:: LPSPACE2 semantic presentation

REAL is non empty V27() V156() V157() V158() V162() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() Element of bool REAL
bool REAL is non empty set
ExtREAL is non empty V157() set
[:NAT,ExtREAL:] is non empty V164() set
bool [:NAT,ExtREAL:] is non empty set
bool ExtREAL is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal V156() V157() V158() V159() V160() V161() V162() set
bool omega is non empty set
bool NAT is non empty set
COMPLEX is non empty V27() V156() V162() set
[:NAT,REAL:] is non empty V163() V164() V165() set
bool [:NAT,REAL:] is non empty set
bool (bool REAL) is non empty set
RAT is non empty V27() V156() V157() V158() V159() V162() set
INT is non empty V27() V156() V157() V158() V159() V160() V162() set
{} is set
the empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative V38() real V156() V157() V158() V159() V160() V161() V162() set is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural ext-real non positive non negative V38() real V156() V157() V158() V159() V160() V161() V162() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
{{},1} is set
[:NAT,COMPLEX:] is non empty V163() set
bool [:NAT,COMPLEX:] is non empty set
K657() is non empty set
[:K657(),K657():] is non empty set
[:[:K657(),K657():],K657():] is non empty set
bool [:[:K657(),K657():],K657():] is non empty set
[:REAL,K657():] is non empty set
[:[:REAL,K657():],K657():] is non empty set
bool [:[:REAL,K657():],K657():] is non empty set
K663() is RLSStruct
the carrier of K663() is set
bool the carrier of K663() is non empty set
K667() is Element of bool the carrier of K663()
[:K667(),K667():] is set
[:[:K667(),K667():],REAL:] is V163() V164() V165() set
bool [:[:K667(),K667():],REAL:] is non empty set
the_set_of_l1RealSequences is Element of bool the carrier of K663()
[:the_set_of_l1RealSequences,REAL:] is V163() V164() V165() set
bool [:the_set_of_l1RealSequences,REAL:] is non empty set
bool RAT is non empty set
[:COMPLEX,COMPLEX:] is non empty V163() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V163() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty V163() V164() V165() set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V163() V164() V165() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is V5( RAT ) non empty V163() V164() V165() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is V5( RAT ) non empty V163() V164() V165() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
[:[:NAT,NAT:],NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:[:NAT,NAT:],NAT:] is non empty set
0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
|.0.| is ext-real V38() real V194() Element of REAL
bool [:NAT,NAT:] is non empty set
- 1 is non empty ext-real non positive negative V38() real Element of REAL
0. is ext-real Element of ExtREAL
K626() is non empty ext-real positive non negative non real Element of ExtREAL
K627() is non empty ext-real non positive negative non real Element of ExtREAL
+infty is non empty ext-real positive non negative non real set
-infty is non empty ext-real non positive negative non real set
X is non empty ext-real positive non negative V38() real set
1 / X is non empty ext-real positive non negative V38() real Element of REAL
X " is non empty ext-real positive non negative V38() real set
1 * (X ") is non empty ext-real positive non negative V38() real set
S is non empty ext-real positive non negative V38() real set
1 / S is non empty ext-real positive non negative V38() real Element of REAL
S " is non empty ext-real positive non negative V38() real set
1 * (S ") is non empty ext-real positive non negative V38() real set
(1 / X) + (1 / S) is non empty ext-real positive non negative V38() real Element of REAL
1 - (1 / X) is ext-real V38() real Element of REAL
- (1 / X) is non empty ext-real non positive negative V38() real set
1 + (- (1 / X)) is ext-real V38() 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 V164() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is Element of S
g is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom g is Element of bool X
Integral (M,g) is ext-real Element of ExtREAL
max+ g is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ g)) is ext-real Element of ExtREAL
max- g is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- g)) is ext-real Element of ExtREAL
(integral+ (M,(max+ g))) - (integral+ (M,(max- g))) is ext-real Element of ExtREAL
dom (max- g) is Element of bool X
dom (max+ g) is Element of bool X
a is Element of X
(max+ g) . a is ext-real set
Integral (M,(max+ g)) is ext-real Element of ExtREAL
max+ (max+ g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (max+ g))) is ext-real Element of ExtREAL
max- (max+ g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (max+ g))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (max+ g)))) - (integral+ (M,(max- (max+ g)))) is ext-real Element of ExtREAL
a is Element of X
(max+ g) . a is ext-real set
g . a is ext-real set
max ((g . a),0) is ext-real set
a is Element of X
(max- g) . a is ext-real set
Integral (M,(max- g)) is ext-real Element of ExtREAL
max+ (max- g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (max- g))) is ext-real Element of ExtREAL
max- (max- g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (max- g))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (max- g)))) - (integral+ (M,(max- (max- g)))) is ext-real Element of ExtREAL
a is Element of X
(max- g) . a is ext-real set
(max+ g) . a is ext-real set
g . a is ext-real set
X is ext-real V38() real set
M is ext-real V38() real Element of REAL
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
S to_power M is ext-real V38() real Element of REAL
X to_power M is ext-real V38() real Element of REAL
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
X to_power S is ext-real V38() real Element of REAL
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
X * S is ext-real V38() real Element of REAL
M is ext-real V38() real Element of REAL
(X * S) to_power M is ext-real V38() real Element of REAL
X to_power M is ext-real V38() real Element of REAL
S to_power M is ext-real V38() real Element of REAL
(X to_power M) * (S to_power M) is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is ext-real V38() real Element of REAL
M is ext-real V38() real Element of REAL
S * M is ext-real V38() real Element of REAL
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x to_power S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(x to_power S) to_power M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x to_power (S * M) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (x to_power S) is Element of bool X
bool X is non empty set
dom x is Element of bool X
dom ((x to_power S) to_power M) is Element of bool X
dom (x to_power (S * M)) is Element of bool X
g is set
((x to_power S) to_power M) . g is ext-real V38() real Element of REAL
(x to_power (S * M)) . g is ext-real V38() real Element of REAL
(x to_power S) . g is ext-real V38() real Element of REAL
((x to_power S) . g) to_power M is ext-real V38() real Element of REAL
x . g is ext-real V38() real Element of REAL
(x . g) to_power S is ext-real V38() real Element of REAL
((x . g) to_power S) to_power M is ext-real V38() real Element of REAL
(x . g) to_power (S * M) is ext-real V38() real Element of REAL
0 to_power M is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is ext-real V38() real Element of REAL
M is ext-real V38() real Element of REAL
S + M is ext-real V38() real Element of REAL
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x to_power S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x to_power M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(x to_power S) (#) (x to_power M) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x to_power (S + M) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (x to_power S) is Element of bool X
bool X is non empty set
dom x is Element of bool X
dom (x to_power M) is Element of bool X
dom ((x to_power S) (#) (x to_power M)) is Element of bool X
(dom (x to_power S)) /\ (dom (x to_power M)) is Element of bool X
dom (x to_power (S + M)) is Element of bool X
g is set
((x to_power S) (#) (x to_power M)) . g is ext-real V38() real Element of REAL
(x to_power (S + M)) . g is ext-real V38() real Element of REAL
(x to_power S) . g is ext-real V38() real Element of REAL
x . g is ext-real V38() real Element of REAL
(x . g) to_power S is ext-real V38() real Element of REAL
(x to_power M) . g is ext-real V38() real Element of REAL
(x . g) to_power M is ext-real V38() real Element of REAL
((x . g) to_power S) * ((x . g) to_power M) is ext-real V38() real Element of REAL
(x . g) to_power (S + M) is ext-real V38() real Element of REAL
0 to_power M is ext-real V38() real Element of REAL
0 * (0 to_power M) is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
S to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (S to_power 1) is Element of bool X
bool X is non empty set
dom S is Element of bool X
M is set
(S to_power 1) . M is ext-real V38() real Element of REAL
S . M is ext-real V38() real Element of REAL
(S . M) to_power 1 is ext-real V38() real Element of REAL
X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
S is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
M is non empty ext-real positive non negative V38() real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X . x is ext-real V38() real Element of REAL
S . x is ext-real V38() real Element of REAL
(S . x) to_power M is ext-real V38() real Element of REAL
1 / M is non empty ext-real positive non negative V38() real Element of REAL
M " is non empty ext-real positive non negative V38() real set
1 * (M ") is non empty ext-real positive non negative V38() real set
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . x is ext-real V38() real Element of REAL
X . x is ext-real V38() real Element of REAL
(X . x) to_power (1 / M) is ext-real V38() real Element of REAL
(S . x) to_power M is ext-real V38() real Element of REAL
((S . x) to_power M) to_power (1 / M) is ext-real V38() real Element of REAL
M * (1 / M) is non empty ext-real positive non negative V38() real Element of REAL
(S . x) to_power (M * (1 / M)) is ext-real V38() real Element of REAL
(S . x) to_power 1 is ext-real V38() real Element of REAL
X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
abs X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (abs X) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums X) . S is ext-real V38() real Element of REAL
(Partial_Sums X) . M is ext-real V38() real Element of REAL
((Partial_Sums X) . S) - ((Partial_Sums X) . M) is ext-real V38() real Element of REAL
- ((Partial_Sums X) . M) is ext-real V38() real set
((Partial_Sums X) . S) + (- ((Partial_Sums X) . M)) is ext-real V38() real set
abs (((Partial_Sums X) . S) - ((Partial_Sums X) . M)) is ext-real V38() real Element of REAL
(Partial_Sums (abs X)) . S is ext-real V38() real Element of REAL
(Partial_Sums (abs X)) . M is ext-real V38() real Element of REAL
((Partial_Sums (abs X)) . S) - ((Partial_Sums (abs X)) . M) is ext-real V38() real Element of REAL
- ((Partial_Sums (abs X)) . M) is ext-real V38() real set
((Partial_Sums (abs X)) . S) + (- ((Partial_Sums (abs X)) . M)) is ext-real V38() real set
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(abs X) . x is ext-real V38() real Element of REAL
X . x is ext-real V38() real Element of REAL
abs (X . x) is ext-real V38() real Element of REAL
abs (((Partial_Sums (abs X)) . S) - ((Partial_Sums (abs X)) . M)) is ext-real V38() real Element of REAL
(((Partial_Sums (abs X)) . S) - ((Partial_Sums (abs X)) . M)) + ((Partial_Sums (abs X)) . M) is ext-real V38() real Element of REAL
X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
S is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim X is ext-real V38() real Element of REAL
lim S is ext-real V38() real Element of REAL
M is non empty ext-real positive non negative V38() real Element of REAL
g is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
a is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
a . b is ext-real V38() real Element of REAL
a - X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
- X is V1() V4( NAT ) Function-like total V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) X is V1() V4( NAT ) Function-like total V163() V164() V165() set
a + (- X) is V1() V4( NAT ) Function-like total V163() V164() V165() set
dom a is V156() V157() V158() V159() V160() V161() Element of bool NAT
dom X is V156() V157() V158() V159() V160() V161() Element of bool NAT
dom (a - X) is V156() V157() V158() V159() V160() V161() Element of bool NAT
dom g is V156() V157() V158() V159() V160() V161() Element of bool NAT
abs (a - X) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
dom (abs (a - X)) is V156() V157() V158() V159() V160() V161() Element of bool NAT
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(abs (a - X)) . b is ext-real V38() real Element of REAL
g . b is ext-real V38() real Element of REAL
X . b is ext-real V38() real Element of REAL
(lim X) - (X . b) is ext-real V38() real Element of REAL
- (X . b) is ext-real V38() real set
(lim X) + (- (X . b)) is ext-real V38() real set
|.((lim X) - (X . b)).| is ext-real V38() real Element of REAL
a . b is ext-real V38() real Element of REAL
(a . b) - (X . b) is ext-real V38() real Element of REAL
(a . b) + (- (X . b)) is ext-real V38() real set
|.((a . b) - (X . b)).| is ext-real V38() real Element of REAL
(a - X) . b is ext-real V38() real Element of REAL
abs ((a - X) . b) is ext-real V38() real Element of REAL
lim (a - X) is ext-real V38() real Element of REAL
a . 0 is ext-real V38() real Element of REAL
(a . 0) - (lim X) is ext-real V38() real Element of REAL
- (lim X) is ext-real V38() real set
(a . 0) + (- (lim X)) is ext-real V38() real set
(lim X) - (lim X) is ext-real V38() real Element of REAL
(lim X) + (- (lim X)) is ext-real V38() real set
lim g is ext-real V38() real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . b is ext-real V38() real Element of REAL
g . b is ext-real V38() real Element of REAL
(g . b) to_power M is ext-real V38() real Element of REAL
X . b is ext-real V38() real Element of REAL
(lim X) - (X . b) is ext-real V38() real Element of REAL
- (X . b) is ext-real V38() real set
(lim X) + (- (X . b)) is ext-real V38() real set
|.((lim X) - (X . b)).| is ext-real V38() real Element of REAL
(lim g) to_power M is ext-real V38() real Element of REAL
X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
S is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums X) . S is ext-real V38() real Element of REAL
abs ((Partial_Sums X) . S) is ext-real V38() real Element of REAL
abs X is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (abs X) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (abs X)) . S is ext-real V38() real Element of REAL
X is non empty ext-real positive non negative V38() real Element of REAL
S is non empty set
S --> 0 is V1() V4(S) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:S,REAL:]
[:S,REAL:] is non empty V163() V164() V165() set
bool [:S,REAL:] is non empty set
(S --> 0) to_power X is V1() V4(S) V5( REAL ) Function-like V163() V164() V165() Element of bool [:S,REAL:]
dom ((S --> 0) to_power X) is Element of bool S
bool S is non empty set
dom (S --> 0) is Element of bool S
M is Element of S
((S --> 0) to_power X) . M is ext-real V38() real Element of REAL
(S --> 0) . M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
((S --> 0) . M) to_power X is ext-real V38() real Element of REAL
0 to_power X is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is set
S | M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (S | M) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs S) | M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (abs (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 (abs S) is Element of bool X
(dom (abs S)) /\ M is Element of bool X
dom ((abs S) | M) is Element of bool X
x is Element of X
(abs (S | M)) . x is ext-real V38() real Element of REAL
((abs S) | M) . x is ext-real V38() real Element of REAL
(S | M) . x is ext-real V38() real Element of REAL
abs ((S | M) . x) is ext-real V38() real Element of REAL
S . x is ext-real V38() real Element of REAL
