExtREAL,x,n)) /\ (c \/ B) is Element of bool X
(dom f) /\ (c \/ B) is Element of bool X
m is Element of X
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,n) is Element of bool X
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) . m is ext-real Element of ExtREAL
(X,ExtREAL,x,n) . m is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
n is Element of X
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) . n is ext-real Element of ExtREAL
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) . n is ext-real Element of ExtREAL
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) . n is ext-real Element of ExtREAL
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) . n is ext-real Element of ExtREAL
n is Element of X
(X,ExtREAL,x,n) . n is ext-real Element of ExtREAL
(X,ExtREAL,x,m) . n is ext-real Element of ExtREAL
n is Element of X
(X,x,n) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,x,n) . m is ext-real Element of ExtREAL
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) . n is ext-real Element of ExtREAL
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,m) . n is ext-real Element of ExtREAL
(X,x,n) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,x,n) . m is ext-real Element of ExtREAL
((X,x,n)) is ext-real Element of ExtREAL
((X,x,n)) is ext-real Element of ExtREAL
f . n is ext-real Element of ExtREAL
(f | (c \/ B)) . n is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KAB . n is ext-real Element of ExtREAL
KAB . m is ext-real Element of ExtREAL
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,m) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,m) is Element of bool X
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,x,n) is Element of bool X
(X,ExtREAL,x,m) - (X,ExtREAL,x,n) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,x,m) - (X,ExtREAL,x,n)) is Element of bool X
x is set
(X,ExtREAL,x,n) . x is ext-real Element of ExtREAL
(X,ExtREAL,x,m) . x is ext-real Element of ExtREAL
(dom (X,ExtREAL,x,m)) /\ (dom (X,ExtREAL,x,n)) is Element of bool X
(X,ExtREAL,x,m) " {+infty} is Element of bool X
(X,ExtREAL,x,n) " {+infty} is Element of bool X
((X,ExtREAL,x,m) " {+infty}) /\ ((X,ExtREAL,x,n) " {+infty}) is Element of bool X
(X,ExtREAL,x,m) " {-infty} is Element of bool X
(X,ExtREAL,x,n) " {-infty} is Element of bool X
((X,ExtREAL,x,m) " {-infty}) /\ ((X,ExtREAL,x,n) " {-infty}) is Element of bool X
(((X,ExtREAL,x,m) " {+infty}) /\ ((X,ExtREAL,x,n) " {+infty})) \/ (((X,ExtREAL,x,m) " {-infty}) /\ ((X,ExtREAL,x,n) " {-infty})) is Element of bool X
((dom (X,ExtREAL,x,m)) /\ (dom (X,ExtREAL,x,n))) \ ((((X,ExtREAL,x,m) " {+infty}) /\ ((X,ExtREAL,x,n) " {+infty})) \/ (((X,ExtREAL,x,m) " {-infty}) /\ ((X,ExtREAL,x,n) " {-infty}))) is Element of bool X
KAB . m is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,x,m)) is ext-real Element of ExtREAL
(dom (X,ExtREAL,x,m)) /\ (dom (X,ExtREAL,x,n)) is Element of bool X
(X,ExtREAL,x,m) | (dom ((X,ExtREAL,x,m) - (X,ExtREAL,x,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,x,n) | (dom ((X,ExtREAL,x,m) - (X,ExtREAL,x,n))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,x,n) | (dom ((X,ExtREAL,x,m) - (X,ExtREAL,x,n))))) is ext-real Element of ExtREAL
(X,S,M,((X,ExtREAL,x,m) | (dom ((X,ExtREAL,x,m) - (X,ExtREAL,x,n))))) is ext-real Element of ExtREAL
n is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
m is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
KB . n is ext-real Element of ExtREAL
KB . m is ext-real Element of ExtREAL
(KB) is ext-real Element of ExtREAL
(KAB) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
M . c is ext-real Element of ExtREAL
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
(X,S,M,f) is ext-real Element of ExtREAL
B is Element of S
B is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
a is Element of S
a is Element of S
a /\ c is M13(X,S)
(dom f) /\ (a /\ c) is Element of bool X
f | (a /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (a /\ c)) is Element of bool X
dom (f | c) is Element of bool X
E is set
(f | c) . E is ext-real Element of ExtREAL
(f | (a /\ c)) . E is ext-real Element of ExtREAL
f . E is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
E is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is ext-real Element of ExtREAL
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,I1,x)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E . x is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((X,ExtREAL,B,DFPG) | c)) is ext-real Element of ExtREAL
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,I1,DFPG)) is ext-real Element of ExtREAL
(E) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,x) is Element of bool X
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,DFPG) is Element of bool X
dom ((X,ExtREAL,B,DFPG) | c) is Element of bool X
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,x) is Element of bool X
(dom (X,ExtREAL,B,x)) /\ c is Element of bool X
x is Element of X
(X,I1,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,I1,x)) is ext-real Element of ExtREAL
(f | c) . x is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,DFPG) | c) is Element of bool X
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,DFPG) is Element of bool X
(X,I1,x) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,I1,DFPG) . x is ext-real Element of ExtREAL
((X,ExtREAL,B,DFPG) | c) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) . x is ext-real Element of ExtREAL
(X,B,x) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,B,x) . DFPG is ext-real Element of ExtREAL
(dom f) /\ c is Element of bool X
((X,B,x)) is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
NFPG is Element of X
(X,ExtREAL,I1,x) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,I1,DFPG) . NFPG is ext-real Element of ExtREAL
(dom f) /\ c is Element of bool X
ff is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,ff) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,ff) | c) is Element of bool X
(X,ExtREAL,I1,ff) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,ff) is Element of bool X
(X,ExtREAL,I1,ff) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,ff) | c) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,ff) . NFPG is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,x) | c) is Element of bool X
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,x) is Element of bool X
(X,ExtREAL,I1,x) . NFPG is ext-real Element of ExtREAL
((X,ExtREAL,B,x) | c) . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,x) . NFPG 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
c /\ B is M13(X,S)
B \ c is M13(X,S)
(c /\ B) \/ (B \ c) is M13(X,S)
f | ((c /\ B) \/ (B \ c)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | ((c /\ B) \/ (B \ c)))) is ext-real Element of ExtREAL
f | (c /\ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c /\ B))) is ext-real Element of ExtREAL
f | (B \ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (B \ c))) is ext-real Element of ExtREAL
(X,S,M,(f | (c /\ B))) + (X,S,M,(f | (B \ c))) is ext-real Element of ExtREAL
a is Element of S
a is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
c is Element of S
B is Element of S
M . B is ext-real Element of ExtREAL
c \ B is M13(X,S)
f | (c \ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \ B))) is ext-real Element of ExtREAL
B \/ (c \ B) is M13(X,S)
B \/ c is M13(X,S)
(dom f) /\ (B \/ (c \ B)) is Element of bool X
f | (B \/ (c \ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | (B \/ (c \ B))) is Element of bool X
I1 is set
(f | (B \/ (c \ B))) . I1 is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
(X,S,M,(f | (B \/ (c \ B)))) is ext-real Element of ExtREAL
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
(X,S,M,(f | B)) + (X,S,M,(f | (c \ B))) is ext-real Element of ExtREAL
0. + (X,S,M,(f | (c \ B))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
(X,S,M,c) is ext-real Element of ExtREAL
(X,S,M,f) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
B is Element of S
B is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
I1 is Element of S
I1 is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
a is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(a) is ext-real Element of ExtREAL
f1 is Element of S
f1 is Element of S
E is Element of X
(X,B,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,B,E)) is ext-real Element of ExtREAL
rng (X,B,E) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,B,E)) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,B,E) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) . E is ext-real Element of ExtREAL
(X,B,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . E is ext-real Element of ExtREAL
rng x is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng x) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,E) is Element of bool X
DFPG is Element of X
(X,B,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,ExtREAL,B,E) . DFPG is ext-real Element of ExtREAL
((X,B,DFPG)) is ext-real Element of ExtREAL
(X,B,DFPG) . E is ext-real Element of ExtREAL
rng (X,B,DFPG) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,B,DFPG)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,E)) is ext-real Element of ExtREAL
DFPG is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng DFPG is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng DFPG) is ext-real Element of ExtREAL
(DFPG) is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,B,NFPG)) is ext-real Element of ExtREAL