abs (S . x) is ext-real V38() real Element of REAL
(abs S) . x is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is set
dom (abs S) is Element of bool X
bool X is non empty set
(abs S) . M is ext-real V38() real Element of REAL
S . M is ext-real V38() real Element of REAL
abs (S . M) is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom S is Element of bool X
bool X is non empty set
dom (abs S) is Element of bool X
M is Element of X
S . M is ext-real V38() real Element of REAL
(abs S) . M is ext-real V38() real Element of REAL
abs (S . M) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
Integral (M,x) is ext-real Element of ExtREAL
R_EAL x is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL x)) is ext-real Element of ExtREAL
max+ (R_EAL x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
max- (R_EAL x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL x))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL x)))) - (integral+ (M,(max- (R_EAL 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
a is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
a is V1() V4(X) V5( REAL ) Function-like non empty total V41(X, REAL ) V163() V164() V165() Element of bool [:X,REAL:]
abs a is V1() V4(X) V5( REAL ) Function-like non empty total V41(X, REAL ) V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom a is Element of bool X
Ef is set
a . Ef is ext-real V38() real Element of REAL
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom (X --> 0) is Element of bool X
Ef is set
(X --> 0) . Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
dom ((abs a) to_power x) is Element of bool X
Ef is Element of X
((abs a) to_power x) . Ef is ext-real V38() real Element of REAL
2 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
M is ext-real V38() real Element of REAL
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
X + S is ext-real V38() real Element of REAL
abs (X + S) is ext-real V38() real Element of REAL
(abs (X + S)) to_power M is ext-real V38() real Element of REAL
abs X is ext-real V38() real Element of REAL
abs S is ext-real V38() real Element of REAL
(abs X) + (abs S) is ext-real V38() real Element of REAL
((abs X) + (abs S)) to_power M is ext-real V38() real Element of REAL
max ((abs X),(abs S)) is ext-real V38() real set
2 * (max ((abs X),(abs S))) is ext-real V38() real Element of REAL
(2 * (max ((abs X),(abs S)))) to_power M is ext-real V38() real Element of REAL
(max ((abs X),(abs S))) + (max ((abs X),(abs S))) is ext-real V38() real set
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
M is ext-real V38() real Element of REAL
max (X,S) is ext-real V38() real set
(max (X,S)) to_power M is ext-real V38() real set
X to_power M is ext-real V38() real Element of REAL
S to_power M is ext-real V38() real Element of REAL
(X to_power M) + (S to_power M) is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
x is ext-real V38() real Element of REAL
(abs S) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is ext-real V38() real Element of REAL
abs M is ext-real V38() real Element of REAL
(abs M) to_power x is ext-real V38() real Element of REAL
((abs M) to_power x) (#) ((abs S) to_power x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M (#) S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (M (#) S) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (M (#) S)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (((abs M) to_power x) (#) ((abs S) to_power x)) is Element of bool X
bool X is non empty set
dom ((abs S) to_power x) is Element of bool X
dom (M (#) S) is Element of bool X
dom S is Element of bool X
dom (abs S) is Element of bool X
dom (abs (M (#) S)) is Element of bool X
dom ((abs (M (#) S)) to_power x) is Element of bool X
g is Element of X
(((abs M) to_power x) (#) ((abs S) to_power x)) . g is ext-real V38() real Element of REAL
((abs (M (#) S)) to_power x) . g is ext-real V38() real Element of REAL
S . g is ext-real V38() real Element of REAL
abs (S . g) is ext-real V38() real Element of REAL
((abs S) to_power x) . g is ext-real V38() real Element of REAL
((abs M) to_power x) * (((abs S) to_power x) . g) is ext-real V38() real Element of REAL
(abs S) . g is ext-real V38() real Element of REAL
((abs S) . g) to_power x is ext-real V38() real Element of REAL
((abs M) to_power x) * (((abs S) . g) to_power x) is ext-real V38() real Element of REAL
(abs (S . g)) to_power x is ext-real V38() real Element of REAL
((abs M) to_power x) * ((abs (S . g)) to_power x) is ext-real V38() real Element of REAL
(abs M) * (abs (S . g)) is ext-real V38() real Element of REAL
((abs M) * (abs (S . g))) to_power x is ext-real V38() real Element of REAL
M * (S . g) is ext-real V38() real Element of REAL
abs (M * (S . g)) is ext-real V38() real Element of REAL
(abs (M * (S . g))) to_power x is ext-real V38() real Element of REAL
(M (#) S) . g is ext-real V38() real Element of REAL
abs ((M (#) S) . g) is ext-real V38() real Element of REAL
(abs ((M (#) S) . g)) to_power x is ext-real V38() real Element of REAL
(abs (M (#) S)) . g is ext-real V38() real Element of REAL
((abs (M (#) S)) . g) to_power x is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
M is ext-real V38() real Element of REAL
x is ext-real V38() real Element of REAL
(abs S) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M to_power x is ext-real V38() real Element of REAL
(M to_power x) (#) ((abs S) to_power x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M (#) (abs S) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(M (#) (abs S)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs M is ext-real V38() real Element of REAL
M (#) S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (M (#) S) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (M (#) S)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is ext-real V38() real set
S to_power M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x is set
S | x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(S | x) to_power M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(S to_power M) | x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((S | x) to_power M) is Element of bool X
bool X is non empty set
dom (S | x) is Element of bool X
dom S is Element of bool X
(dom S) /\ x is Element of bool X
dom (S to_power M) is Element of bool X
(dom (S to_power M)) /\ x is Element of bool X
dom ((S to_power M) | x) is Element of bool X
g is Element of X
((S | x) to_power M) . g is ext-real V38() real Element of REAL
(S | x) . g is ext-real V38() real Element of REAL
((S | x) . g) to_power M is ext-real V38() real set
S . g is ext-real V38() real Element of REAL
(S . g) to_power M is ext-real V38() real set
(S to_power M) . g is ext-real V38() real Element of REAL
((S to_power M) | x) . g is ext-real V38() real Element of REAL
M is ext-real V38() real Element of REAL
X is ext-real V38() real Element of REAL
S is ext-real V38() real Element of REAL
X + S is ext-real V38() real Element of REAL
abs (X + S) is ext-real V38() real Element of REAL
(abs (X + S)) to_power M is ext-real V38() real Element of REAL
2 to_power M is ext-real V38() real Element of REAL
abs X is ext-real V38() real Element of REAL
(abs X) to_power M is ext-real V38() real Element of REAL
abs S is ext-real V38() real Element of REAL
(abs S) to_power M is ext-real V38() real Element of REAL
((abs X) to_power M) + ((abs S) to_power M) is ext-real V38() real Element of REAL
(2 to_power M) * (((abs X) to_power M) + ((abs S) to_power M)) is ext-real V38() real Element of REAL
max ((abs X),(abs S)) is ext-real V38() real set
2 * (max ((abs X),(abs S))) is ext-real V38() real Element of REAL
(2 * (max ((abs X),(abs S)))) to_power M is ext-real V38() real Element of REAL
(max ((abs X),(abs S))) to_power M is ext-real V38() real set
(2 to_power M) * ((max ((abs X),(abs S))) to_power M) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs g) to_power x) + ((abs a) to_power x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom Ef is Element of bool X
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
g is V1() V4(X) V5( REAL ) Function-like non empty total V41(X, REAL ) V163() V164() V165() Element of bool [:X,REAL:]
dom g is Element of bool X
b is Element of X
g . b is ext-real V38() real Element of REAL
b is set
g . b is ext-real V38() real Element of REAL
abs g is V1() V4(X) V5( REAL ) Function-like non empty total V41(X, REAL ) V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
b is set
g . b is ext-real V38() real Element of REAL
X is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is ext-real V38() real Element of REAL
2 to_power S is ext-real V38() real Element of REAL
M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom M is Element of bool X
bool X is non empty set
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
(dom M) /\ (dom x) is Element of bool X
M + x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (M + x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (M + x)) to_power S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs M) to_power S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs x) to_power S is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs M) to_power S) + ((abs x) to_power S) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(2 to_power S) (#) (((abs M) to_power S) + ((abs x) to_power S)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is Element of X
((abs (M + x)) to_power S) . g is ext-real V38() real Element of REAL
((2 to_power S) (#) (((abs M) to_power S) + ((abs x) to_power S))) . g is ext-real V38() real Element of REAL
dom (M + x) is Element of bool X
dom (abs (M + x)) is Element of bool X
dom ((abs (M + x)) to_power S) is Element of bool X
(abs (M + x)) . g is ext-real V38() real Element of REAL
((abs (M + x)) . g) to_power S is ext-real V38() real Element of REAL
(M + x) . g is ext-real V38() real Element of REAL
abs ((M + x) . g) is ext-real V38() real Element of REAL
(abs ((M + x) . g)) to_power S is ext-real V38() real Element of REAL
M . g is ext-real V38() real Element of REAL
x . g is ext-real V38() real Element of REAL
(M . g) + (x . g) is ext-real V38() real Element of REAL
abs ((M . g) + (x . g)) is ext-real V38() real Element of REAL
(abs ((M . g) + (x . g))) to_power S is ext-real V38() real Element of REAL
dom (abs M) is Element of bool X
dom (abs x) is Element of bool X
dom ((abs M) to_power S) is Element of bool X
dom ((abs x) to_power S) is Element of bool X
abs (M . g) is ext-real V38() real Element of REAL
(abs (M . g)) to_power S is ext-real V38() real Element of REAL
(abs M) . g is ext-real V38() real Element of REAL
((abs M) . g) to_power S is ext-real V38() real Element of REAL
abs (x . g) is ext-real V38() real Element of REAL
(abs (x . g)) to_power S is ext-real V38() real Element of REAL
(abs x) . g is ext-real V38() real Element of REAL
((abs x) . g) to_power S is ext-real V38() real Element of REAL
((abs M) to_power S) . g is ext-real V38() real Element of REAL
((abs x) to_power S) . g is ext-real V38() real Element of REAL
dom (((abs M) to_power S) + ((abs x) to_power S)) is Element of bool X
(dom ((abs M) to_power S)) /\ (dom ((abs x) to_power S)) is Element of bool X
((abs (M . g)) to_power S) + ((abs (x . g)) to_power S) is ext-real V38() real Element of REAL
(2 to_power S) * (((abs (M . g)) to_power S) + ((abs (x . g)) to_power S)) is ext-real V38() real Element of REAL
(((abs M) to_power S) + ((abs x) to_power S)) . g is ext-real V38() real Element of REAL
(2 to_power S) * ((((abs M) to_power S) + ((abs x) to_power S)) . g) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
ND + E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (ND + E) is Element of bool X
Ef /\ EE is Element of S
abs (ND + E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (ND + E)) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs ND) to_power a) + ((abs E) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
2 to_power a is ext-real V38() real Element of REAL
(2 to_power a) (#) (((abs ND) to_power a) + ((abs E) to_power a)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Ef /\ EE) ` is Element of bool X
X \ (Ef /\ EE) is set
X \ Ef is Element of bool X
X \ EE is Element of bool X
(X \ Ef) \/ (X \ EE) is Element of bool X
(Ef `) \/ (EE `) is Element of bool X
M . ((Ef /\ EE) `) is ext-real set
G is Element of S
G ` is Element of bool X
X \ G is set
M . (G `) is ext-real set
Gp is Element of S
Gp ` is Element of bool X
X \ Gp is set
M . (Gp `) is ext-real set
dom (abs ND) is Element of bool X
dom (abs E) is Element of bool X
dom (abs (ND + E)) is Element of bool X
abs ((abs (ND + E)) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom ((abs (ND + E)) to_power a) is Element of bool X
dom ((2 to_power a) (#) (((abs ND) to_power a) + ((abs E) to_power a))) is Element of bool X
dom (((abs ND) to_power a) + ((abs E) to_power a)) is Element of bool X
dom ((abs ND) to_power a) is Element of bool X
dom ((abs E) to_power a) is Element of bool X
(dom ((abs ND) to_power a)) /\ (dom ((abs E) to_power a)) is Element of bool X
(dom (abs ND)) /\ (dom ((abs E) to_power a)) is Element of bool X
(dom (abs ND)) /\ (dom (abs E)) is Element of bool X
G is Element of X
(abs ((abs (ND + E)) to_power a)) . G is ext-real V38() real Element of REAL
((2 to_power a) (#) (((abs ND) to_power a) + ((abs E) to_power a))) . G is ext-real V38() real Element of REAL
((abs (ND + E)) to_power a) . G is ext-real V38() real Element of REAL
abs (((abs (ND + E)) to_power a) . G) is ext-real V38() real Element of REAL
G is Element of S
G ` is Element of bool X
X \ G is set
M . (G `) is ext-real set
Gp is Element of S
Gp ` is Element of bool X
X \ Gp is set
M . (Gp `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is ext-real V38() real Element of REAL
g (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
g (#) b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (g (#) b) is Element of bool X
abs g is ext-real V38() real Element of REAL
(abs g) to_power a is ext-real V38() real Element of REAL
((abs g) to_power a) (#) ((abs b) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
abs (g (#) b) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (g (#) b)) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x - g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- g is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) g is V1() V4(X) Function-like V163() V164() V165() set
x + (- g) is V1() V4(X) Function-like V163() V164() V165() set
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(- 1) (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
dom (abs b) is Element of bool X
abs (abs b) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (abs b)) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool the carrier of (RLSp_PFunct X)
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
a is Element of the carrier of (RLSp_PFunct X)
b is Element of the carrier of (RLSp_PFunct X)
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom Ef is Element of bool X
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a + b is Element of the carrier of (RLSp_PFunct X)
the addF of (RLSp_PFunct X) is V1() V4([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):] is non empty set
[:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the addF of (RLSp_PFunct X) . (a,b) is Element of the carrier of (RLSp_PFunct X)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom E is Element of bool X
(dom ND) /\ (dom Ef) is Element of bool X
EE is set
E . EE is ext-real V38() real Element of REAL
ND . EE is ext-real V38() real Element of REAL
Ef . EE is ext-real V38() real Element of REAL
(ND . EE) + (Ef . EE) is ext-real V38() real Element of REAL
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
M . (E0 `) is ext-real set
ND + Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
b is Element of the carrier of (RLSp_PFunct X)
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is ext-real V38() real Element of REAL
a * b is Element of the carrier of (RLSp_PFunct X)
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) . (a,b) is set
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom Ef is Element of bool X
E is Element of X
Ef . E is ext-real V38() real Element of REAL