x . NFPG is ext-real Element of ExtREAL
rng a is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng a) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . E is ext-real Element of ExtREAL
(X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,E) is Element of bool X
DFPG is Element of X
(X,I1,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,ExtREAL,B,E) . DFPG is ext-real Element of ExtREAL
((X,I1,DFPG)) is ext-real Element of ExtREAL
(X,B,DFPG) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(X,B,DFPG) . E is ext-real Element of ExtREAL
rng (X,B,DFPG) is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng (X,B,DFPG)) is ext-real Element of ExtREAL
((X,B,DFPG)) is ext-real Element of ExtREAL
c . DFPG is ext-real Element of ExtREAL
f . DFPG is ext-real Element of ExtREAL
NFPG is Element of S
(X,S,M,(X,ExtREAL,B,E)) is ext-real Element of ExtREAL
DFPG is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
rng DFPG is non empty ext-real-membered V93() Element of bool ExtREAL
sup (rng DFPG) is ext-real Element of ExtREAL
(DFPG) is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
DFPG . NFPG is ext-real Element of ExtREAL
(X,ExtREAL,I1,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,I1,NFPG)) is ext-real Element of ExtREAL
a . NFPG 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
c (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) f)) is ext-real Element of ExtREAL
R_EAL c is ext-real Element of ExtREAL
(R_EAL c) * (X,S,M,f) is ext-real Element of ExtREAL
PFuncs (X,ExtREAL) is non empty functional set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
B is Element of S
B is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
x is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
(x) is ext-real Element of ExtREAL
I1 is Relation-like NAT -defined PFuncs (X,ExtREAL) -valued Function-like V32( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,a) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
a is Element of S
a is Element of S
dom (c (#) f) is Element of bool X
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,f1) is Element of bool X
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,I1,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,B,f1) is Element of bool X
f1 is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x . f1 is ext-real Element of ExtREAL
x . E is ext-real Element of ExtREAL
(X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,B,f1)) is ext-real Element of ExtREAL
(X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,B,E)) is ext-real Element of ExtREAL
dom (X,ExtREAL,B,f1) is Element of bool X
dom (X,ExtREAL,B,E) is Element of bool X
(X,ExtREAL,B,E) - (X,ExtREAL,B,f1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((X,ExtREAL,B,E) - (X,ExtREAL,B,f1)) is Element of bool X
x is set
(X,ExtREAL,B,f1) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,E) . x is ext-real Element of ExtREAL
(dom (X,ExtREAL,B,E)) /\ (dom (X,ExtREAL,B,f1)) is Element of bool X
(X,ExtREAL,B,E) " {+infty} is Element of bool X
(X,ExtREAL,B,f1) " {+infty} is Element of bool X
((X,ExtREAL,B,E) " {+infty}) /\ ((X,ExtREAL,B,f1) " {+infty}) is Element of bool X
(X,ExtREAL,B,E) " {-infty} is Element of bool X
(X,ExtREAL,B,f1) " {-infty} is Element of bool X
((X,ExtREAL,B,E) " {-infty}) /\ ((X,ExtREAL,B,f1) " {-infty}) is Element of bool X
(((X,ExtREAL,B,E) " {+infty}) /\ ((X,ExtREAL,B,f1) " {+infty})) \/ (((X,ExtREAL,B,E) " {-infty}) /\ ((X,ExtREAL,B,f1) " {-infty})) is Element of bool X
((dom (X,ExtREAL,B,E)) /\ (dom (X,ExtREAL,B,f1))) \ ((((X,ExtREAL,B,E) " {+infty}) /\ ((X,ExtREAL,B,f1) " {+infty})) \/ (((X,ExtREAL,B,E) " {-infty}) /\ ((X,ExtREAL,B,f1) " {-infty}))) is Element of bool X
(dom (X,ExtREAL,B,E)) /\ (dom (X,ExtREAL,B,f1)) is Element of bool X
(X,ExtREAL,B,E) | (dom ((X,ExtREAL,B,E) - (X,ExtREAL,B,f1))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,f1) | (dom ((X,ExtREAL,B,E) - (X,ExtREAL,B,f1))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f1 is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . E is ext-real Element of ExtREAL
(X,ExtREAL,I1,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,I1,E)) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
f1 . E is ext-real Element of ExtREAL
x . E is ext-real Element of ExtREAL
(R_EAL c) * (x . E) is ext-real Element of ExtREAL
(X,ExtREAL,B,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(X,ExtREAL,I1,x)) is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,E)) is ext-real Element of ExtREAL
(R_EAL c) * (X,S,M,(X,ExtREAL,B,E)) is ext-real Element of ExtREAL
E is Element of X
(X,I1,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
((X,I1,E)) is ext-real Element of ExtREAL
(c (#) f) . E is ext-real Element of ExtREAL
x is set
(X,B,E) is Relation-like NAT -defined ExtREAL -valued Function-like V32( NAT , ExtREAL ) V59() Element of bool [:NAT,ExtREAL:]
dom (X,B,E) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,B,E) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) . E is ext-real Element of ExtREAL
(X,B,E) . x is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c (#) (X,ExtREAL,B,DFPG)) is Element of bool X
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,DFPG) is Element of bool X
(X,I1,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,I1,x) . E is ext-real Element of ExtREAL
(c (#) (X,ExtREAL,B,DFPG)) . E is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . E is ext-real Element of ExtREAL
(R_EAL c) * ((X,ExtREAL,B,x) . E) is ext-real Element of ExtREAL
(X,B,E) . x is ext-real Element of ExtREAL
(R_EAL c) * ((X,B,E) . x) is ext-real Element of ExtREAL
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,B,E) . DFPG is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,DFPG) . E is ext-real Element of ExtREAL
(X,B,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,B,x) . E is ext-real Element of ExtREAL
((X,B,E)) is ext-real Element of ExtREAL
(R_EAL c) * ((X,B,E)) is ext-real Element of ExtREAL
f . E is ext-real Element of ExtREAL
(R_EAL c) * (f . E) is ext-real Element of ExtREAL
E is set
dom x is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x . x is ext-real Element of ExtREAL
(X,S,M,(X,ExtREAL,B,x)) is ext-real Element of ExtREAL
x . E is ext-real Element of ExtREAL
(f1) is ext-real Element of ExtREAL
(R_EAL c) * (x) is ext-real Element of ExtREAL
E is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
x is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
(X,ExtREAL,I1,E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,ExtREAL,I1,x) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of X
(X,ExtREAL,I1,E) . x is ext-real Element of ExtREAL
(X,ExtREAL,I1,x) . x is ext-real Element of ExtREAL
NFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c (#) (X,ExtREAL,B,NFPG)) is Element of bool X
(X,ExtREAL,I1,NFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,NFPG) is Element of bool X
(X,ExtREAL,I1,NFPG) . x is ext-real Element of ExtREAL
(c (#) (X,ExtREAL,B,NFPG)) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,NFPG) . x is ext-real Element of ExtREAL
(R_EAL c) * ((X,ExtREAL,B,NFPG) . x) is ext-real Element of ExtREAL
DFPG is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of NAT
(X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c (#) (X,ExtREAL,B,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c (#) (X,ExtREAL,B,DFPG)) is Element of bool X
(X,ExtREAL,I1,DFPG) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (X,ExtREAL,I1,DFPG) is Element of bool X
(X,ExtREAL,I1,DFPG) . x is ext-real Element of ExtREAL
(c (#) (X,ExtREAL,B,DFPG)) . x is ext-real Element of ExtREAL
(X,ExtREAL,B,DFPG) . x is ext-real Element of ExtREAL
(R_EAL c) * ((X,ExtREAL,B,DFPG) . x) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
c is set
f . c is ext-real Element of ExtREAL
0 (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (0 (#) f) is Element of bool X
c is set
(0 (#) f) . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
(R_EAL 0) * (f . c) is ext-real Element of ExtREAL
(X,S,M,(0 (#) f)) is ext-real Element of ExtREAL
(R_EAL 0) * (X,S,M,f) is ext-real Element of ExtREAL
c is Element of S
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 V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
(X,S,M,f) is ext-real Element of ExtREAL
dom (max+ f) is Element of bool X
c is set
f . c is ext-real Element of ExtREAL
(max+ f) . c is ext-real Element of ExtREAL
max ((f . c),0) is ext-real set
dom (max- f) is Element of bool X
c is Element of X
(max+ f) . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
(max- f) . c is ext-real Element of ExtREAL
(X,S,M,f) - (R_EAL 0) is ext-real Element of ExtREAL
- (R_EAL 0) is ext-real set
(X,S,M,f) + (- (R_EAL 0)) is ext-real set
c is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
(X,S,M,f) is ext-real Element of ExtREAL
(X,S,M,f) is ext-real Element of ExtREAL
dom f is Element of bool X
c is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
(X,S,M,f) is ext-real Element of ExtREAL
c is Element of S
c is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
c \/ B is M13(X,S)
f | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
max+ (f | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | (c \/ B)))) is ext-real Element of ExtREAL
max- (f | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | (c \/ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (c \/ B)))) - (X,S,M,(max- (f | (c \/ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (c \/ B)))) is ext-real set
(X,S,M,(max+ (f | (c \/ B)))) + (- (X,S,M,(max- (f | (c \/ B))))) is ext-real set
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
x is Element of S
x is Element of S
dom (f | c) is Element of bool X
x /\ c is M13(X,S)
I1 is M13(X,S)
(dom f) /\ I1 is Element of bool X
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | I1) is Element of bool X
a is set
(f | c) . a is ext-real Element of ExtREAL
(f | I1) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(X,S,M,(f | c)) is ext-real Element of ExtREAL
I1 is Element of S
dom (f | (c \/ B)) is Element of bool X
x /\ (c \/ B) is M13(X,S)
I1 is Element of S
(dom f) /\ I1 is Element of bool X
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | I1) is Element of bool X
a is set
(f | (c \/ B)) . a is ext-real Element of ExtREAL
(f | I1) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
I1 is Element of S
dom (f | B) is Element of bool X
x /\ B is M13(X,S)
I1 is M13(X,S)
(dom f) /\ I1 is Element of bool X
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | I1) is Element of bool X
a is set
(f | B) . a is ext-real Element of ExtREAL
(f | I1) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(X,S,M,(f | B)) is ext-real Element of ExtREAL
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
I1 is Element of S
I1 is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
B is Element of S
B is Element of S
dom (f | c) is Element of bool X
B /\ c is M13(X,S)
x is M13(X,S)
(dom f) /\ x is Element of bool X
f | x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | x) is Element of bool X
I1 is set
(f | c) . I1 is ext-real Element of ExtREAL
(f | x) . I1 is ext-real Element of ExtREAL
f . I1 is ext-real Element of ExtREAL
(X,S,M,(f | c)) is ext-real Element of ExtREAL
x is Element of S
x is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
B is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
x is Element of S
x is Element of S
dom (f | c) is Element of bool X
x /\ c is M13(X,S)
I1 is M13(X,S)
(dom f) /\ I1 is Element of bool X
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | I1) is Element of bool X
a is set
(f | c) . a is ext-real Element of ExtREAL
(f | I1) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
dom (f | B) is Element of bool X
x /\ B is M13(X,S)
I1 is M13(X,S)
(dom f) /\ I1 is Element of bool X
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | I1) is Element of bool X
a is set
(f | B) . a is ext-real Element of ExtREAL
(f | I1) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
(X,S,M,(f | c)) is ext-real Element of ExtREAL
(X,S,M,(f | B)) is ext-real Element of ExtREAL
I1 is Element of S
I1 is Element of S
I1 is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Element of S
M . c is ext-real Element of ExtREAL
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ f) is Element of bool X
(max+ f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | c)) is ext-real Element of ExtREAL
B is Element of S
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- f) is Element of bool X
(X,S,M,((max+ f) | c)) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | c)) + (- (X,S,M,(max- (f | c)))) is ext-real set
(max- f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | c)) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | c)) - (X,S,M,((max- f) | c)) is ext-real Element of ExtREAL
- (X,S,M,((max- f) | c)) is ext-real set
(X,S,M,((max+ f) | c)) + (- (X,S,M,((max- f) | c))) is ext-real set
0. - 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() Element of ExtREAL
- 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
- 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
0. + (- 0.) is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
0. + (- 0.) is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
0. - 0. is epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() set
B is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
c is Element of S
B is Element of S
M . B is ext-real Element of ExtREAL
c \ B is M13(X,S)
f | (c \ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \ B))) is ext-real Element of ExtREAL
max+ (f | (c \ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | (c \ B)))) is ext-real Element of ExtREAL
max- (f | (c \ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | (c \ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (c \ B)))) - (X,S,M,(max- (f | (c \ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (c \ B)))) is ext-real set
(X,S,M,(max+ (f | (c \ B)))) + (- (X,S,M,(max- (f | (c \ B))))) is ext-real set
dom (max+ f) is Element of bool X
dom (max- f) is Element of bool X
(max+ f) | (c \ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | (c \ B))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (c \ B))) - (X,S,M,(max- (f | (c \ B)))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (c \ B))) + (- (X,S,M,(max- (f | (c \ B))))) is ext-real set
(max- f) | (c \ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | (c \ B))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (c \ B))) - (X,S,M,((max- f) | (c \ B))) is ext-real Element of ExtREAL
- (X,S,M,((max- f) | (c \ B))) is ext-real set
(X,S,M,((max+ f) | (c \ B))) + (- (X,S,M,((max- f) | (c \ B)))) is ext-real set
(X,S,M,(max+ f)) - (X,S,M,((max- f) | (c \ B))) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) + (- (X,S,M,((max- f) | (c \ B)))) 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 V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,f) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
dom f is Element of bool X
c is Element of S
dom (max+ f) is Element of bool X
dom (max- f) is Element of bool X
B is complex real ext-real Element of REAL
x is complex real ext-real Element of REAL
B - x is complex real ext-real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
c is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
dom f is Element of bool X
B is Element of S
B /\ c is M13(X,S)
(dom f) /\ (B /\ c) is Element of bool X
f | (B /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | B) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ f) | (B /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max+ f) | B) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ f) is Element of bool X
(max+ f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | c)) is ext-real Element of ExtREAL
(max- f) | (B /\ c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max- f) | B) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- f) is Element of bool X
(max- f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | c)) is ext-real Element of ExtREAL
dom (f | c) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
B is Element of S
c \/ B is M13(X,S)
f | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (c \/ B))) is ext-real Element of ExtREAL
max+ (f | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | (c \/ B)))) is ext-real Element of ExtREAL
max- (f | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | (c \/ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (c \/ B)))) - (X,S,M,(max- (f | (c \/ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (c \/ B)))) is ext-real set
(X,S,M,(max+ (f | (c \/ B)))) + (- (X,S,M,(max- (f | (c \/ B))))) is ext-real set
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
dom f is Element of bool X
x is Element of S
x /\ (c \/ B) is M13(X,S)
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- f) is Element of bool X
(max- f) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- f) | x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max- f) | x) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- f) | (x /\ (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | (c \/ B))) is ext-real Element of ExtREAL
(max- f) | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | c)) is ext-real Element of ExtREAL
(max- f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | B)) is ext-real Element of ExtREAL
(X,S,M,((max- f) | c)) + (X,S,M,((max- f) | B)) is ext-real Element of ExtREAL
f1 is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
f1 - a is complex real ext-real Element of REAL
dom (max+ f) is Element of bool X
(max+ f) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ f) | x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max+ f) | x) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ f) | (x /\ (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | (c \/ B))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | c)) is ext-real Element of ExtREAL
(max+ f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | B)) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | c)) + (X,S,M,((max+ f) | B)) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) + (X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
E is complex real ext-real Element of REAL
E + f1 is complex real ext-real Element of REAL
(X,S,M,(max- (f | c))) + (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
x is complex real ext-real Element of REAL
x + a is complex real ext-real Element of REAL
f | x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | x) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((f | x) | (c \/ B))) is ext-real Element of ExtREAL
max+ ((f | x) | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((f | x) | (c \/ B)))) is ext-real Element of ExtREAL
max- ((f | x) | (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((f | x) | (c \/ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | x) | (c \/ B)))) - (X,S,M,(max- ((f | x) | (c \/ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f | x) | (c \/ B)))) is ext-real set