ND . E is ext-real V38() real Element of REAL
a * (ND . E) is ext-real V38() real Element of REAL
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
E is set
Ef . E is ext-real V38() real Element of REAL
ND . E is ext-real V38() real Element of REAL
a * (ND . E) is ext-real V38() real Element of REAL
a (#) ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool the carrier of (RLSp_PFunct X)
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the carrier of (RLSp_PFunct X) is non empty set
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict RLSStruct
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict RLSStruct
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(RLSp_PFunct X)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,a),(In ((0. (RLSp_PFunct X)),(X,S,M,a))),(add| ((X,S,M,a),(RLSp_PFunct X))),(Mult_ (X,S,M,a)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
b is Element of the carrier of (X,S,M,a)
ND is Element of the carrier of (X,S,M,a)
b + ND is Element of the carrier of (X,S,M,a)
the addF of (X,S,M,a) is V1() V4([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):] is non empty set
[:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the addF of (X,S,M,a) . (b,ND) is Element of the carrier of (X,S,M,a)
Ef is Element of the carrier of (RLSp_PFunct X)
E is Element of the carrier of (RLSp_PFunct X)
Ef + E is Element of the carrier of (RLSp_PFunct X)
the addF of (RLSp_PFunct X) is V1() V4([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):] is non empty set
[:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the addF of (RLSp_PFunct X) . (Ef,E) is Element of the carrier of (RLSp_PFunct X)
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom EE is Element of bool X
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom E1 is Element of bool X
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom E0 is Element of bool X
(dom E1) /\ (dom E0) is Element of bool X
E2 is Element of X
EE . E2 is ext-real V38() real Element of REAL
E1 . E2 is ext-real V38() real Element of REAL
E0 . E2 is ext-real V38() real Element of REAL
(E1 . E2) + (E0 . E2) is ext-real V38() real Element of REAL
E2 is set
EE . E2 is ext-real V38() real Element of REAL
x . E2 is ext-real V38() real Element of REAL
g . E2 is ext-real V38() real Element of REAL
(x . E2) + (g . E2) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is ext-real V38() real Element of REAL
g (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(RLSp_PFunct X)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,a),(In ((0. (RLSp_PFunct X)),(X,S,M,a))),(add| ((X,S,M,a),(RLSp_PFunct X))),(Mult_ (X,S,M,a)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
b is Element of the carrier of (X,S,M,a)
g * b is Element of the carrier of (X,S,M,a)
the Mult of (X,S,M,a) is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[:REAL, the carrier of (X,S,M,a):] is non empty set
[:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) . (g,b) is set
ND is Element of the carrier of (RLSp_PFunct X)
g * ND is Element of the carrier of (RLSp_PFunct X)
the Mult of (RLSp_PFunct X) . (g,ND) is set
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
dom Ef is Element of bool X
dom x is Element of bool X
E is set
Ef . E is ext-real V38() real Element of REAL
x . E is ext-real V38() real Element of REAL
g * (x . E) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
(X --> 0) | (dom x) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,g)) is Element of (X,S,M,g)
add| ((X,S,M,g),(RLSp_PFunct X)) is V1() V4([:(X,S,M,g),(X,S,M,g):]) V5((X,S,M,g)) Function-like non empty total V41([:(X,S,M,g),(X,S,M,g):],(X,S,M,g)) Element of bool [:[:(X,S,M,g),(X,S,M,g):],(X,S,M,g):]
[:(X,S,M,g),(X,S,M,g):] is non empty set
[:[:(X,S,M,g),(X,S,M,g):],(X,S,M,g):] is non empty set
bool [:[:(X,S,M,g),(X,S,M,g):],(X,S,M,g):] is non empty set
Mult_ (X,S,M,g) is V1() V4([:REAL,(X,S,M,g):]) V5((X,S,M,g)) Function-like non empty total V41([:REAL,(X,S,M,g):],(X,S,M,g)) Element of bool [:[:REAL,(X,S,M,g):],(X,S,M,g):]
[:REAL,(X,S,M,g):] is non empty set
[:[:REAL,(X,S,M,g):],(X,S,M,g):] is non empty set
bool [:[:REAL,(X,S,M,g):],(X,S,M,g):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,g):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,g),(In ((0. (RLSp_PFunct X)),(X,S,M,g))),(add| ((X,S,M,g),(RLSp_PFunct X))),(Mult_ (X,S,M,g)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,g) is non empty set
a is Element of the carrier of (X,S,M,g)
(- 1) * a is Element of the carrier of (X,S,M,g)
the Mult of (X,S,M,g) is V1() V4([:REAL, the carrier of (X,S,M,g):]) V5( the carrier of (X,S,M,g)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g)) Element of bool [:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):]
[:REAL, the carrier of (X,S,M,g):] is non empty set
[:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
the Mult of (X,S,M,g) . ((- 1),a) is set
a + ((- 1) * a) is Element of the carrier of (X,S,M,g)
the addF of (X,S,M,g) is V1() V4([: the carrier of (X,S,M,g), the carrier of (X,S,M,g):]) V5( the carrier of (X,S,M,g)) Function-like non empty total V41([: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g)) Element of bool [:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):]
[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):] is non empty set
[:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
bool [:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
the addF of (X,S,M,g) . (a,((- 1) * a)) is Element of the carrier of (X,S,M,g)
b is Element of the carrier of (RLSp_PFunct X)
(- 1) * b is Element of the carrier of (RLSp_PFunct X)
the Mult of (RLSp_PFunct X) . ((- 1),b) is set
b + ((- 1) * b) is Element of the carrier of (RLSp_PFunct X)
the addF of (RLSp_PFunct X) is V1() V4([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):] is non empty set
[:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[: the carrier of (RLSp_PFunct X), the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the addF of (RLSp_PFunct X) . (b,((- 1) * b)) is Element of the carrier of (RLSp_PFunct X)
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom Ef is Element of bool X
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
(Ef `) ` is Element of bool X
X \ (Ef `) is set
ND | Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X --> 0) | Ef is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
((X --> 0) | Ef) | Ef is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b₁ a.e.= X --> 0,M ) } is set
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
bool the carrier of (X,S,M,x) is non empty set
g is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty Element of bool the carrier of (X,S,M,x)
the carrier of (X,S,M,x) is non empty set
bool the carrier of (X,S,M,x) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b₁ a.e.= X --> 0,M ) } is set
b is Element of the carrier of (X,S,M,x)
ND is Element of the carrier of (X,S,M,x)
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b + ND is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (b,ND) is Element of the carrier of (X,S,M,x)
Ef + E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X --> 0) + (X --> 0) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
ND is Element of the carrier of (X,S,M,x)
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is ext-real V38() real Element of REAL
b * ND is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . (b,ND) is set
b (#) Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b (#) (X --> 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty Element of bool the carrier of (X,S,M,x)
the carrier of (X,S,M,x) is non empty set
bool the carrier of (X,S,M,x) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b₁ a.e.= X --> 0,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (X,S,M,x)
bool the carrier of (X,S,M,x) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b₁ a.e.= X --> 0,M ) } is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (X,S,M,x)
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
bool the carrier of (X,S,M,x) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b₁ a.e.= X --> 0,M ) } is set
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
In ((0. (X,S,M,x)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(X,S,M,x)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
RLSStruct(# (X,S,M,x),(In ((0. (X,S,M,x)),(X,S,M,x))),(add| ((X,S,M,x),(X,S,M,x))),(Mult_ (X,S,M,x)) #) is non empty strict RLSStruct
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (X,S,M,a)
(X,S,M,a) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(RLSp_PFunct X)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,a),(In ((0. (RLSp_PFunct X)),(X,S,M,a))),(add| ((X,S,M,a),(RLSp_PFunct X))),(Mult_ (X,S,M,a)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
bool the carrier of (X,S,M,a) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & b₁ a.e.= X --> 0,M ) } is set
0. (X,S,M,a) is V52((X,S,M,a)) Element of the carrier of (X,S,M,a)
the ZeroF of (X,S,M,a) is Element of the carrier of (X,S,M,a)
In ((0. (X,S,M,a)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(X,S,M,a)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
RLSStruct(# (X,S,M,a),(In ((0. (X,S,M,a)),(X,S,M,a))),(add| ((X,S,M,a),(X,S,M,a))),(Mult_ (X,S,M,a)) #) is non empty strict RLSStruct
the carrier of (X,S,M,a) is non empty set
b is Element of the carrier of (X,S,M,a)
ND is Element of the carrier of (X,S,M,a)
b + ND is Element of the carrier of (X,S,M,a)
the addF of (X,S,M,a) is V1() V4([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):] is non empty set
[:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the addF of (X,S,M,a) . (b,ND) is Element of the carrier of (X,S,M,a)
Ef is Element of the carrier of (X,S,M,a)
E is Element of the carrier of (X,S,M,a)
Ef + E is Element of the carrier of (X,S,M,a)
the addF of (X,S,M,a) is V1() V4([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):] is non empty set
[:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[: the carrier of (X,S,M,a), the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the addF of (X,S,M,a) . (Ef,E) is Element of the carrier of (X,S,M,a)
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is ext-real V38() real Element of REAL
g (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (X,S,M,a)
(X,S,M,a) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(RLSp_PFunct X)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,a),(In ((0. (RLSp_PFunct X)),(X,S,M,a))),(add| ((X,S,M,a),(RLSp_PFunct X))),(Mult_ (X,S,M,a)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
bool the carrier of (X,S,M,a) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & b₁ a.e.= X --> 0,M ) } is set
0. (X,S,M,a) is V52((X,S,M,a)) Element of the carrier of (X,S,M,a)
the ZeroF of (X,S,M,a) is Element of the carrier of (X,S,M,a)
In ((0. (X,S,M,a)),(X,S,M,a)) is Element of (X,S,M,a)
add| ((X,S,M,a),(X,S,M,a)) is V1() V4([:(X,S,M,a),(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:(X,S,M,a),(X,S,M,a):],(X,S,M,a)) Element of bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):]
[:(X,S,M,a),(X,S,M,a):] is non empty set
[:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:(X,S,M,a),(X,S,M,a):],(X,S,M,a):] is non empty set
Mult_ (X,S,M,a) is V1() V4([:REAL,(X,S,M,a):]) V5((X,S,M,a)) Function-like non empty total V41([:REAL,(X,S,M,a):],(X,S,M,a)) Element of bool [:[:REAL,(X,S,M,a):],(X,S,M,a):]
[:REAL,(X,S,M,a):] is non empty set
[:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
bool [:[:REAL,(X,S,M,a):],(X,S,M,a):] is non empty set
[:REAL, the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) | [:REAL,(X,S,M,a):] is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
RLSStruct(# (X,S,M,a),(In ((0. (X,S,M,a)),(X,S,M,a))),(add| ((X,S,M,a),(X,S,M,a))),(Mult_ (X,S,M,a)) #) is non empty strict RLSStruct
the carrier of (X,S,M,a) is non empty set
b is Element of the carrier of (X,S,M,a)
g * b is Element of the carrier of (X,S,M,a)
the Mult of (X,S,M,a) is V1() V4([:REAL, the carrier of (X,S,M,a):]) V5( the carrier of (X,S,M,a)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a)) Element of bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):]
[:REAL, the carrier of (X,S,M,a):] is non empty set
[:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,a):], the carrier of (X,S,M,a):] is non empty set
the Mult of (X,S,M,a) . (g,b) is set
ND is Element of the carrier of (X,S,M,a)
g * ND is Element of the carrier of (X,S,M,a)
the Mult of (X,S,M,a) . (g,ND) is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power g is_integrable_on M ) } is set
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,g) & x a.e.= b<sub>1</sub>,M ) } is set
bool (X,S,M,g) is non empty set
a is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(X,S,M,g,a) is Element of bool (X,S,M,a)
bool (X,S,M,a) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & g a.e.= b<sub>1</sub>,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x,a) is Element of bool (X,S,M,a)
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
bool (X,S,M,a) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & x a.e.= b₁,M ) } is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom g is Element of bool X
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is Element of S
M . E0 is ext-real Element of ExtREAL
E0 ` is Element of bool X
X \ E0 is set
g | (E0 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x | (E0 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef \/ E0 is Element of S
E2 is Element of S
E2 ` is Element of bool X
X \ E2 is set
x | (E2 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(x | (E0 `)) | (E2 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g | (E2 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(g | (E0 `)) | (E2 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (abs x) is Element of bool X
dom ((abs x) to_power a) is Element of bool X
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (max+ (R_EAL ((abs x) to_power a))) is Element of bool X
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (max- (R_EAL ((abs x) to_power a))) is Element of bool X
dom (R_EAL ((abs x) to_power a)) is Element of bool X
G is Element of X
(max+ (R_EAL ((abs x) to_power a))) . G is ext-real set
Gp is Element of X
(max- (R_EAL ((abs x) to_power a))) . Gp is ext-real set
E1 /\ E2 is Element of S
E1 \ E2 is Element of S
(E1 /\ E2) \/ (E1 \ E2) is Element of S
(max+ (R_EAL ((abs x) to_power a))) | (dom (max+ (R_EAL ((abs x) to_power a)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max- (R_EAL ((abs x) to_power a))) | (dom (max- (R_EAL ((abs x) to_power a)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
G is Element of S
(max+ (R_EAL ((abs x) to_power a))) | G is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL ((abs x) to_power a))) | G)) is ext-real Element of ExtREAL
Gp is Element of S
(max+ (R_EAL ((abs x) to_power a))) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL ((abs x) to_power a))) | Gp)) is ext-real Element of ExtREAL
(integral+ (M,((max+ (R_EAL ((abs x) to_power a))) | G))) + (integral+ (M,((max+ (R_EAL ((abs x) to_power a))) | Gp))) is ext-real Element of ExtREAL
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(max- (R_EAL ((abs x) to_power a))) | G is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL ((abs x) to_power a))) | G)) is ext-real Element of ExtREAL
(max- (R_EAL ((abs x) to_power a))) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL ((abs x) to_power a))) | Gp)) is ext-real Element of ExtREAL
(integral+ (M,((max- (R_EAL ((abs x) to_power a))) | G))) + (integral+ (M,((max- (R_EAL ((abs x) to_power a))) | Gp))) is ext-real Element of ExtREAL
M . E2 is ext-real Element of ExtREAL
(max+ (R_EAL ((abs x) to_power a))) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL ((abs x) to_power a))) | E2)) is ext-real Element of ExtREAL
(max- (R_EAL ((abs x) to_power a))) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL ((abs x) to_power a))) | E2)) is ext-real Element of ExtREAL
E1 /\ (E2 `) is Element of bool X
x | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(g | (E2 `)) | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs x) | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (x | Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (g | Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs x) | Gp) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs x) to_power a) | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs g) | Gp) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs g) to_power a) | Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(R_EAL ((abs x) to_power a)) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ ((R_EAL ((abs x) to_power a)) | Gp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
R_EAL ((abs g) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max+ (R_EAL ((abs g) to_power a))) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(R_EAL ((abs g) to_power a)) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ ((R_EAL ((abs g) to_power a)) | Gp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- ((R_EAL ((abs x) to_power a)) | Gp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max- (R_EAL ((abs g) to_power a))) | Gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- ((R_EAL ((abs g) to_power a)) | Gp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
integral+ (M,(max- (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
integral+ (M,(max+ ((R_EAL ((abs g) to_power a)) | Gp))) is ext-real Element of ExtREAL
integral+ (M,(max- ((R_EAL ((abs g) to_power a)) | Gp))) is ext-real Element of ExtREAL
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power g is_integrable_on M ) } is set
(X,S,M,x,g) is Element of bool (X,S,M,g)
bool (X,S,M,g) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,g) & x a.e.= b<sub>1</sub>,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g,a) is Element of bool (X,S,M,a)
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
bool (X,S,M,a) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & g a.e.= b₁,M ) } is set
(X,S,M,x,a) is Element of bool (X,S,M,a)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & x a.e.= b₁,M ) } is set
b is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom g is Element of bool X
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(X,S,M,x,a) is Element of bool (X,S,M,a)
bool (X,S,M,a) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & x a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,g,a) is Element of bool (X,S,M,a)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & g a.e.= b<sub>1</sub>,M ) } is set
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x,a) is Element of bool (X,S,M,a)
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
bool (X,S,M,a) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & x a.e.= b₁,M ) } is set
(X,S,M,g,a) is Element of bool (X,S,M,a)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & g a.e.= b₁,M ) } is set
b is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x,a) is Element of bool (X,S,M,a)
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
bool (X,S,M,a) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & x a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,g,a) is Element of bool (X,S,M,a)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & g a.e.= b<sub>1</sub>,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
(X,S,M,x,a) is Element of bool (X,S,M,a)
bool (X,S,M,a) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & x a.e.= b₁,M ) } is set
(X,S,M,g,a) is Element of bool (X,S,M,a)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,a) & g a.e.= b₁,M ) } is set
dom x is Element of bool X
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom g is Element of bool X
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom a is Element of bool X
x + a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom b is Element of bool X
g + b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x,ND) is Element of bool (X,S,M,ND)
(X,S,M,ND) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power ND is_integrable_on M ) } is set
bool (X,S,M,ND) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & x a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,a,ND) is Element of bool (X,S,M,ND)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & a a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,g,ND) is Element of bool (X,S,M,ND)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & g a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,b,ND) is Element of bool (X,S,M,ND)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & b a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,(x + a),ND) is Element of bool (X,S,M,ND)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & x + a a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,(g + b),ND) is Element of bool (X,S,M,ND)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,ND) & g + b a.e.= b<sub>1</sub>,M ) } is set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x + a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g + b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,ND) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power ND is_integrable_on M ) } is set
(X,S,M,x,ND) is Element of bool (X,S,M,ND)
bool (X,S,M,ND) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & x a.e.= b₁,M ) } is set
(X,S,M,g,ND) is Element of bool (X,S,M,ND)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & g a.e.= b₁,M ) } is set
(X,S,M,a,ND) is Element of bool (X,S,M,ND)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & a a.e.= b₁,M ) } is set
(X,S,M,b,ND) is Element of bool (X,S,M,ND)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & b a.e.= b₁,M ) } is set
(X,S,M,(x + a),ND) is Element of bool (X,S,M,ND)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & x + a a.e.= b₁,M ) } is set
(X,S,M,(g + b),ND) is Element of bool (X,S,M,ND)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,ND) & g + b a.e.= b₁,M ) } is set
dom g is Element of bool X
dom b is Element of bool X
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom g is Element of bool X
a is ext-real V38() real Element of REAL
a (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x,b) is Element of bool (X,S,M,b)
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power b is_integrable_on M ) } is set
bool (X,S,M,b) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,b) & x a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,g,b) is Element of bool (X,S,M,b)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,b) & g a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,(a (#) x),b) is Element of bool (X,S,M,b)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,b) & a (#) x a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,(a (#) g),b) is Element of bool (X,S,M,b)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,b) & a (#) g a.e.= b<sub>1</sub>,M ) } is set
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is ext-real V38() real Element of REAL
a (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power b is_integrable_on M ) } is set
(X,S,M,x,b) is Element of bool (X,S,M,b)
bool (X,S,M,b) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & x a.e.= b₁,M ) } is set
(X,S,M,g,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & g a.e.= b₁,M ) } is set
(X,S,M,(a (#) x),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & a (#) x a.e.= b₁,M ) } is set
(X,S,M,(a (#) g),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & a (#) g a.e.= b₁,M ) } is set
dom x is Element of bool X
dom g is Element of bool X
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
{ (X,S,M,b<sub>1</sub>,x) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,x) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
bool (X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
a is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,b,x) is Element of bool (X,S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & b a.e.= b<sub>1</sub>,M ) } is set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),x) is Element of bool (X,S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & X --> 0 a.e.= b<sub>1</sub>,M ) } is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
a is Element of (X,S,M,x)
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,ND,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & ND a.e.= b₁,M ) } is set
dom ND is Element of bool X
b is Element of (X,S,M,x)
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,Ef,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & Ef a.e.= b₁,M ) } is set
dom Ef is Element of bool X
ND + Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(ND + Ef),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & ND + Ef a.e.= b₁,M ) } is set
EE is Element of (X,S,M,x)
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
G is Element of S
G ` is Element of bool X
X \ G is set
M . (G `) is ext-real set
Gp is Element of S
Gp ` is Element of bool X
X \ Gp is set
M . (Gp `) is ext-real set
E1 is Element of (X,S,M,x)
E0 + E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E0 + E2),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 + E2 a.e.= b₁,M ) } is set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 + E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E0 + E2),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 + E2 a.e.= b₁,M ) } is set
a is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
b is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
ND is Element of (X,S,M,x)
Ef is Element of (X,S,M,x)
b . (ND,Ef) is Element of (X,S,M,x)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E + EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E + EE),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E + EE a.e.= b₁,M ) } is set
g is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
a is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
b is Element of (X,S,M,x)
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,Ef,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & Ef a.e.= b₁,M ) } is set
ND is Element of (X,S,M,x)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E a.e.= b₁,M ) } is set
g . (b,ND) is Element of (X,S,M,x)
Ef + E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(Ef + E),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & Ef + E a.e.= b₁,M ) } is set
a . (b,ND) is Element of (X,S,M,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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,(X --> 0),x) is Element of bool (X,S,M,x)
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & X --> 0 a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b<sub>1</sub>,x) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,x) } is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
b is Element of (X,S,M,x)
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,ND,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & ND a.e.= b₁,M ) } is set
dom ND is Element of bool X
a is ext-real V38() real Element of REAL
a (#) ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(a (#) ND),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & a (#) ND a.e.= b₁,M ) } is set
E is Element of (X,S,M,x)
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a (#) E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
M . (E0 `) is ext-real set
EE is Element of (X,S,M,x)
(X,S,M,(a (#) E1),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & a (#) E1 a.e.= b₁,M ) } is set
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a (#) E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(a (#) E1),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & a (#) E1 a.e.= b₁,M ) } is set
a is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
b is ext-real V38() real Element of REAL
ND is Element of (X,S,M,x)
a . (b,ND) is Element of (X,S,M,x)
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b (#) Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(b (#) Ef),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b (#) Ef a.e.= b₁,M ) } is set
a is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
b is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
Ef is Element of (X,S,M,x)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E a.e.= b₁,M ) } is set
ND is ext-real V38() real Element of REAL
a . (ND,Ef) is Element of (X,S,M,x)
ND (#) E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(ND (#) E),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & ND (#) E a.e.= b₁,M ) } is set
b . (ND,Ef) is Element of (X,S,M,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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
(X,S,M,x) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
(X,S,M,x) is Element of (X,S,M,x)
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & X --> 0 a.e.= b₁,M ) } is set
(X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is non empty strict RLSStruct
the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is non empty set
the addF of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is V1() V4([: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):]) V5( the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)) Function-like non empty total V41([: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)) Element of bool [:[: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):]
[: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
[:[: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #), the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
0. RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is V52( RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)) Element of the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)
the ZeroF of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is Element of the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)
the Mult of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #) is V1() V4([:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):]) V5( the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)) Function-like non empty total V41([:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #)) Element of bool [:[:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):]
[:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):], the carrier of RLSStruct(# (X,S,M,x),(X,S,M,x),(X,S,M,x),(X,S,M,x) #):] is non empty set
g is strict RLSStruct
the carrier of g is set
the addF of g is V1() V4([: the carrier of g, the carrier of g:]) V5( the carrier of g) Function-like V41([: the carrier of g, the carrier of g:], the carrier of g) Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
[: the carrier of g, the carrier of g:] is set
[:[: the carrier of g, the carrier of g:], the carrier of g:] is set
bool [:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
0. g is V52(g) Element of the carrier of g
the ZeroF of g is Element of the carrier of g
the Mult of g is V1() V4([:REAL, the carrier of g:]) V5( the carrier of g) Function-like V41([:REAL, the carrier of g:], the carrier of g) Element of bool [:[:REAL, the carrier of g:], the carrier of g:]
[:REAL, the carrier of g:] is set
[:[:REAL, the carrier of g:], the carrier of g:] is set
bool [:[:REAL, the carrier of g:], the carrier of g:] is non empty set
a is strict RLSStruct
the carrier of a is set
the addF of a is V1() V4([: the carrier of a, the carrier of a:]) V5( the carrier of a) Function-like V41([: the carrier of a, the carrier of a:], the carrier of a) Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
0. a is V52(a) Element of the carrier of a
the ZeroF of a is Element of the carrier of a
the Mult of a is V1() V4([:REAL, the carrier of a:]) V5( the carrier of a) Function-like V41([:REAL, the carrier of a:], the carrier of a) Element of bool [:[:REAL, the carrier of a:], the carrier of a:]
[:REAL, the carrier of a:] is set
[:[:REAL, the carrier of a:], the carrier of a:] is set
bool [:[:REAL, the carrier of a:], the carrier of a:] is non empty 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is strict RLSStruct
the carrier of (X,S,M,x) is set
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b<sub>1</sub>,x) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,x) } is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty strict RLSStruct
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
(X,S,M,x) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
the carrier of (X,S,M,x) is non empty set