(X,S,M,(max+ ((f | x) | (c \/ B)))) + (- (X,S,M,(max- ((f | x) | (c \/ B))))) is ext-real set
f | (x /\ (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | (x /\ (c \/ B)))) is ext-real Element of ExtREAL
max+ (f | (x /\ (c \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | (x /\ (c \/ B))))) is ext-real Element of ExtREAL
max- (f | (x /\ (c \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | (x /\ (c \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (x /\ (c \/ B))))) - (X,S,M,(max- (f | (x /\ (c \/ B))))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (x /\ (c \/ B))))) is ext-real set
(X,S,M,(max+ (f | (x /\ (c \/ B))))) + (- (X,S,M,(max- (f | (x /\ (c \/ B)))))) is ext-real set
(X,S,M,((max+ f) | (x /\ (c \/ B)))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (x /\ (c \/ B)))) - (X,S,M,(max- (f | (x /\ (c \/ B))))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (x /\ (c \/ B)))) + (- (X,S,M,(max- (f | (x /\ (c \/ B)))))) is ext-real set
(X,S,M,((max- f) | (x /\ (c \/ B)))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | (x /\ (c \/ B)))) - (X,S,M,((max- f) | (x /\ (c \/ B)))) is ext-real Element of ExtREAL
- (X,S,M,((max- f) | (x /\ (c \/ B)))) is ext-real set
(X,S,M,((max+ f) | (x /\ (c \/ B)))) + (- (X,S,M,((max- f) | (x /\ (c \/ B))))) is ext-real set
(E + f1) - (x + a) is complex real ext-real Element of REAL
E - x is complex real ext-real Element of REAL
(E - x) + (f1 - a) is complex real ext-real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
B is Element of S
c is Element of S
(dom f) \ c is Element of bool X
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(X,S,M,(f | c)) + (X,S,M,(f | B)) is ext-real Element of ExtREAL
c \/ B is M13(X,S)
c \/ (dom f) is Element of bool X
(dom f) /\ (c \/ B) is Element of bool X
f | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | (dom f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | (dom f)) | (c \/ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | ((dom f) /\ (c \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
|.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom |.f.| is Element of bool X
max- |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- |.f.|) is Element of bool X
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- f) is Element of bool X
c is set
|.f.| . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
|.(f . c).| is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ f) is Element of bool X
(max+ f) + (max- f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
c is Element of S
max+ |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ |.f.|) is Element of bool X
B is set
|.f.| . B is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
|.(f . B).| is ext-real Element of ExtREAL
(max+ |.f.|) . B is ext-real Element of ExtREAL
max ((|.f.| . B),0) is ext-real set
B is Element of X
(max+ |.f.|) . B is ext-real Element of ExtREAL
|.f.| . B is ext-real Element of ExtREAL
(max- |.f.|) . B is ext-real Element of ExtREAL
(X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
(X,S,M,((max+ f) + (max- f))) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) + (X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ |.f.|)) is ext-real Element of ExtREAL
(X,S,M,|.f.|) is ext-real Element of ExtREAL
(X,S,M,(max+ |.f.|)) - (X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
- (X,S,M,(max- |.f.|)) is ext-real set
(X,S,M,(max+ |.f.|)) + (- (X,S,M,(max- |.f.|))) is ext-real set
B is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
|.(X,S,M,f).| is ext-real Element of ExtREAL
|.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,|.f.|) is ext-real Element of ExtREAL
max+ |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ |.f.|)) is ext-real Element of ExtREAL
max- |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
(X,S,M,(max+ |.f.|)) - (X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
- (X,S,M,(max- |.f.|)) is ext-real set
(X,S,M,(max+ |.f.|)) + (- (X,S,M,(max- |.f.|))) is ext-real set
|.((X,S,M,(max+ f)) - (X,S,M,(max- f))).| is ext-real Element of ExtREAL
|.(X,S,M,(max+ f)).| is ext-real Element of ExtREAL
|.(X,S,M,(max- f)).| is ext-real Element of ExtREAL
|.(X,S,M,(max+ f)).| + |.(X,S,M,(max- f)).| is ext-real Element of ExtREAL
dom f is Element of bool X
dom (max+ f) is Element of bool X
c is set
dom |.f.| is Element of bool X
|.f.| . c is ext-real Element of ExtREAL
f . c is ext-real Element of ExtREAL
|.(f . c).| is ext-real Element of ExtREAL
dom (max- f) is Element of bool X
(max+ f) + (max- f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
(X,S,M,(max+ f)) + |.(X,S,M,(max- f)).| is ext-real Element of ExtREAL
(X,S,M,((max+ f) + (max- f))) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) + (X,S,M,(max- 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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
|.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,|.f.|) is ext-real Element of ExtREAL
max+ |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ |.f.|)) is ext-real Element of ExtREAL
max- |.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
(X,S,M,(max+ |.f.|)) - (X,S,M,(max- |.f.|)) is ext-real Element of ExtREAL
- (X,S,M,(max- |.f.|)) is ext-real set
(X,S,M,(max+ |.f.|)) + (- (X,S,M,(max- |.f.|))) is ext-real set
(X,S,M,c) is ext-real Element of ExtREAL
max+ c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ c)) is ext-real Element of ExtREAL
max- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- c)) is ext-real Element of ExtREAL
(X,S,M,(max+ c)) - (X,S,M,(max- c)) is ext-real Element of ExtREAL
- (X,S,M,(max- c)) is ext-real set
(X,S,M,(max+ c)) + (- (X,S,M,(max- c))) is ext-real set
B is set
f . B is ext-real Element of ExtREAL
|.(f . B).| is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
dom (max+ c) is Element of bool X
B is set
c . B is ext-real Element of ExtREAL
(max+ c) . B is ext-real Element of ExtREAL
max ((c . B),0) is ext-real set
dom |.f.| is Element of bool X
dom (max+ |.f.|) is Element of bool X
B is set
|.f.| . B is ext-real Element of ExtREAL
f . B is ext-real Element of ExtREAL
|.(f . B).| is ext-real Element of ExtREAL
(max+ |.f.|) . B is ext-real Element of ExtREAL
max ((|.f.| . B),0) is ext-real set
B is Element of S
B is Element of S
x is Element of X
|.f.| . x is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
|.(f . x).| is ext-real Element of ExtREAL
x is set
|.f.| . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
|.(f . x).| is ext-real Element of ExtREAL
(X,S,M,|.f.|) is ext-real Element of ExtREAL
(X,S,M,c) is ext-real Element of ExtREAL
x is Element of S
dom (max- |.f.|) is Element of bool X
x is Element of X
(max+ |.f.|) . x is ext-real Element of ExtREAL
|.f.| . x is ext-real Element of ExtREAL
(max- |.f.|) . x is ext-real Element of ExtREAL
x is Element of S
x is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
integral (X,S,M,f) is ext-real Element of ExtREAL
M . (dom f) is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
c * (M . (dom f)) is ext-real set
B is set
f . B is ext-real Element of ExtREAL
B is Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like V113() FinSequence of S
x is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
I1 is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
dom I1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
dom B is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
M * B is Relation-like NAT -defined ExtREAL -valued Function-like finite FinSequence-like FinSubsequence-like V59() FinSequence of ExtREAL
Sum I1 is ext-real Element of ExtREAL
rng B is finite Element of bool S
bool S is non empty set
union (rng B) is set
a is epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational set
I1 . a is ext-real Element of ExtREAL
(M * B) . a is ext-real Element of ExtREAL
c * ((M * B) . a) is ext-real set
B . a is set
x . a is ext-real Element of ExtREAL
(x . a) * ((M * B) . a) is ext-real Element of ExtREAL
M . (B . a) is ext-real Element of ExtREAL
f1 is set
f . f1 is ext-real Element of ExtREAL
dom (M * B) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded Element of bool NAT
R_EAL c is ext-real Element of ExtREAL
Sum (M * B) is ext-real Element of ExtREAL
(R_EAL c) * (Sum (M * B)) is ext-real Element of ExtREAL
M . (union (rng B)) is ext-real Element of ExtREAL
(R_EAL c) * (M . (union (rng B))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
M . (dom f) is ext-real Element of ExtREAL
c is complex real ext-real Element of REAL
R_EAL c is ext-real Element of ExtREAL
(R_EAL c) * (M . (dom f)) is ext-real Element of ExtREAL
integral (X,S,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
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f " {+infty} is Element of bool X
f " {-infty} is Element of bool X
M . (f " {+infty}) is ext-real Element of ExtREAL
M . (f " {-infty}) is ext-real Element of ExtREAL
(f " {+infty}) \/ (f " {-infty}) is Element of bool X
M . ((f " {+infty}) \/ (f " {-infty})) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
dom f is Element of bool X
c is Element of S
eq_dom (f,+infty) is Element of bool X
eq_dom (f,-infty) is Element of bool X
B is set
x is set
c /\ (eq_dom (f,+infty)) is Element of bool X
c /\ (eq_dom (f,-infty)) is Element of bool X
B is Element of S
c \ B is M13(X,S)
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
dom (max+ f) is Element of bool X