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
(X,S,M,x) is Element of (X,S,M,x)
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & X --> 0 a.e.= b₁,M ) } is set
[:REAL, the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
Ef is Element of the carrier of (X,S,M,x)
E is Element of the carrier of (X,S,M,x)
Ef + E is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Ef,E) is Element of the carrier of (X,S,M,x)
E + Ef is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (E,Ef) is Element of the carrier of (X,S,M,x)
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,EE,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & EE a.e.= b₁,M ) } is set
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E1,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E1 a.e.= b₁,M ) } is set
EE + E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(EE + E1),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & EE + E1 a.e.= b₁,M ) } is set
Ef is Element of the carrier of (X,S,M,x)
E is Element of the carrier of (X,S,M,x)
Ef + E is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Ef,E) is Element of the carrier of (X,S,M,x)
EE is Element of the carrier of (X,S,M,x)
(Ef + E) + EE is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((Ef + E),EE) is Element of the carrier of (X,S,M,x)
E + EE is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (E,EE) is Element of the carrier of (X,S,M,x)
Ef + (E + EE) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Ef,(E + EE)) is Element of the carrier of (X,S,M,x)
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E1,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E1 a.e.= b₁,M ) } is set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E0,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 a.e.= b₁,M ) } is set
E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E2,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E2 a.e.= b₁,M ) } is set
(X,S,M,x) . (Ef,E) is set
E1 + E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E1 + E0),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E1 + E0 a.e.= b₁,M ) } is set
(X,S,M,x) . (E,EE) is set
E0 + E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E0 + E2),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 + E2 a.e.= b₁,M ) } is set
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
G is Element of the carrier of (X,S,M,x)
Gp is Element of the carrier of (X,S,M,x)
G + Gp is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (G,Gp) is Element of the carrier of (X,S,M,x)
ExtGp is Element of the carrier of (X,S,M,x)
Gp + ExtGp is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Gp,ExtGp) is Element of the carrier of (X,S,M,x)
E1 + (E0 + E2) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
G + (Gp + ExtGp) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (G,(Gp + ExtGp)) is Element of the carrier of (X,S,M,x)
(G + Gp) + ExtGp is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((G + Gp),ExtGp) is Element of the carrier of (X,S,M,x)
(E1 + E0) + E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E1 + (E0 + E2)),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E1 + (E0 + E2) a.e.= b₁,M ) } is set
Ef is Element of the carrier of (X,S,M,x)
Ef + (0. (X,S,M,x)) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Ef,(0. (X,S,M,x))) is Element of the carrier of (X,S,M,x)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E a.e.= b₁,M ) } is set
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
E + (X --> 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E1 is Element of the carrier of (X,S,M,x)
E0 is Element of the carrier of (X,S,M,x)
E1 + E0 is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (E1,E0) is Element of the carrier of (X,S,M,x)
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
E1 + (0. (X,S,M,x)) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (E1,(0. (X,S,M,x))) is Element of the carrier of (X,S,M,x)
Ef is Element of the carrier of (X,S,M,x)
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E a.e.= b₁,M ) } is set
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
(- 1) (#) E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,((- 1) (#) E),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & (- 1) (#) E a.e.= b₁,M ) } is set
E0 is Element of the carrier of (X,S,M,x)
Ef + E0 is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Ef,E0) is Element of the carrier of (X,S,M,x)
EE is Element of the carrier of (X,S,M,x)
(- 1) * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . ((- 1),EE) is set
EE + ((- 1) * EE) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (EE,((- 1) * EE)) is Element of the carrier of (X,S,M,x)
E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E + ((- 1) (#) E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E + ((- 1) (#) E)),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E + ((- 1) (#) E) a.e.= b₁,M ) } is set
Ef is ext-real V38() real set
E is ext-real V38() real set
EE is Element of the carrier of (X,S,M,x)
G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,G,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & G a.e.= b₁,M ) } is set
E1 is Element of the carrier of (X,S,M,x)
Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,Gp,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & Gp a.e.= b₁,M ) } is set
(X,S,M,x) . (EE,E1) is set
G + Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(G + Gp),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & G + Gp a.e.= b₁,M ) } is set
EE + E1 is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (EE,E1) is Element of the carrier of (X,S,M,x)
(X,S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
0. (RLSp_PFunct X) is V52( RLSp_PFunct X) Element of the carrier of (RLSp_PFunct X)
the ZeroF of (RLSp_PFunct X) is Element of the carrier of (RLSp_PFunct X)
In ((0. (RLSp_PFunct X)),(X,S,M,x)) is Element of (X,S,M,x)
add| ((X,S,M,x),(RLSp_PFunct X)) is V1() V4([:(X,S,M,x),(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:(X,S,M,x),(X,S,M,x):],(X,S,M,x)) Element of bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):]
[:(X,S,M,x),(X,S,M,x):] is non empty set
[:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:(X,S,M,x),(X,S,M,x):],(X,S,M,x):] is non empty set
Mult_ (X,S,M,x) is V1() V4([:REAL,(X,S,M,x):]) V5((X,S,M,x)) Function-like non empty total V41([:REAL,(X,S,M,x):],(X,S,M,x)) Element of bool [:[:REAL,(X,S,M,x):],(X,S,M,x):]
[:REAL,(X,S,M,x):] is non empty set
[:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
bool [:[:REAL,(X,S,M,x):],(X,S,M,x):] is non empty set
[:REAL, the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like non empty total V41([:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X)) Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
[:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):] is non empty set
the Mult of (RLSp_PFunct X) | [:REAL,(X,S,M,x):] is V1() V4([:REAL, the carrier of (RLSp_PFunct X):]) V5( the carrier of (RLSp_PFunct X)) Function-like Element of bool [:[:REAL, the carrier of (RLSp_PFunct X):], the carrier of (RLSp_PFunct X):]
RLSStruct(# (X,S,M,x),(In ((0. (RLSp_PFunct X)),(X,S,M,x))),(add| ((X,S,M,x),(RLSp_PFunct X))),(Mult_ (X,S,M,x)) #) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
E2 is ext-real V38() real Element of REAL
E2 (#) G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ExtGp is Element of the carrier of (X,S,M,x)
E2 * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . (E2,ExtGp) is set
E0 is ext-real V38() real Element of REAL
E0 (#) G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,ExtGp) is set
E0 (#) Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ExtGp is Element of the carrier of (X,S,M,x)
E0 * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,ExtGp) is set
E0 + E2 is ext-real V38() real Element of REAL
(E0 + E2) (#) G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E0 + E2) * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . ((E0 + E2),ExtGp) is set
1 (#) G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
1 * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (1,ExtGp) is set
ExtGp + ExtGp is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (ExtGp,ExtGp) is Element of the carrier of (X,S,M,x)
E0 (#) (G + Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 * (ExtGp + ExtGp) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,(ExtGp + ExtGp)) is set
(E0 * ExtGp) + (E0 * ExtGp) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((E0 * ExtGp),(E0 * ExtGp)) is Element of the carrier of (X,S,M,x)
(E0 (#) G) + (E0 (#) Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E0 * ExtGp) + (E2 * ExtGp) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((E0 * ExtGp),(E2 * ExtGp)) is Element of the carrier of (X,S,M,x)
(E0 (#) G) + (E2 (#) G) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 (#) (E2 (#) G) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 * (E2 * ExtGp) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,(E2 * ExtGp)) is set
E0 * E2 is ext-real V38() real Element of REAL
(E0 * E2) * ExtGp is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . ((E0 * E2),ExtGp) is set
(E0 * E2) (#) G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,x) . (E0,EE) is set
(X,S,M,(E0 (#) G),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 (#) G a.e.= b₁,M ) } is set
(X,S,M,x) . (E0,E1) is set
(X,S,M,(E0 (#) Gp),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 (#) Gp a.e.= b₁,M ) } is set
(X,S,M,x) . (E2,EE) is set
(X,S,M,(E2 (#) G),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E2 (#) G a.e.= b₁,M ) } is set
E0 * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,EE) is set
E0 * E1 is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,E1) is set
E2 * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E2,EE) is set
E0 * (EE + E1) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E0,(EE + E1)) is set
(X,S,M,((E0 (#) G) + (E0 (#) Gp)),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & (E0 (#) G) + (E0 (#) Gp) a.e.= b₁,M ) } is set
Ef * (EE + E1) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (Ef,(EE + E1)) is set
Ef * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (Ef,EE) is set
Ef * E1 is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (Ef,E1) is set
(Ef * EE) + (Ef * E1) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((Ef * EE),(Ef * E1)) is Element of the carrier of (X,S,M,x)
(E0 + E2) * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . ((E0 + E2),EE) is set
(X,S,M,((E0 (#) G) + (E2 (#) G)),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & (E0 (#) G) + (E2 (#) G) a.e.= b₁,M ) } is set
Ef + E is ext-real V38() real set
(Ef + E) * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . ((Ef + E),EE) is set
E * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (E,EE) is set
(Ef * EE) + (E * EE) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((Ef * EE),(E * EE)) is Element of the carrier of (X,S,M,x)
Ef * E is ext-real V38() real set
(Ef * E) * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . ((Ef * E),EE) is set
(X,S,M,(E0 (#) (E2 (#) G)),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E0 (#) (E2 (#) G) a.e.= b₁,M ) } is set
Ef * (E * EE) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (Ef,(E * EE)) is set
1 * EE is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) . (1,EE) is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power a))))) is ext-real Element of ExtREAL
(abs g) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs g) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs g) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs g) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs g) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs g) to_power a))))) is ext-real Element of ExtREAL
dom x is Element of bool X
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom Ef is Element of bool X
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
dom g is Element of bool X
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom (abs x) is Element of bool X
dom (abs g) is Element of bool X
EE is Element of S
M . EE is ext-real Element of ExtREAL
EE ` is Element of bool X
X \ EE is set
x | (EE `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g | (EE `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is Element of S
E0 \/ EE is Element of S
Ef \ (E0 \/ EE) is Element of S
E1 is Element of S
E1 \/ EE is Element of S
E \ (E1 \/ EE) is Element of S
E1 \/ E0 is Element of S
(E1 \/ E0) \/ EE is Element of S
M . (E1 \/ EE) is ext-real Element of ExtREAL
M . (E0 \/ EE) is ext-real Element of ExtREAL
X \ E1 is Element of bool X
X /\ Ef is Element of bool X
X \ E0 is Element of bool X
X /\ E is Element of bool X
Ef \ E0 is Element of S
(Ef \ E0) \ EE is Element of S
E \ E1 is Element of S
(E \ E1) \ EE is Element of S
X \ (E1 \/ E0) is Element of bool X
(X \ (E1 \/ E0)) \ EE is Element of bool X
X \ ((E1 \/ E0) \/ EE) is Element of bool X
dom ((abs x) to_power a) is Element of bool X
dom ((abs g) to_power a) is Element of bool X
((abs x) to_power a) | (Ef \ (E0 \/ EE)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((abs x) to_power a) | (Ef \ (E0 \/ EE)))) is ext-real Element of ExtREAL
R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE))))) is ext-real Element of ExtREAL
max+ (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE)))))) is ext-real Element of ExtREAL
max- (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE))))))) - (integral+ (M,(max- (R_EAL (((abs x) to_power a) | (Ef \ (E0 \/ EE))))))) is ext-real Element of ExtREAL
((abs g) to_power a) | (E \ (E1 \/ EE)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((abs g) to_power a) | (E \ (E1 \/ EE)))) is ext-real Element of ExtREAL
R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE))))) is ext-real Element of ExtREAL
max+ (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE)))))) is ext-real Element of ExtREAL
max- (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE))))))) - (integral+ (M,(max- (R_EAL (((abs g) to_power a) | (E \ (E1 \/ EE))))))) is ext-real Element of ExtREAL
dom (((abs x) to_power a) | (Ef \ (E0 \/ EE))) is Element of bool X
(dom ((abs x) to_power a)) /\ (Ef \ (E0 \/ EE)) is Element of bool X
((abs g) to_power a) | (Ef \ (E0 \/ EE)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (((abs g) to_power a) | (Ef \ (E0 \/ EE))) is Element of bool X
(dom ((abs g) to_power a)) /\ (Ef \ (E0 \/ EE)) is Element of bool X
Ef /\ Ef is Element of S
(Ef /\ Ef) \ (E0 \/ EE) is Element of S
E /\ E is Element of S
(E /\ E) \ (E1 \/ EE) is Element of S
ExtGp is Element of X
((E1 \/ E0) \/ EE) ` is Element of bool X
X \ ((E1 \/ E0) \/ EE) is set
x . ExtGp is ext-real V38() real Element of REAL
(x | (EE `)) . ExtGp is ext-real V38() real Element of REAL
g . ExtGp is ext-real V38() real Element of REAL
(g | (EE `)) . ExtGp is ext-real V38() real Element of REAL
(((abs x) to_power a) | (Ef \ (E0 \/ EE))) . ExtGp is ext-real V38() real Element of REAL
((abs x) to_power a) . ExtGp is ext-real V38() real Element of REAL
(((abs g) to_power a) | (Ef \ (E0 \/ EE))) . ExtGp is ext-real V38() real Element of REAL
((abs g) to_power a) . ExtGp is ext-real V38() real Element of REAL
(abs x) . ExtGp is ext-real V38() real Element of REAL
((abs x) . ExtGp) to_power a is ext-real V38() real Element of REAL
(abs g) . ExtGp is ext-real V38() real Element of REAL
((abs g) . ExtGp) to_power a is ext-real V38() real Element of REAL
abs (x . ExtGp) is ext-real V38() real Element of REAL
(abs (x . ExtGp)) to_power a is ext-real V38() real Element of REAL
abs (g . ExtGp) is ext-real V38() real Element of REAL
(abs (g . ExtGp)) to_power a is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power g)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power g))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power g))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power g))))) is ext-real Element of ExtREAL
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (R_EAL ((abs x) to_power g)) is Element of bool X
a is Element of S
dom ((abs x) to_power g) is Element of bool X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
b is Element of the carrier of (X,S,M,a)
b is Element of the carrier of (X,S,M,a)
(X,S,M,a) is non empty Element of bool (bool (X,S,M,a))