I1 is Element of S
(dom (max+ f)) /\ I1 is Element of bool X
(max+ f) | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c \ I1 is M13(X,S)
I1 \/ (c \ I1) is M13(X,S)
(max+ f) | (I1 \/ (c \ I1)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max+ f) | (I1 \/ (c \ I1))) is Element of bool X
B \/ (c \ B) is M13(X,S)
(max- f) | (B \/ (c \ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max- f) | (B \/ (c \ B))) is Element of bool X
f1 is Element of X
((max+ f) | (I1 \/ (c \ I1))) . f1 is ext-real Element of ExtREAL
(max+ f) . f1 is ext-real Element of ExtREAL
E is Element of X
((max- f) | (B \/ (c \ B))) . E is ext-real Element of ExtREAL
(max- f) . E is ext-real Element of ExtREAL
(dom (max+ f)) /\ (dom (max+ f)) is Element of bool X
(X,S,M,((max+ f) | I1)) is ext-real Element of ExtREAL
(max+ f) | (c \ I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max+ f) | (c \ I1))) is ext-real Element of ExtREAL
(X,S,M,((max+ f) | I1)) + (X,S,M,((max+ f) | (c \ I1))) is ext-real Element of ExtREAL
M . I1 is ext-real Element of ExtREAL
dom ((max+ f) | I1) is Element of bool X
f1 is complex real ext-real Element of REAL
R_EAL f1 is ext-real Element of ExtREAL
(R_EAL f1) * (M . I1) is ext-real Element of ExtREAL
E is set
E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom E is Element of bool X
x is set
E . x is ext-real Element of ExtREAL
X /\ (dom ((max+ f) | I1)) is Element of bool X
(X,S,M,E) is ext-real Element of ExtREAL
M . (dom E) is ext-real Element of ExtREAL
(R_EAL f1) * (M . (dom E)) is ext-real Element of ExtREAL
x is Element of X
E . x is ext-real Element of ExtREAL
((max+ f) | I1) . x is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
max ((f . x),0) is ext-real set
(max+ f) . x is ext-real Element of ExtREAL
chi (I1,X) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (chi (I1,X)) is Element of bool X
(dom (chi (I1,X))) /\ I1 is Element of bool X
(chi (I1,X)) | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((chi (I1,X)) | I1) is Element of bool X
f1 (#) ((chi (I1,X)) | I1) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of X
E . x is ext-real Element of ExtREAL
(f1 (#) ((chi (I1,X)) | I1)) . x is ext-real Element of ExtREAL
dom (f1 (#) ((chi (I1,X)) | I1)) is Element of bool X
((chi (I1,X)) | I1) . x is ext-real Element of ExtREAL
(R_EAL f1) * (((chi (I1,X)) | I1) . x) is ext-real Element of ExtREAL
(chi (I1,X)) . x is ext-real Element of ExtREAL
(R_EAL f1) * ((chi (I1,X)) . x) is ext-real Element of ExtREAL
dom (f1 (#) ((chi (I1,X)) | I1)) is Element of bool X
(X,S,M,E) is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
(R_EAL 1) * (M . I1) is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
(R_EAL 1) * (M . I1) is ext-real Element of ExtREAL
E is complex real ext-real Element of REAL
2 * E is complex real ext-real Element of REAL
f1 is complex real ext-real Element of REAL
(2 * E) / f1 is complex real ext-real Element of REAL
R_EAL ((2 * E) / f1) is ext-real Element of ExtREAL
(R_EAL ((2 * E) / f1)) * (M . I1) is ext-real Element of ExtREAL
((2 * E) / f1) * f1 is complex real ext-real Element of REAL
dom (max- f) is Element of bool X
(dom (max- f)) /\ B is Element of bool X
(max- f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(dom (max- f)) /\ (dom (max- f)) is Element of bool X
(X,S,M,((max- f) | B)) is ext-real Element of ExtREAL
(max- f) | (c \ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((max- f) | (c \ B))) is ext-real Element of ExtREAL
(X,S,M,((max- f) | B)) + (X,S,M,((max- f) | (c \ B))) is ext-real Element of ExtREAL
M . B is ext-real Element of ExtREAL
dom ((max- f) | B) is Element of bool X
E is complex real ext-real Element of REAL
R_EAL E is ext-real Element of ExtREAL
(R_EAL E) * (M . B) is ext-real Element of ExtREAL
x is set
x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom x is Element of bool X
DFPG is set
x . DFPG is ext-real Element of ExtREAL
X /\ (dom ((max- f) | B)) is Element of bool X
(X,S,M,x) is ext-real Element of ExtREAL
M . (dom x) is ext-real Element of ExtREAL
(R_EAL E) * (M . (dom x)) is ext-real Element of ExtREAL
chi (B,X) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (chi (B,X)) is Element of bool X
(dom (chi (B,X))) /\ B is Element of bool X
(chi (B,X)) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((chi (B,X)) | B) is Element of bool X
E (#) ((chi (B,X)) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
DFPG is Element of X
x . DFPG is ext-real Element of ExtREAL
(E (#) ((chi (B,X)) | B)) . DFPG is ext-real Element of ExtREAL
dom (E (#) ((chi (B,X)) | B)) is Element of bool X
((chi (B,X)) | B) . DFPG is ext-real Element of ExtREAL
(R_EAL E) * (((chi (B,X)) | B) . DFPG) is ext-real Element of ExtREAL
(chi (B,X)) . DFPG is ext-real Element of ExtREAL
(R_EAL E) * ((chi (B,X)) . DFPG) is ext-real Element of ExtREAL
DFPG is Element of X
x . DFPG is ext-real Element of ExtREAL
((max- f) | B) . DFPG is ext-real Element of ExtREAL
f . DFPG is ext-real Element of ExtREAL
- (f . DFPG) is ext-real Element of ExtREAL
max ((- (f . DFPG)),0) is ext-real set
(max- f) . DFPG is ext-real Element of ExtREAL
dom (E (#) ((chi (B,X)) | B)) is Element of bool X
(X,S,M,x) is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
(R_EAL 1) * (M . B) is ext-real Element of ExtREAL
R_EAL 1 is ext-real Element of ExtREAL
(R_EAL 1) * (M . B) is ext-real Element of ExtREAL
x is complex real ext-real Element of REAL
2 * x is complex real ext-real Element of REAL
E is complex real ext-real Element of REAL
(2 * x) / E is complex real ext-real Element of REAL
R_EAL ((2 * x) / E) is ext-real Element of ExtREAL
(R_EAL ((2 * x) / E)) * (M . B) is ext-real Element of ExtREAL
((2 * x) / E) * E is complex real ext-real Element of REAL
f1 is measure_zero of M
f1 \/ B is M13(X,S)
M . (f1 \/ B) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ c)) is ext-real Element of ExtREAL
dom c is Element of bool X
dom (max+ c) is Element of bool X
B is set
(max+ c) . B is ext-real Element of ExtREAL
c . B is ext-real Element of ExtREAL
max ((c . B),0) is ext-real set
B is Element of S
dom f is Element of bool X
x is Element of S
x /\ B is M13(X,S)
dom (f + c) is Element of bool X
(X,S,M,(f + c)) is ext-real Element of ExtREAL
I1 is Element of S
f | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | I1)) is ext-real Element of ExtREAL
c | I1 is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | I1)) is ext-real Element of ExtREAL
(X,S,M,(f | I1)) + (X,S,M,(c | I1)) is ext-real Element of ExtREAL
c | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | B)) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) + (X,S,M,(max+ c)) is ext-real Element of ExtREAL
dom (max+ f) is Element of bool X
a is set
(max+ f) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
max ((f . a),0) is ext-real set
max+ (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ (f + c)) is Element of bool X
a is set
(f + c) . a is ext-real Element of ExtREAL
f . a is ext-real Element of ExtREAL
c . a is ext-real Element of ExtREAL
(f . a) + (c . a) is ext-real Element of ExtREAL
(max+ (f + c)) . a is ext-real Element of ExtREAL
max (((f + c) . a),0) is ext-real set
a is Element of X
max- (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- (f + c)) is Element of bool X
(max+ (f + c)) . a is ext-real Element of ExtREAL
(f + c) . a is ext-real Element of ExtREAL
(max- (f + c)) . a is ext-real Element of ExtREAL
f | x is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | x)) is ext-real Element of ExtREAL
(X,S,M,(max+ (f + c))) is ext-real Element of ExtREAL
(X,S,M,(max- (f + c))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
f " {-infty} is Element of bool X
dom c is Element of bool X
dom f is Element of bool X
c " {-infty} is Element of bool X
c " {+infty} is Element of bool X
f " {+infty} is Element of bool X
B is Element of S
x is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is Element of S
B is Element of S
|.f.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
|.f.| | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
|.c.| is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is Element of S
|.c.| | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
(dom f) /\ (dom c) is Element of bool X
f " {-infty} is Element of bool X
c " {+infty} is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
f " {+infty} is Element of bool X
c " {-infty} is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
dom |.c.| is Element of bool X
(dom |.c.|) /\ B is Element of bool X
dom |.f.| is Element of bool X
(dom |.f.|) /\ B is Element of bool X
dom (|.f.| | B) is Element of bool X
dom (|.c.| | B) is Element of bool X
(dom (|.f.| | B)) /\ (dom (|.c.| | B)) is Element of bool X
B /\ B is M13(X,S)
(|.f.| | B) + (|.c.| | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((|.f.| | B) + (|.c.| | B)) is Element of bool X
x is Element of X
f . x is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
(f . x) + (c . x) is ext-real Element of ExtREAL
|.((f . x) + (c . x)).| is ext-real Element of ExtREAL
|.(f . x).| is ext-real Element of ExtREAL
|.(c . x).| is ext-real Element of ExtREAL
|.(f . x).| + |.(c . x).| is ext-real Element of ExtREAL
|.f.| . x is ext-real Element of ExtREAL
(|.f.| . x) + |.(c . x).| is ext-real Element of ExtREAL
|.c.| . x is ext-real Element of ExtREAL
(|.f.| . x) + (|.c.| . x) is ext-real Element of ExtREAL
(|.f.| | B) . x is ext-real Element of ExtREAL
((|.f.| | B) . x) + (|.c.| . x) is ext-real Element of ExtREAL