bool (X,S,M,a) is non empty set
bool (bool (X,S,M,a)) is non empty set
{ (X,S,M,b<sub>1</sub>,a) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,a) } is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,ND,a) is Element of bool (X,S,M,a)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,a) & ND a.e.= b<sub>1</sub>,M ) } is set
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,g) is non empty set
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
a is Element of the carrier of (X,S,M,g)
(X,S,M,g) is non empty Element of bool (bool (X,S,M,g))
bool (X,S,M,g) is non empty set
bool (bool (X,S,M,g)) is non empty set
{ (X,S,M,b₁,g) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,g) } is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,b,g) is Element of bool (X,S,M,g)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,g) & b a.e.= b₁,M ) } is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,a) is non empty set
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power a))))) is ext-real Element of ExtREAL
(abs g) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs g) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs g) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs g) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs g) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs g) to_power a))))) is ext-real Element of ExtREAL
b is Element of the carrier of (X,S,M,a)
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
g is Element of the carrier of (X,S,M,x)
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,a,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & a a.e.= b₁,M ) } is set
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs a) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs a) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs a) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs a) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs a) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs a) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs a) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs a) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs a) to_power x))))) is ext-real Element of ExtREAL
b is ext-real V38() real Element of REAL
b to_power (1 / x) is ext-real V38() real Element of REAL
g is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
a is Element of the carrier of (X,S,M,x)
g . a is ext-real V38() real Element of REAL
g is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
a is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
b is Element of the carrier of (X,S,M,x)
g . b is ext-real V38() real Element of REAL
a . b is ext-real V38() real Element of REAL
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is ext-real V38() real Element of REAL
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs ND) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs ND) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs ND) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ND) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ND) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ND) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ND) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ND) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs ND) to_power x))))) is ext-real Element of ExtREAL
Ef to_power (1 / x) is ext-real V38() real Element of REAL
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is ext-real V38() real Element of REAL
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs E) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs E) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs E) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs E) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs E) to_power x))))) is ext-real Element of ExtREAL
EE to_power (1 / x) is ext-real V38() real Element of REAL
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
the carrier of (X,S,M,x) is non empty set
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
g is Element of the carrier of (X,S,M,x)
||.g.|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . g is ext-real V38() real Element of REAL
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
bool (X,S,M,x) is non empty set
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b<sub>1</sub>,x) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,x) } is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,a,x) is Element of bool (X,S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & a a.e.= b<sub>1</sub>,M ) } is set
(X,S,M,x) . g is ext-real V38() real Element of REAL
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs a) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs a) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs a) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs a) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs a) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs a) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs a) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs a) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs a) to_power x))))) is ext-real Element of ExtREAL
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs b) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs b) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs b) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs b) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs b) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs b) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs b) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs b) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs b) to_power x))))) is ext-real Element of ExtREAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / x) is ext-real V38() real Element of REAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / x) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is ext-real V38() real Element of REAL
a (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,b) is non empty NORMSTR
(X,S,M,b) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,b) is non empty set
the ZeroF of (X,S,M,b) is Element of the carrier of (X,S,M,b)
the addF of (X,S,M,b) is V1() V4([: the carrier of (X,S,M,b), the carrier of (X,S,M,b):]) V5( the carrier of (X,S,M,b)) Function-like non empty total V41([: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b)) Element of bool [:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):]
[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):] is non empty set
[:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
bool [:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
the Mult of (X,S,M,b) is V1() V4([:REAL, the carrier of (X,S,M,b):]) V5( the carrier of (X,S,M,b)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b)) Element of bool [:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):]
[:REAL, the carrier of (X,S,M,b):] is non empty set
[:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
(X,S,M,b) is V1() V4( the carrier of (X,S,M,b)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,b), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,b),REAL:]
[: the carrier of (X,S,M,b),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,b),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,b), the ZeroF of (X,S,M,b), the addF of (X,S,M,b), the Mult of (X,S,M,b),(X,S,M,b) #) is strict NORMSTR
the carrier of (X,S,M,b) is non empty set
ND is Element of the carrier of (X,S,M,b)
a * ND is Element of the carrier of (X,S,M,b)
the Mult of (X,S,M,b) is V1() V4([:REAL, the carrier of (X,S,M,b):]) V5( the carrier of (X,S,M,b)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b)) Element of bool [:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):]
[:REAL, the carrier of (X,S,M,b):] is non empty set
[:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
the Mult of (X,S,M,b) . (a,ND) is set
Ef is Element of the carrier of (X,S,M,b)
ND + Ef is Element of the carrier of (X,S,M,b)
the addF of (X,S,M,b) is V1() V4([: the carrier of (X,S,M,b), the carrier of (X,S,M,b):]) V5( the carrier of (X,S,M,b)) Function-like non empty total V41([: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b)) Element of bool [:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):]
[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):] is non empty set
[:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
bool [:[: the carrier of (X,S,M,b), the carrier of (X,S,M,b):], the carrier of (X,S,M,b):] is non empty set
the addF of (X,S,M,b) . (ND,Ef) is Element of the carrier of (X,S,M,b)
(X,S,M,b) is non empty Element of bool (bool (X,S,M,b))
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power b is_integrable_on M ) } is set
bool (X,S,M,b) is non empty set
bool (bool (X,S,M,b)) is non empty set
{ (X,S,M,b₁,b) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,b) } is set
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,EE,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & EE a.e.= b₁,M ) } is set
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,E1,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & E1 a.e.= b₁,M ) } is set
(X,S,M,b) is V1() V4([:(X,S,M,b),(X,S,M,b):]) V5((X,S,M,b)) Function-like non empty total V41([:(X,S,M,b),(X,S,M,b):],(X,S,M,b)) Element of bool [:[:(X,S,M,b),(X,S,M,b):],(X,S,M,b):]
[:(X,S,M,b),(X,S,M,b):] is non empty set
[:[:(X,S,M,b),(X,S,M,b):],(X,S,M,b):] is non empty set
bool [:[:(X,S,M,b),(X,S,M,b):],(X,S,M,b):] is non empty set
(X,S,M,b) . (ND,Ef) is set
EE + E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(EE + E1),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & EE + E1 a.e.= b₁,M ) } is set
(X,S,M,x,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & x a.e.= b₁,M ) } is set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,g,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & g a.e.= b₁,M ) } is set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(x + g),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & x + g a.e.= b₁,M ) } is set
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,EE,b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & EE a.e.= b₁,M ) } is set
(X,S,M,b) is V1() V4([:REAL,(X,S,M,b):]) V5((X,S,M,b)) Function-like non empty total V41([:REAL,(X,S,M,b):],(X,S,M,b)) Element of bool [:[:REAL,(X,S,M,b):],(X,S,M,b):]
[:REAL,(X,S,M,b):] is non empty set
[:[:REAL,(X,S,M,b):],(X,S,M,b):] is non empty set
bool [:[:REAL,(X,S,M,b):],(X,S,M,b):] is non empty set
(X,S,M,b) . (a,ND) is set
a (#) EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(a (#) EE),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & a (#) EE a.e.= b₁,M ) } is set
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(a (#) x),b) is Element of bool (X,S,M,b)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,b) & a (#) x a.e.= b₁,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty NORMSTR
(X,S,M,g) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,g) is non empty set
the ZeroF of (X,S,M,g) is Element of the carrier of (X,S,M,g)
the addF of (X,S,M,g) is V1() V4([: the carrier of (X,S,M,g), the carrier of (X,S,M,g):]) V5( the carrier of (X,S,M,g)) Function-like non empty total V41([: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g)) Element of bool [:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):]
[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):] is non empty set
[:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
bool [:[: the carrier of (X,S,M,g), the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
the Mult of (X,S,M,g) is V1() V4([:REAL, the carrier of (X,S,M,g):]) V5( the carrier of (X,S,M,g)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g)) Element of bool [:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):]
[:REAL, the carrier of (X,S,M,g):] is non empty set
[:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,g):], the carrier of (X,S,M,g):] is non empty set
(X,S,M,g) is V1() V4( the carrier of (X,S,M,g)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,g), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,g),REAL:]
[: the carrier of (X,S,M,g),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,g),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,g), the ZeroF of (X,S,M,g), the addF of (X,S,M,g), the Mult of (X,S,M,g),(X,S,M,g) #) is strict NORMSTR
the carrier of (X,S,M,g) is non empty set
(X,S,M,x,g) is Element of bool (X,S,M,g)
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power g is_integrable_on M ) } is set
bool (X,S,M,g) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,g) & x a.e.= b<sub>1</sub>,M ) } is set
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power g)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power g))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power g))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power g))))) is ext-real Element of ExtREAL
1 / g is non empty ext-real positive non negative V38() real Element of REAL
g " is non empty ext-real positive non negative V38() real set
1 * (g ") is non empty ext-real positive non negative V38() real set
a is Element of the carrier of (X,S,M,g)
||.a.|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,g) is V1() V4( the carrier of (X,S,M,g)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,g), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,g),REAL:]
[: the carrier of (X,S,M,g),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,g),REAL:] is non empty set
the U9 of (X,S,M,g) . a is ext-real V38() real Element of REAL
(X,S,M,g) is non empty Element of bool (bool (X,S,M,g))
bool (bool (X,S,M,g)) is non empty set
{ (X,S,M,b<sub>1</sub>,g) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,g) } is set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,b,g) is Element of bool (X,S,M,g)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,g) & b a.e.= b<sub>1</sub>,M ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
x is Element of S
M . x is ext-real Element of ExtREAL
dom (X --> 0) is Element of bool X
x ` is Element of bool X
X \ x is set
g is Element of X
(X --> 0) . g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power g)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power g))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power g)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power g))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power g))))) is ext-real Element of ExtREAL
dom x is Element of bool X
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom b is Element of bool X
dom (abs x) is Element of bool X
ND is Element of S
ND ` is Element of bool X
X \ ND is set
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
M . Ef is ext-real Element of ExtREAL
Ef ` is Element of bool X
X \ Ef is set
((abs x) to_power g) | (Ef `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X --> 0) | (Ef `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
ND \/ Ef is Element of S
M . (ND \/ Ef) is ext-real Element of ExtREAL
(ND \/ Ef) ` is Element of bool X
X \ (ND \/ Ef) is set
dom (X --> 0) is Element of bool X
(X --> 0) | ((ND \/ Ef) `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
dom ((X --> 0) | ((ND \/ Ef) `)) is Element of bool X
x | ((ND \/ Ef) `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (x | ((ND \/ Ef) `)) is Element of bool X
EE is set
(x | ((ND \/ Ef) `)) . EE is ext-real V38() real Element of REAL
((X --> 0) | ((ND \/ Ef) `)) . EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x . EE is ext-real V38() real Element of REAL
abs (x . EE) is ext-real V38() real Element of REAL
(abs (x . EE)) to_power g is ext-real V38() real Element of REAL
(abs x) . EE is ext-real V38() real Element of REAL
((abs x) . EE) to_power g is ext-real V38() real Element of REAL
((abs x) to_power g) . EE is ext-real V38() real Element of REAL
((X --> 0) | (Ef `)) . EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(X --> 0) . EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
abs (X --> 0) is V1() V4(X) V5( REAL ) V5( RAT ) Function-like V163() V164() V165() V166() nonnegative Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(abs (X --> 0)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (X --> 0)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (X --> 0)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs (X --> 0)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (X --> 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (X --> 0)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (X --> 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (X --> 0)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (X --> 0)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (X --> 0)) to_power x))))) is ext-real Element of ExtREAL
dom (X --> 0) is Element of bool X
g is set
(X --> 0) . g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(X --> 0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((X --> 0) to_power x)) is ext-real Element of ExtREAL
R_EAL ((X --> 0) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((X --> 0) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((X --> 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((X --> 0) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((X --> 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((X --> 0) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((X --> 0) to_power x))))) - (integral+ (M,(max- (R_EAL ((X --> 0) to_power x))))) is ext-real Element of ExtREAL
Integral (M,(X --> 0)) is ext-real Element of ExtREAL
R_EAL (X --> 0) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (X --> 0))) is ext-real Element of ExtREAL
max+ (R_EAL (X --> 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (X --> 0)))) is ext-real Element of ExtREAL
max- (R_EAL (X --> 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (X --> 0)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (X --> 0))))) - (integral+ (M,(max- (R_EAL (X --> 0))))) is ext-real Element of ExtREAL
Integral (M,(abs (X --> 0))) is ext-real Element of ExtREAL
R_EAL (abs (X --> 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (X --> 0)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (X --> 0)))))) - (integral+ (M,(max- (R_EAL (abs (X --> 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,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
x (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
1 / a is non empty ext-real positive non negative V38() real Element of REAL
a " is non empty ext-real positive non negative V38() real set
1 * (a ") is non empty ext-real positive non negative V38() real set
b is non empty ext-real positive non negative V38() real Element of REAL
1 / b is non empty ext-real positive non negative V38() real Element of REAL
b " is non empty ext-real positive non negative V38() real set
1 * (b ") is non empty ext-real positive non negative V38() real set
(1 / a) + (1 / b) is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power b is_integrable_on M ) } is set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
ND (#) E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(1 / a) (#) ((abs ND) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(1 / b) (#) ((abs E) to_power b) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((1 / a) (#) ((abs ND) to_power a)) + ((1 / b) (#) ((abs E) to_power b)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (abs ND) is Element of bool X
dom (abs E) is Element of bool X
dom ((abs ND) to_power a) is Element of bool X
dom ((abs E) to_power b) is Element of bool X
dom (ND (#) E) is Element of bool X
(dom ((abs ND) to_power a)) /\ (dom ((abs E) to_power b)) is Element of bool X
Ef /\ EE is Element of S
I is Element of S
I ` is Element of bool X
X \ I is set
Ir is Element of S
Ir ` is Element of bool X
X \ Ir is set
KPAbsFMF is Element of S
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
M . (KPAbsFMF `) is ext-real set
PAbsFMF is Element of S
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
M . (PAbsFMF `) is ext-real set
dom ((1 / a) (#) ((abs ND) to_power a)) is Element of bool X
dom ((1 / b) (#) ((abs E) to_power b)) is Element of bool X
dom (((1 / a) (#) ((abs ND) to_power a)) + ((1 / b) (#) ((abs E) to_power b))) is Element of bool X
(Ef /\ EE) ` is Element of bool X
X \ (Ef /\ EE) is set
(Ef `) \/ (EE `) is Element of bool X
I \/ Ir is Element of S
M . ((Ef /\ EE) `) is ext-real set
KPAbsFMF is Element of X
(ND (#) E) . KPAbsFMF is ext-real V38() real Element of REAL
abs ((ND (#) E) . KPAbsFMF) is ext-real V38() real Element of REAL
(((1 / a) (#) ((abs ND) to_power a)) + ((1 / b) (#) ((abs E) to_power b))) . KPAbsFMF is ext-real V38() real Element of REAL
abs (ND (#) E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) (#) (abs E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (ND (#) E)) . KPAbsFMF is ext-real V38() real Element of REAL
(abs ND) . KPAbsFMF is ext-real V38() real Element of REAL
(abs E) . KPAbsFMF is ext-real V38() real Element of REAL
((abs ND) . KPAbsFMF) * ((abs E) . KPAbsFMF) is ext-real V38() real Element of REAL
((abs ND) . KPAbsFMF) to_power a is ext-real V38() real Element of REAL
(((abs ND) . KPAbsFMF) to_power a) / a is ext-real V38() real Element of REAL
(((abs ND) . KPAbsFMF) to_power a) * (a ") is ext-real V38() real set
((abs ND) to_power a) . KPAbsFMF is ext-real V38() real Element of REAL
(1 / a) * (((abs ND) to_power a) . KPAbsFMF) is ext-real V38() real Element of REAL
((abs E) . KPAbsFMF) to_power b is ext-real V38() real Element of REAL
(((abs E) . KPAbsFMF) to_power b) / b is ext-real V38() real Element of REAL
(((abs E) . KPAbsFMF) to_power b) * (b ") is ext-real V38() real set
((abs E) to_power b) . KPAbsFMF is ext-real V38() real Element of REAL
(1 / b) * (((abs E) to_power b) . KPAbsFMF) is ext-real V38() real Element of REAL
((1 / a) (#) ((abs ND) to_power a)) . KPAbsFMF is ext-real V38() real Element of REAL
((1 / b) (#) ((abs E) to_power b)) . KPAbsFMF is ext-real V38() real Element of REAL
(((1 / a) (#) ((abs ND) to_power a)) . KPAbsFMF) + (((1 / b) (#) ((abs E) to_power b)) . KPAbsFMF) is ext-real V38() real Element of REAL
KPAbsFMF is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
PAbsFMF is Element of S
M . PAbsFMF is ext-real Element of ExtREAL
dom KPAbsFMF is Element of bool X
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
PAbsFMF is Element of S
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
RF0 is Element of S
RF0 ` is Element of bool X
X \ RF0 is set
M . (RF0 `) is ext-real set
QAbsFMF is Element of S
QAbsFMF ` is Element of bool X
X \ QAbsFMF is set
M . (QAbsFMF `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
x (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (x (#) g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (x (#) g))) is ext-real Element of ExtREAL
R_EAL (abs (x (#) g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs (x (#) g)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (x (#) g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (x (#) g))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (x (#) g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (x (#) g))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (x (#) g)))))) - (integral+ (M,(max- (R_EAL (abs (x (#) g)))))) is ext-real Element of ExtREAL
a is non empty ext-real positive non negative V38() real Element of REAL
1 / a is non empty ext-real positive non negative V38() real Element of REAL
a " is non empty ext-real positive non negative V38() real set
1 * (a ") is non empty ext-real positive non negative V38() real set
b is non empty ext-real positive non negative V38() real Element of REAL
1 / b is non empty ext-real positive non negative V38() real Element of REAL
b " is non empty ext-real positive non negative V38() real set
1 * (b ") is non empty ext-real positive non negative V38() real set
(1 / a) + (1 / b) is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power b is_integrable_on M ) } is set
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs x) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power a))))) is ext-real Element of ExtREAL
(abs g) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs g) to_power b)) is ext-real Element of ExtREAL
R_EAL ((abs g) to_power b) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs g) to_power b))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs g) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power b)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs g) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs g) to_power b)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs g) to_power b))))) - (integral+ (M,(max- (R_EAL ((abs g) to_power b))))) is ext-real Element of ExtREAL
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
Integral (M,((abs ND) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs ND) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs ND) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ND) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ND) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ND) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ND) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ND) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs ND) to_power a))))) is ext-real Element of ExtREAL
Integral (M,((abs E) to_power b)) is ext-real Element of ExtREAL
R_EAL ((abs E) to_power b) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs E) to_power b))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs E) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs E) to_power b)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs E) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs E) to_power b)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs E) to_power b))))) - (integral+ (M,(max- (R_EAL ((abs E) to_power b))))) is ext-real Element of ExtREAL
E2 is Element of S
E2 ` is Element of bool X
X \ E2 is set
M . (E2 `) is ext-real set
G is Element of S
G ` is Element of bool X
X \ G is set
M . (G `) is ext-real set
Gp is Element of S
Gp ` is Element of bool X
X \ Gp is set
M . (Gp `) is ext-real set
ExtGp is Element of S
ExtGp ` is Element of bool X
X \ ExtGp is set
M . (ExtGp `) is ext-real set
dom (abs ND) is Element of bool X
dom (abs E) is Element of bool X
E2 is ext-real V38() real Element of REAL
E2 to_power (1 / a) is ext-real V38() real Element of REAL
G is ext-real V38() real Element of REAL
G to_power (1 / b) is ext-real V38() real Element of REAL
Ef /\ EE is Element of S
(Ef /\ EE) ` is Element of bool X
X \ (Ef /\ EE) is set
(Ef `) \/ (EE `) is Element of bool X
Gp is Element of S
ExtGp is Element of S
ND (#) E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (ND (#) E) is Element of bool X
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
KPAbsFMF is Element of S
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
M . (KPAbsFMF `) is ext-real set
PAbsFMF is Element of S
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
M . (PAbsFMF `) is ext-real set
abs (ND (#) E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (ND (#) E))) is ext-real Element of ExtREAL
R_EAL (abs (ND (#) E)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (ND (#) E)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (ND (#) E))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (ND (#) E))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (ND (#) E))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (ND (#) E))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (ND (#) E)))))) - (integral+ (M,(max- (R_EAL (abs (ND (#) E)))))) is ext-real Element of ExtREAL
KPAbsFMF is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
PAbsFMF is Element of S
M . PAbsFMF is ext-real Element of ExtREAL
dom KPAbsFMF is Element of bool X
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
KPAbsFMF is Element of S
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
M . (KPAbsFMF `) is ext-real set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
KPAbsFMF is Element of S
M . KPAbsFMF is ext-real Element of ExtREAL
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
ND | (KPAbsFMF `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X --> 0) | (KPAbsFMF `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(Ef /\ EE) \ KPAbsFMF is Element of S
((Ef /\ EE) \ KPAbsFMF) ` is Element of bool X
X \ ((Ef /\ EE) \ KPAbsFMF) is set
((Ef /\ EE) `) \/ KPAbsFMF is Element of bool X
PAbsFMF is Element of S
M . PAbsFMF is ext-real Element of ExtREAL
dom (X --> 0) is Element of bool X
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
(X --> 0) | (PAbsFMF `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
dom ((X --> 0) | (PAbsFMF `)) is Element of bool X
(ND (#) E) | (PAbsFMF `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((ND (#) E) | (PAbsFMF `)) is Element of bool X
RF0 is set
((ND (#) E) | (PAbsFMF `)) . RF0 is ext-real V38() real Element of REAL
((X --> 0) | (PAbsFMF `)) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND . RF0 is ext-real V38() real Element of REAL
(ND | (KPAbsFMF `)) . RF0 is ext-real V38() real Element of REAL
(X --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
((X --> 0) | (KPAbsFMF `)) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(ND (#) E) . RF0 is ext-real V38() real Element of REAL
E . RF0 is ext-real V38() real Element of REAL
(ND . RF0) * (E . RF0) is ext-real V38() real Element of REAL
(PAbsFMF `) --> 0 is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of PFuncs (X,NAT)
PFuncs (X,NAT) is functional non empty PartFunc-set of X, NAT
((PAbsFMF `) --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X /\ (PAbsFMF `) is Element of bool X
(X /\ (PAbsFMF `)) --> 0 is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of PFuncs (X,NAT)
((X /\ (PAbsFMF `)) --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
abs (X --> 0) is V1() V4(X) V5( REAL ) V5( RAT ) Function-like V163() V164() V165() V166() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (X --> 0))) is ext-real Element of ExtREAL
R_EAL (abs (X --> 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (X --> 0)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (X --> 0)))))) - (integral+ (M,(max- (R_EAL (abs (X --> 0)))))) is ext-real Element of ExtREAL
(E2 to_power (1 / a)) * (G to_power (1 / b)) is ext-real V38() real Element of REAL
0 * (G to_power (1 / b)) is ext-real V38() real Element of REAL
RF0 is Element of S
RF0 ` is Element of bool X
X \ RF0 is set
M . (RF0 `) is ext-real set
QAbsFMF is Element of S
QAbsFMF ` is Element of bool X
X \ QAbsFMF is set
M . (QAbsFMF `) is ext-real set
KPAbsFMF is Element of S
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
M . (KPAbsFMF `) is ext-real set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
KPAbsFMF is Element of S
M . KPAbsFMF is ext-real Element of ExtREAL
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
E | (KPAbsFMF `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X --> 0) | (KPAbsFMF `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(Ef /\ EE) \ KPAbsFMF is Element of S
((Ef /\ EE) \ KPAbsFMF) ` is Element of bool X
X \ ((Ef /\ EE) \ KPAbsFMF) is set
((Ef /\ EE) `) \/ KPAbsFMF is Element of bool X
PAbsFMF is Element of S
M . PAbsFMF is ext-real Element of ExtREAL
dom (X --> 0) is Element of bool X
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
(X --> 0) | (PAbsFMF `) is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
dom ((X --> 0) | (PAbsFMF `)) is Element of bool X
(ND (#) E) | (PAbsFMF `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((ND (#) E) | (PAbsFMF `)) is Element of bool X
RF0 is set
((ND (#) E) | (PAbsFMF `)) . RF0 is ext-real V38() real Element of REAL
((X --> 0) | (PAbsFMF `)) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E . RF0 is ext-real V38() real Element of REAL
(E | (KPAbsFMF `)) . RF0 is ext-real V38() real Element of REAL
(X --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
((X --> 0) | (KPAbsFMF `)) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(ND (#) E) . RF0 is ext-real V38() real Element of REAL
ND . RF0 is ext-real V38() real Element of REAL
(ND . RF0) * (E . RF0) is ext-real V38() real Element of REAL
(PAbsFMF `) --> 0 is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of PFuncs (X,NAT)
PFuncs (X,NAT) is functional non empty PartFunc-set of X, NAT
((PAbsFMF `) --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X /\ (PAbsFMF `) is Element of bool X