(|.c.| | B) . x is ext-real Element of ExtREAL
((|.f.| | B) . x) + ((|.c.| | B) . x) is ext-real Element of ExtREAL
((|.f.| | B) + (|.c.| | B)) . x is ext-real Element of ExtREAL
(f + c) . x is ext-real Element of ExtREAL
|.((f + c) . x).| is ext-real Element of ExtREAL
x is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom c is Element of bool X
dom f is Element of bool X
dom (f + c) is Element of bool X
B is Element of S
x is Element of S
B is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
dom c is Element of bool X
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
(X,S,M,c) is ext-real Element of ExtREAL
max+ c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ c)) is ext-real Element of ExtREAL
max- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- c)) is ext-real Element of ExtREAL
(X,S,M,(max+ c)) - (X,S,M,(max- c)) is ext-real Element of ExtREAL
- (X,S,M,(max- c)) is ext-real set
(X,S,M,(max+ c)) + (- (X,S,M,(max- c))) is ext-real set
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
f " {+infty} is Element of bool X
c " {-infty} is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
c " {+infty} is Element of bool X
(c " {+infty}) \/ (c " {-infty}) is Element of bool X
f " {-infty} is Element of bool X
(f " {+infty}) \/ (f " {-infty}) is Element of bool X
x is Element of S
B is Element of S
x \/ B is M13(X,S)
a is Element of S
a \ (x \/ B) is M13(X,S)
f | (a \ (x \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c | (a \ (x \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
NFPG is set
x is set
ff is set
KB is set
DFPG is Element of S
DFPG \ (a \ (x \/ B)) is M13(X,S)
(dom f) /\ (dom c) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
a \ (a \ (x \/ B)) is M13(X,S)
a /\ (x \/ B) is M13(X,S)
(f + c) | (a \ (x \/ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(f | (a \ (x \/ B))) + (c | (a \ (x \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
DFPG \ (DFPG \ (a \ (x \/ B))) is M13(X,S)
DFPG /\ (a \ (x \/ B)) is M13(X,S)
dom ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))) is Element of bool X
dom (c | (a \ (x \/ B))) is Element of bool X
x is set
(c | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
ff is set
c . ff is ext-real Element of ExtREAL
[:(dom c),ExtREAL:] is V59() set
bool [:(dom c),ExtREAL:] is non empty set
(dom c) /\ (a \ (x \/ B)) is Element of bool X
c . x is ext-real Element of ExtREAL
ff is Relation-like dom c -defined ExtREAL -valued Function-like V32( dom c, ExtREAL ) V59() Element of bool [:(dom c),ExtREAL:]
ff . x is ext-real Element of ExtREAL
ff " {-infty} is Element of bool (dom c)
bool (dom c) is non empty set
ff " {+infty} is Element of bool (dom c)
x is set
(c | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
- +infty is non empty ext-real non positive negative Element of ExtREAL
x is Element of X
(c | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
|.((c | (a \ (x \/ B))) . x).| is ext-real Element of ExtREAL
dom (f | (a \ (x \/ B))) is Element of bool X
x is set
(f | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
ff is set
f . ff is ext-real Element of ExtREAL
[:(dom f),ExtREAL:] is V59() set
bool [:(dom f),ExtREAL:] is non empty set
(dom f) /\ (a \ (x \/ B)) is Element of bool X
f . x is ext-real Element of ExtREAL
ff is Relation-like dom f -defined ExtREAL -valued Function-like V32( dom f, ExtREAL ) V59() Element of bool [:(dom f),ExtREAL:]
ff . x is ext-real Element of ExtREAL
ff " {-infty} is Element of bool (dom f)
bool (dom f) is non empty set
ff " {+infty} is Element of bool (dom f)
x is set
(f | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max+ (f | (a \ (x \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B))))) is Element of bool X
(dom (f | (a \ (x \/ B)))) /\ (dom (c | (a \ (x \/ B)))) is Element of bool X
max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
max- (f | (a \ (x \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B))))) is Element of bool X
dom (max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) is Element of bool X
(dom c) /\ (a \ (x \/ B)) is Element of bool X
max- (c | (a \ (x \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- (c | (a \ (x \/ B)))) is Element of bool X
x is Element of S
max+ (c | (a \ (x \/ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max+ (c | (a \ (x \/ B)))) is Element of bool X
(dom f) /\ (a \ (x \/ B)) is Element of bool X
M . B is ext-real Element of ExtREAL
M . x is ext-real Element of ExtREAL
M . (x \/ B) is ext-real Element of ExtREAL
(X,S,M,(f | (a \ (x \/ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (a \ (x \/ B))))) - (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real set
(X,S,M,(max+ (f | (a \ (x \/ B))))) + (- (X,S,M,(max- (f | (a \ (x \/ B)))))) is ext-real set
M . (DFPG \ (a \ (x \/ B))) is ext-real Element of ExtREAL
x is set
(c | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
dom (max+ (f | (a \ (x \/ B)))) is Element of bool X
x is set
(c | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
dom (max- (f | (a \ (x \/ B)))) is Element of bool X
(X,S,M,(max+ (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
x is set
(f | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
x is set
(f | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
((max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B))))) + (max- (c | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
((max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B))))) + (max+ (c | (a \ (x \/ B)))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) is Element of bool X
(X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
x is Element of X
(f | (a \ (x \/ B))) . x is ext-real Element of ExtREAL
|.((f | (a \ (x \/ B))) . x).| is ext-real Element of ExtREAL
(X,S,M,(((max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B))))) + (max- (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,((max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,((max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max- (f | (a \ (x \/ B)))))) + (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
((X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B)))))) + (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(((max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B))))) + (max+ (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,((max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,((max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) + (max+ (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max- (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real set
(X,S,M,(max- (c | (a \ (x \/ B))))) + (- (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real set
((X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B)))))) + ((X,S,M,(max- (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (c | (a \ (x \/ B)))))) - (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (c | (a \ (x \/ B)))))) + (- (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real set
(X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c | (a \ (x \/ B))))) + (- (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real set
((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
((X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B)))))) + 0. is ext-real Element of ExtREAL
- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
((- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + ((X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + ((X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + ((X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B)))))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
((- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + ((X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(R_EAL 0) + ((X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- (f | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
((- (X,S,M,(max- (f | (a \ (x \/ B)))))) + (X,S,M,(max- (f | (a \ (x \/ B)))))) + ((- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- (f | (a \ (x \/ B)))))) + ((X,S,M,(max+ (f | (a \ (x \/ B))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(- (X,S,M,(max- (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B))))) is ext-real Element of ExtREAL
((- (X,S,M,(max- (f | (a \ (x \/ B)))))) + (X,S,M,(max+ (f | (a \ (x \/ B)))))) + ((X,S,M,(max+ (c | (a \ (x \/ B))))) - (X,S,M,(max- (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
(R_EAL 0) + ((- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) + (X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) is ext-real Element of ExtREAL
(X,S,M,((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) - (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) is ext-real set
(X,S,M,(max+ ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B)))))) + (- (X,S,M,(max- ((f | (a \ (x \/ B))) + (c | (a \ (x \/ B))))))) is ext-real set
(X,S,M,(c | (a \ (x \/ B)))) is ext-real Element of ExtREAL
(X,S,M,(f | (a \ (x \/ B)))) + (X,S,M,(c | (a \ (x \/ B)))) is ext-real Element of ExtREAL
x is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
dom c is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f + c)) is ext-real Element of ExtREAL
max+ (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f + c))) is ext-real Element of ExtREAL
max- (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f + c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f + c))) - (X,S,M,(max- (f + c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f + c))) is ext-real set
(X,S,M,(max+ (f + c))) + (- (X,S,M,(max- (f + c)))) is ext-real set
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
(X,S,M,c) is ext-real Element of ExtREAL
max+ c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ c)) is ext-real Element of ExtREAL
max- c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- c)) is ext-real Element of ExtREAL
(X,S,M,(max+ c)) - (X,S,M,(max- c)) is ext-real Element of ExtREAL
- (X,S,M,(max- c)) is ext-real set
(X,S,M,(max+ c)) + (- (X,S,M,(max- c))) is ext-real set
(X,S,M,f) + (X,S,M,c) is ext-real Element of ExtREAL
dom (f + c) is Element of bool X
B is Element of S
x is Element of S
(dom f) \ x is Element of bool X
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f | B) is Element of bool X
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(dom c) \ x is Element of bool X
c | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (c | B) is Element of bool X
(X,S,M,(c | B)) is ext-real Element of ExtREAL
max+ (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (c | B))) is ext-real Element of ExtREAL
max- (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (c | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c | B))) - (X,S,M,(max- (c | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c | B))) is ext-real set
(X,S,M,(max+ (c | B))) + (- (X,S,M,(max- (c | B)))) is ext-real set
I1 is Element of S
(dom (f + c)) \ I1 is Element of bool X
M . x is ext-real Element of ExtREAL
M . I1 is ext-real Element of ExtREAL
(f + c) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f + c) | B) is Element of bool X
(f | B) + (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((f + c) | B)) is ext-real Element of ExtREAL
max+ ((f + c) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((f + c) | B))) is ext-real Element of ExtREAL
max- ((f + c) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((f + c) | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f + c) | B))) - (X,S,M,(max- ((f + c) | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f + c) | B))) is ext-real set
(X,S,M,(max+ ((f + c) | B))) + (- (X,S,M,(max- ((f + c) | B)))) is ext-real set
(X,S,M,(f | B)) + (X,S,M,(c | B)) is ext-real Element of ExtREAL
a is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom f is Element of bool X
dom c is Element of bool X
(dom f) /\ (dom c) is Element of bool X
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f + c)) is ext-real Element of ExtREAL
max+ (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f + c))) is ext-real Element of ExtREAL
max- (f + c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f + c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f + c))) - (X,S,M,(max- (f + c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f + c))) is ext-real set
(X,S,M,(max+ (f + c))) + (- (X,S,M,(max- (f + c)))) is ext-real set
B is Element of S
x is Element of S
x /\ B is M13(X,S)
c | (x /\ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f | (x /\ B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
E is M13(X,S)
f | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | E)) is ext-real Element of ExtREAL
max+ (f | E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | E))) is ext-real Element of ExtREAL
max- (f | E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | E))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | E))) - (X,S,M,(max- (f | E))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | E))) is ext-real set
(X,S,M,(max+ (f | E))) + (- (X,S,M,(max- (f | E)))) is ext-real set
c | E is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | E)) is ext-real Element of ExtREAL
max+ (c | E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (c | E))) is ext-real Element of ExtREAL
max- (c | E) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (c | E))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c | E))) - (X,S,M,(max- (c | E))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c | E))) is ext-real set
(X,S,M,(max+ (c | E))) + (- (X,S,M,(max- (c | E)))) is ext-real set
(X,S,M,(f | E)) + (X,S,M,(c | E)) is ext-real Element of ExtREAL
dom (f | (x /\ B)) is Element of bool X
(dom f) /\ (x /\ B) is Element of bool X
x /\ x is M13(X,S)
(x /\ x) /\ B is M13(X,S)
(f | (x /\ B)) " {+infty} is Element of bool X
f " {+infty} is Element of bool X
E /\ (f " {+infty}) is Element of bool X
(c | (x /\ B)) " {-infty} is Element of bool X
c " {-infty} is Element of bool X
E /\ (c " {-infty}) is Element of bool X
((f | (x /\ B)) " {+infty}) /\ ((c | (x /\ B)) " {-infty}) is Element of bool X
(f " {+infty}) /\ E is Element of bool X
((f " {+infty}) /\ E) /\ E is Element of bool X
(((f " {+infty}) /\ E) /\ E) /\ (c " {-infty}) is Element of bool X
E /\ E is M13(X,S)
(f " {+infty}) /\ (E /\ E) is Element of bool X
((f " {+infty}) /\ (E /\ E)) /\ (c " {-infty}) is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
E /\ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
(c | (x /\ B)) " {+infty} is Element of bool X
c " {+infty} is Element of bool X
E /\ (c " {+infty}) is Element of bool X
(f | (x /\ B)) " {-infty} is Element of bool X
f " {-infty} is Element of bool X
E /\ (f " {-infty}) is Element of bool X
((f | (x /\ B)) " {-infty}) /\ ((c | (x /\ B)) " {+infty}) is Element of bool X
(f " {-infty}) /\ E is Element of bool X
((f " {-infty}) /\ E) /\ E is Element of bool X
(((f " {-infty}) /\ E) /\ E) /\ (c " {+infty}) is Element of bool X
(f " {-infty}) /\ (E /\ E) is Element of bool X
((f " {-infty}) /\ (E /\ E)) /\ (c " {+infty}) is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
E /\ ((f " {-infty}) /\ (c " {+infty})) is Element of bool X
(((f | (x /\ B)) " {-infty}) /\ ((c | (x /\ B)) " {+infty})) \/ (((f | (x /\ B)) " {+infty}) /\ ((c | (x /\ B)) " {-infty})) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
E /\ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
dom (c | (x /\ B)) is Element of bool X
(dom c) /\ (x /\ B) is Element of bool X
B /\ B is M13(X,S)
(B /\ B) /\ x is M13(X,S)
(f | (x /\ B)) + (c | (x /\ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom ((f | (x /\ B)) + (c | (x /\ B))) is Element of bool X
(dom (f | (x /\ B))) /\ (dom (c | (x /\ B))) is Element of bool X
((dom (f | (x /\ B))) /\ (dom (c | (x /\ B)))) \ ((((f | (x /\ B)) " {-infty}) /\ ((c | (x /\ B)) " {+infty})) \/ (((f | (x /\ B)) " {+infty}) /\ ((c | (x /\ B)) " {-infty}))) is Element of bool X
E \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
dom (f + c) is Element of bool X
x is set
((f | (x /\ B)) + (c | (x /\ B))) . x is ext-real Element of ExtREAL
(f + c) . x is ext-real Element of ExtREAL
(f | (x /\ B)) . x is ext-real Element of ExtREAL
(c | (x /\ B)) . x is ext-real Element of ExtREAL
((f | (x /\ B)) . x) + ((c | (x /\ B)) . x) is ext-real Element of ExtREAL
f . x is ext-real Element of ExtREAL
(f . x) + ((c | (x /\ B)) . x) is ext-real Element of ExtREAL
c . x is ext-real Element of ExtREAL
(f . x) + (c . x) is ext-real Element of ExtREAL
(X,S,M,((f | (x /\ B)) + (c | (x /\ B)))) is ext-real Element of ExtREAL
max+ ((f | (x /\ B)) + (c | (x /\ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((f | (x /\ B)) + (c | (x /\ B))))) is ext-real Element of ExtREAL
max- ((f | (x /\ B)) + (c | (x /\ B))) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((f | (x /\ B)) + (c | (x /\ B))))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | (x /\ B)) + (c | (x /\ B))))) - (X,S,M,(max- ((f | (x /\ B)) + (c | (x /\ B))))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f | (x /\ B)) + (c | (x /\ B))))) is ext-real set
(X,S,M,(max+ ((f | (x /\ B)) + (c | (x /\ B))))) + (- (X,S,M,(max- ((f | (x /\ B)) + (c | (x /\ B)))))) is ext-real set
(X,S,M,(f | (x /\ B))) is ext-real Element of ExtREAL
max+ (f | (x /\ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | (x /\ B)))) is ext-real Element of ExtREAL
max- (f | (x /\ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | (x /\ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | (x /\ B)))) - (X,S,M,(max- (f | (x /\ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | (x /\ B)))) is ext-real set
(X,S,M,(max+ (f | (x /\ B)))) + (- (X,S,M,(max- (f | (x /\ B))))) is ext-real set
(X,S,M,(c | (x /\ B))) is ext-real Element of ExtREAL
max+ (c | (x /\ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (c | (x /\ B)))) is ext-real Element of ExtREAL
max- (c | (x /\ B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (c | (x /\ B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c | (x /\ B)))) - (X,S,M,(max- (c | (x /\ B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c | (x /\ B)))) is ext-real set
(X,S,M,(max+ (c | (x /\ B)))) + (- (X,S,M,(max- (c | (x /\ B))))) is ext-real set
(X,S,M,(f | (x /\ B))) + (X,S,M,(c | (x /\ B))) 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,f) is ext-real Element of ExtREAL
max+ f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ f)) is ext-real Element of ExtREAL
max- f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(max+ f)) - (X,S,M,(max- f)) is ext-real Element of ExtREAL
- (X,S,M,(max- f)) is ext-real set
(X,S,M,(max+ f)) + (- (X,S,M,(max- f))) is ext-real set
c is complex real ext-real Element of REAL
c (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) f)) is ext-real Element of ExtREAL
max+ (c (#) f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (c (#) f))) is ext-real Element of ExtREAL
max- (c (#) f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (c (#) f))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c (#) f))) - (X,S,M,(max- (c (#) f))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c (#) f))) is ext-real set
(X,S,M,(max+ (c (#) f))) + (- (X,S,M,(max- (c (#) f)))) is ext-real set
R_EAL c is ext-real Element of ExtREAL
(R_EAL c) * (X,S,M,f) is ext-real Element of ExtREAL
dom f is Element of bool X
B is Element of S
dom (max- f) is Element of bool X
dom (c (#) f) is Element of bool X
dom (max+ f) is Element of bool X
I1 is complex real ext-real Element of REAL
c * I1 is complex real ext-real Element of REAL
(R_EAL c) * (X,S,M,(max+ f)) is ext-real Element of ExtREAL
c (#) (max+ f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) (max+ f))) is ext-real Element of ExtREAL
x is complex real ext-real Element of REAL
c * x is complex real ext-real Element of REAL
(R_EAL c) * (X,S,M,(max- f)) is ext-real Element of ExtREAL
c (#) (max- f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) (max- f))) is ext-real Element of ExtREAL
(X,S,M,(c (#) (max+ f))) - (X,S,M,(max- (c (#) f))) is ext-real Element of ExtREAL
(X,S,M,(c (#) (max+ f))) + (- (X,S,M,(max- (c (#) f)))) is ext-real set
(X,S,M,(c (#) (max+ f))) - (X,S,M,(c (#) (max- f))) is ext-real Element of ExtREAL
- (X,S,M,(c (#) (max- f))) is ext-real set
(X,S,M,(c (#) (max+ f))) + (- (X,S,M,(c (#) (max- f)))) is ext-real set
((R_EAL c) * (X,S,M,(max+ f))) - (X,S,M,(c (#) (max- f))) is ext-real Element of ExtREAL
((R_EAL c) * (X,S,M,(max+ f))) + (- (X,S,M,(c (#) (max- f)))) is ext-real set
((R_EAL c) * (X,S,M,(max+ f))) - ((R_EAL c) * (X,S,M,(max- f))) is ext-real Element of ExtREAL
- ((R_EAL c) * (X,S,M,(max- f))) is ext-real set
((R_EAL c) * (X,S,M,(max+ f))) + (- ((R_EAL c) * (X,S,M,(max- f)))) is ext-real set
- c is complex real ext-real Element of REAL
- (- c) is complex real ext-real Element of REAL
a is complex real ext-real Element of REAL
- a is complex real ext-real Element of REAL
a (#) (max- f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
a (#) (max+ f) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is complex real ext-real Element of REAL
a * x is complex real ext-real Element of REAL
R_EAL a is ext-real Element of ExtREAL
(R_EAL a) * (X,S,M,(max- f)) is ext-real Element of ExtREAL
(X,S,M,(a (#) (max- f))) is ext-real Element of ExtREAL
I1 is complex real ext-real Element of REAL
a * I1 is complex real ext-real Element of REAL
(R_EAL a) * (X,S,M,(max+ f)) is ext-real Element of ExtREAL
(X,S,M,(a (#) (max+ f))) is ext-real Element of ExtREAL
- (R_EAL a) is ext-real Element of ExtREAL
((R_EAL a) * (X,S,M,(max- f))) - (X,S,M,(a (#) (max+ f))) is ext-real Element of ExtREAL
- (X,S,M,(a (#) (max+ f))) is ext-real set
((R_EAL a) * (X,S,M,(max- f))) + (- (X,S,M,(a (#) (max+ f)))) is ext-real set
((R_EAL a) * (X,S,M,(max- f))) - ((R_EAL a) * (X,S,M,(max+ f))) is ext-real Element of ExtREAL
- ((R_EAL a) * (X,S,M,(max+ f))) is ext-real set
((R_EAL a) * (X,S,M,(max- f))) + (- ((R_EAL a) * (X,S,M,(max+ f)))) is ext-real set
(X,S,M,(max- f)) - (X,S,M,(max+ f)) is ext-real Element of ExtREAL
- (X,S,M,(max+ f)) is ext-real set
(X,S,M,(max- f)) + (- (X,S,M,(max+ f))) is ext-real set
(R_EAL a) * ((X,S,M,(max- f)) - (X,S,M,(max+ f))) is ext-real Element of ExtREAL
- ((X,S,M,(max+ f)) - (X,S,M,(max- f))) is ext-real Element of ExtREAL
(R_EAL a) * (- ((X,S,M,(max+ f)) - (X,S,M,(max- f)))) is ext-real Element of ExtREAL
(R_EAL a) * ((X,S,M,(max+ f)) - (X,S,M,(max- f))) is ext-real Element of ExtREAL
- ((R_EAL a) * ((X,S,M,(max+ f)) - (X,S,M,(max- 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
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Element of S
f | c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | c)) is ext-real Element of ExtREAL
max+ (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | c))) is ext-real Element of ExtREAL
max- (f | c) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | c))) - (X,S,M,(max- (f | c))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | c))) is ext-real set
(X,S,M,(max+ (f | c))) + (- (X,S,M,(max- (f | c)))) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
f + c is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
dom (f + c) is Element of bool X
B is Element of S
(X,S,M,(f + c),B) is ext-real Element of ExtREAL
(f + c) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((f + c) | B)) is ext-real Element of ExtREAL
max+ ((f + c) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((f + c) | B))) is ext-real Element of ExtREAL
max- ((f + c) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((f + c) | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f + c) | B))) - (X,S,M,(max- ((f + c) | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f + c) | B))) is ext-real set
(X,S,M,(max+ ((f + c) | B))) + (- (X,S,M,(max- ((f + c) | B)))) is ext-real set
(X,S,M,f,B) is ext-real Element of ExtREAL
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(X,S,M,c,B) is ext-real Element of ExtREAL
c | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c | B)) is ext-real Element of ExtREAL
max+ (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (c | B))) is ext-real Element of ExtREAL
max- (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (c | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (c | B))) - (X,S,M,(max- (c | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (c | B))) is ext-real set
(X,S,M,(max+ (c | B))) + (- (X,S,M,(max- (c | B)))) is ext-real set
(X,S,M,f,B) + (X,S,M,c,B) is ext-real Element of ExtREAL
dom (f | B) is Element of bool X
dom f is Element of bool X
(dom f) /\ B is Element of bool X
dom c is Element of bool X
(dom f) /\ (dom c) is Element of bool X
f " {-infty} is Element of bool X
c " {+infty} is Element of bool X
(f " {-infty}) /\ (c " {+infty}) is Element of bool X
f " {+infty} is Element of bool X
c " {-infty} is Element of bool X
(f " {+infty}) /\ (c " {-infty}) is Element of bool X
((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty})) is Element of bool X
((dom f) /\ (dom c)) \ (((f " {-infty}) /\ (c " {+infty})) \/ ((f " {+infty}) /\ (c " {-infty}))) is Element of bool X
(f | B) + (c | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((f | B) + (c | B))) is ext-real Element of ExtREAL
max+ ((f | B) + (c | B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((f | B) + (c | B)))) is ext-real Element of ExtREAL
max- ((f | B) + (c | B)) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((f | B) + (c | B)))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((f | B) + (c | B)))) - (X,S,M,(max- ((f | B) + (c | B)))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((f | B) + (c | B)))) is ext-real set
(X,S,M,(max+ ((f | B) + (c | B)))) + (- (X,S,M,(max- ((f | B) + (c | B))))) is ext-real set
dom (c | B) is Element of bool X
(dom c) /\ B 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 V59() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V59() set
bool [:S,ExtREAL:] is non empty set
M is Relation-like S -defined ExtREAL -valued Function-like V32(S, ExtREAL ) V59() V67() nonnegative sigma-additive () Element of bool [:S,ExtREAL:]
f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
c is complex real ext-real Element of REAL
c (#) f is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
R_EAL c is ext-real Element of ExtREAL
B is Element of S
f | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(c (#) f),B) is ext-real Element of ExtREAL
(c (#) f) | B is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,((c (#) f) | B)) is ext-real Element of ExtREAL
max+ ((c (#) f) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ ((c (#) f) | B))) is ext-real Element of ExtREAL
max- ((c (#) f) | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- ((c (#) f) | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ ((c (#) f) | B))) - (X,S,M,(max- ((c (#) f) | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- ((c (#) f) | B))) is ext-real set
(X,S,M,(max+ ((c (#) f) | B))) + (- (X,S,M,(max- ((c (#) f) | B)))) is ext-real set
(X,S,M,f,B) is ext-real Element of ExtREAL
(X,S,M,(f | B)) is ext-real Element of ExtREAL
max+ (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max+ (f | B))) is ext-real Element of ExtREAL
max- (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
(X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
(X,S,M,(max+ (f | B))) - (X,S,M,(max- (f | B))) is ext-real Element of ExtREAL
- (X,S,M,(max- (f | B))) is ext-real set
(X,S,M,(max+ (f | B))) + (- (X,S,M,(max- (f | B)))) is ext-real set
(R_EAL c) * (X,S,M,f,B) is ext-real Element of ExtREAL
dom ((c (#) f) | B) is Element of bool X
c (#) (f | B) is Relation-like X -defined ExtREAL -valued Function-like V59() Element of bool [:X,ExtREAL:]
x is set
((c (#) f) | B) . x is ext-real Element of ExtREAL
(c (#) (f | B)) . x is ext-real Element of ExtREAL
(c (#) f) . x is ext-real Element of ExtREAL
dom (c (#) f) is Element of bool X
(dom (c (#) f)) /\ B is Element of bool X
dom f is Element of bool X
(dom f) /\ B is Element of bool X
dom (f | B) is Element of bool X
f . x is ext-real Element of ExtREAL
(R_EAL c) * (f . x) is ext-real Element of ExtREAL
(f | B) . x is ext-real Element of ExtREAL
(R_EAL c) * ((f | B) . x) is ext-real Element of ExtREAL
dom (c (#) (f | B)) is Element of bool X
dom (c (#) f) is Element of bool X
(dom (c (#) f)) /\ B is Element of bool X
dom f is Element of bool X
(dom f) /\ B is Element of bool X
dom (f | B) is Element of bool X
dom (c (#) (f | B)) is Element of bool X