(X /\ (PAbsFMF `)) --> 0 is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of PFuncs (X,NAT)
((X /\ (PAbsFMF `)) --> 0) . RF0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
abs (X --> 0) is V1() V4(X) V5( REAL ) V5( RAT ) Function-like V163() V164() V165() V166() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (X --> 0))) is ext-real Element of ExtREAL
R_EAL (abs (X --> 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (X --> 0)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (X --> 0))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (X --> 0))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (X --> 0)))))) - (integral+ (M,(max- (R_EAL (abs (X --> 0)))))) is ext-real Element of ExtREAL
(E2 to_power (1 / a)) * (G to_power (1 / b)) is ext-real V38() real Element of REAL
(E2 to_power (1 / a)) * 0 is ext-real V38() real Element of REAL
RF0 is Element of S
RF0 ` is Element of bool X
X \ RF0 is set
M . (RF0 `) is ext-real set
QAbsFMF is Element of S
QAbsFMF ` is Element of bool X
X \ QAbsFMF is set
M . (QAbsFMF `) is ext-real set
(E2 to_power (1 / a)) * (G to_power (1 / b)) is ext-real V38() real Element of REAL
1 / ((E2 to_power (1 / a)) * (G to_power (1 / b))) is ext-real V38() real Element of REAL
((E2 to_power (1 / a)) * (G to_power (1 / b))) " is ext-real V38() real set
1 * (((E2 to_power (1 / a)) * (G to_power (1 / b))) ") is ext-real V38() real set
abs (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) is ext-real V38() real Element of REAL
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
1 / (E2 to_power (1 / a)) is ext-real V38() real Element of REAL
(E2 to_power (1 / a)) " is ext-real V38() real set
1 * ((E2 to_power (1 / a)) ") is ext-real V38() real set
(1 / (E2 to_power (1 / a))) (#) (abs ND) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
1 / (G to_power (1 / b)) is ext-real V38() real Element of REAL
(G to_power (1 / b)) " is ext-real V38() real set
1 * ((G to_power (1 / b)) ") is ext-real V38() real set
(1 / (G to_power (1 / b))) (#) (abs E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((1 / (G to_power (1 / b))) (#) (abs E)) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((1 / (E2 to_power (1 / a))) (#) (abs ND)) is Element of bool X
dom ((1 / (G to_power (1 / b))) (#) (abs E)) is Element of bool X
dom ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) is Element of bool X
dom (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a) is Element of bool X
dom ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is Element of bool X
dom (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b) is Element of bool X
dom (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is Element of bool X
(dom (abs ND)) /\ (dom (abs E)) is Element of bool X
(1 / (E2 to_power (1 / a))) to_power a is ext-real V38() real Element of REAL
((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(1 / (G to_power (1 / b))) to_power b is ext-real V38() real Element of REAL
((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Br is Element of S
Br ` is Element of bool X
X \ Br is set
M . (Br `) is ext-real set
LExtGp is Element of S
LExtGp ` is Element of bool X
X \ LExtGp is set
M . (LExtGp `) is ext-real set
dom ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)) is Element of bool X
(dom ND) /\ (dom E) is Element of bool X
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (abs (ND (#) E)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (abs (ND (#) E))) is Element of bool X
dom (abs (ND (#) E)) is Element of bool X
Br is Element of X
((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)) . Br is ext-real V38() real Element of REAL
abs (((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)) . Br) is ext-real V38() real Element of REAL
(((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) . Br is ext-real V38() real Element of REAL
(abs ND) . Br is ext-real V38() real Element of REAL
(abs E) . Br is ext-real V38() real Element of REAL
(1 / (E2 to_power (1 / a))) * ((abs ND) . Br) is ext-real V38() real Element of REAL
(1 / (G to_power (1 / b))) * ((abs E) . Br) is ext-real V38() real Element of REAL
((1 / (E2 to_power (1 / a))) * ((abs ND) . Br)) * ((1 / (G to_power (1 / b))) * ((abs E) . Br)) is ext-real V38() real Element of REAL
((1 / (E2 to_power (1 / a))) * ((abs ND) . Br)) to_power a is ext-real V38() real Element of REAL
(((1 / (E2 to_power (1 / a))) * ((abs ND) . Br)) to_power a) / a is ext-real V38() real Element of REAL
(((1 / (E2 to_power (1 / a))) * ((abs ND) . Br)) to_power a) * (a ") is ext-real V38() real set
((1 / (G to_power (1 / b))) * ((abs E) . Br)) to_power b is ext-real V38() real Element of REAL
(((1 / (G to_power (1 / b))) * ((abs E) . Br)) to_power b) / b is ext-real V38() real Element of REAL
(((1 / (G to_power (1 / b))) * ((abs E) . Br)) to_power b) * (b ") is ext-real V38() real set
((((1 / (E2 to_power (1 / a))) * ((abs ND) . Br)) to_power a) / a) + ((((1 / (G to_power (1 / b))) * ((abs E) . Br)) to_power b) / b) is ext-real V38() real Element of REAL
(abs ND) (#) (abs E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((abs ND) (#) (abs E)) is Element of bool X
((abs ND) (#) (abs E)) . Br is ext-real V38() real Element of REAL
((abs ND) . Br) * ((abs E) . Br) is ext-real V38() real Element of REAL
(1 / (E2 to_power (1 / a))) * (1 / (G to_power (1 / b))) is ext-real V38() real Element of REAL
((1 / (E2 to_power (1 / a))) * (1 / (G to_power (1 / b)))) * ((abs ND) . Br) is ext-real V38() real Element of REAL
(((1 / (E2 to_power (1 / a))) * (1 / (G to_power (1 / b)))) * ((abs ND) . Br)) * ((abs E) . Br) is ext-real V38() real Element of REAL
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * ((abs ND) . Br) is ext-real V38() real Element of REAL
((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * ((abs ND) . Br)) * ((abs E) . Br) is ext-real V38() real Element of REAL
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * (((abs ND) (#) (abs E)) . Br) is ext-real V38() real Element of REAL
(abs (ND (#) E)) . Br is ext-real V38() real Element of REAL
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * ((abs (ND (#) E)) . Br) is ext-real V38() real Element of REAL
((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (abs (ND (#) E))) . Br is ext-real V38() real Element of REAL
abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E))) . Br is ext-real V38() real Element of REAL
((1 / (E2 to_power (1 / a))) (#) (abs ND)) . Br is ext-real V38() real Element of REAL
((1 / (G to_power (1 / b))) (#) (abs E)) . Br is ext-real V38() real Element of REAL
(((1 / (E2 to_power (1 / a))) (#) (abs ND)) . Br) to_power a is ext-real V38() real Element of REAL
(((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a) . Br is ext-real V38() real Element of REAL
(((1 / (G to_power (1 / b))) (#) (abs E)) . Br) to_power b is ext-real V38() real Element of REAL
(((1 / (G to_power (1 / b))) (#) (abs E)) to_power b) . Br is ext-real V38() real Element of REAL
((((1 / (E2 to_power (1 / a))) (#) (abs ND)) . Br) to_power a) / a is ext-real V38() real Element of REAL
((((1 / (E2 to_power (1 / a))) (#) (abs ND)) . Br) to_power a) * (a ") is ext-real V38() real set
((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) . Br is ext-real V38() real Element of REAL
((((1 / (G to_power (1 / b))) (#) (abs E)) . Br) to_power b) / b is ext-real V38() real Element of REAL
((((1 / (G to_power (1 / b))) (#) (abs E)) . Br) to_power b) * (b ") is ext-real V38() real set
((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) . Br is ext-real V38() real Element of REAL
abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)))) is ext-real Element of ExtREAL
R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E))))) is ext-real Element of ExtREAL
max+ (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)))))) is ext-real Element of ExtREAL
max- (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E))))))) - (integral+ (M,(max- (R_EAL (abs ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) (#) (ND (#) E))))))) is ext-real Element of ExtREAL
Integral (M,(((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is ext-real Element of ExtREAL
R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))))) - (integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) + ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))))) is ext-real Element of ExtREAL
(dom ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) /\ (dom ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is Element of bool X
X /\ Ef is Element of bool X
X /\ EE is Element of bool X
X \ Gp is Element of bool X
X \ ExtGp is Element of bool X
Ef \ (Ef /\ EE) is Element of S
Ef \ EE is Element of S
(X \ Gp) \ X is Element of bool X
(X \ Gp) /\ ExtGp is Element of bool X
((X \ Gp) \ X) \/ ((X \ Gp) /\ ExtGp) is Element of bool X
Gp \/ X is set
X \ (Gp \/ X) is Element of bool X
(X \ (Gp \/ X)) \/ ((X \ Gp) /\ ExtGp) is Element of bool X
X \ X is Element of bool X
(X \ X) \/ ((X \ Gp) /\ ExtGp) is Element of bool X
{} \/ ((X \ Gp) /\ ExtGp) is set
EE \ (Ef /\ EE) is Element of S
EE \ Ef is Element of S
(X \ ExtGp) \ X is Element of bool X
(X \ ExtGp) /\ Gp is Element of bool X
((X \ ExtGp) \ X) \/ ((X \ ExtGp) /\ Gp) is Element of bool X
ExtGp \/ X is set
X \ (ExtGp \/ X) is Element of bool X
(X \ (ExtGp \/ X)) \/ ((X \ ExtGp) /\ Gp) is Element of bool X
(X \ X) \/ ((X \ ExtGp) /\ Gp) is Element of bool X
{} \/ ((X \ ExtGp) /\ Gp) is set
Ef /\ ExtGp is Element of S
EE /\ Gp is Element of S
M . (Ef /\ ExtGp) is ext-real Element of ExtREAL
M . (EE /\ Gp) is ext-real Element of ExtREAL
Ef /\ (Ef /\ EE) is Element of S
EE /\ (Ef /\ EE) is Element of S
Ef \ (Ef /\ ExtGp) is Element of S
EE \ (EE /\ Gp) is Element of S
R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is Element of bool X
EE is Element of S
R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is Element of bool X
EE is Element of S
- a is non empty ext-real non positive negative V38() real Element of REAL
(E2 to_power (1 / a)) to_power (- a) is ext-real V38() real Element of REAL
(1 / a) * (- a) is non empty ext-real non positive negative V38() real Element of REAL
E2 to_power ((1 / a) * (- a)) is ext-real V38() real Element of REAL
1 * (1 / a) is non empty ext-real positive non negative V38() real Element of REAL
(1 * (1 / a)) * a is non empty ext-real positive non negative V38() real Element of REAL
- ((1 * (1 / a)) * a) is non empty ext-real non positive negative V38() real Element of REAL
E2 to_power (- ((1 * (1 / a)) * a)) is ext-real V38() real Element of REAL
E2 to_power (- 1) is ext-real V38() real Element of REAL
1 / E2 is ext-real V38() real Element of REAL
E2 " is ext-real V38() real set
1 * (E2 ") is ext-real V38() real set
(1 / E2) to_power 1 is ext-real V38() real Element of REAL
R_EAL (1 / E2) is ext-real Element of ExtREAL
R_EAL E2 is ext-real Element of ExtREAL
(R_EAL (1 / E2)) * (R_EAL E2) is ext-real Element of ExtREAL
(1 / E2) * E2 is ext-real V38() real Element of REAL
1 / G is ext-real V38() real Element of REAL
G " is ext-real V38() real set
1 * (G ") is ext-real V38() real set
R_EAL (1 / G) is ext-real Element of ExtREAL
R_EAL G is ext-real Element of ExtREAL
(R_EAL (1 / G)) * (R_EAL G) is ext-real Element of ExtREAL
(1 / G) * G is ext-real V38() real Element of REAL
- b is non empty ext-real non positive negative V38() real Element of REAL
(G to_power (1 / b)) to_power (- b) is ext-real V38() real Element of REAL
(1 / b) * (- b) is non empty ext-real non positive negative V38() real Element of REAL
G to_power ((1 / b) * (- b)) is ext-real V38() real Element of REAL
1 * (1 / b) is non empty ext-real positive non negative V38() real Element of REAL
(1 * (1 / b)) * b is non empty ext-real positive non negative V38() real Element of REAL
- ((1 * (1 / b)) * b) is non empty ext-real non positive negative V38() real Element of REAL
G to_power (- ((1 * (1 / b)) * b)) is ext-real V38() real Element of REAL
G to_power (- 1) is ext-real V38() real Element of REAL
(1 / G) to_power 1 is ext-real V38() real Element of REAL
((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE))) is ext-real Element of ExtREAL
R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE)))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE))))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE)))))) - (integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | (Ef /\ EE)))))) is ext-real Element of ExtREAL
Integral (M,((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is ext-real Element of ExtREAL
Integral (M,(R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)))) is ext-real Element of ExtREAL
max+ (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))))) is ext-real Element of ExtREAL
max- (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)))))) - (integral+ (M,(max- (R_EAL ((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)))))) is ext-real Element of ExtREAL
R_EAL (1 / a) is ext-real Element of ExtREAL
Integral (M,(((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) is ext-real Element of ExtREAL
R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))))) - (integral+ (M,(max- (R_EAL (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))))) is ext-real Element of ExtREAL
(R_EAL (1 / a)) * (Integral (M,(((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a))) is ext-real Element of ExtREAL
Integral (M,(((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a))) is ext-real Element of ExtREAL
R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a)))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a))))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a)))))) - (integral+ (M,(max- (R_EAL (((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a)))))) is ext-real Element of ExtREAL
(R_EAL (1 / a)) * (Integral (M,(((1 / (E2 to_power (1 / a))) to_power a) (#) ((abs ND) to_power a)))) is ext-real Element of ExtREAL
R_EAL ((1 / (E2 to_power (1 / a))) to_power a) is ext-real Element of ExtREAL
(R_EAL ((1 / (E2 to_power (1 / a))) to_power a)) * (Integral (M,((abs ND) to_power a))) is ext-real Element of ExtREAL
(R_EAL (1 / a)) * ((R_EAL ((1 / (E2 to_power (1 / a))) to_power a)) * (Integral (M,((abs ND) to_power a)))) is ext-real Element of ExtREAL
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE))) is ext-real Element of ExtREAL
R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE)))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE))))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE)))))) - (integral+ (M,(max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | (Ef /\ EE)))))) is ext-real Element of ExtREAL
Integral (M,((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is ext-real Element of ExtREAL
Integral (M,(R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is ext-real Element of ExtREAL
max+ (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))) is ext-real Element of ExtREAL
max- (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))))) - (integral+ (M,(max- (R_EAL ((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))))) is ext-real Element of ExtREAL
R_EAL (1 / b) is ext-real Element of ExtREAL
Integral (M,(((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is ext-real Element of ExtREAL
R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))) - (integral+ (M,(max- (R_EAL (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))))) is ext-real Element of ExtREAL
(R_EAL (1 / b)) * (Integral (M,(((1 / (G to_power (1 / b))) (#) (abs E)) to_power b))) is ext-real Element of ExtREAL
Integral (M,(((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b))) is ext-real Element of ExtREAL
R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b)))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b))))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b)))))) - (integral+ (M,(max- (R_EAL (((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b)))))) is ext-real Element of ExtREAL
(R_EAL (1 / b)) * (Integral (M,(((1 / (G to_power (1 / b))) to_power b) (#) ((abs E) to_power b)))) is ext-real Element of ExtREAL
R_EAL ((1 / (G to_power (1 / b))) to_power b) is ext-real Element of ExtREAL
(R_EAL ((1 / (G to_power (1 / b))) to_power b)) * (Integral (M,((abs E) to_power b))) is ext-real Element of ExtREAL
(R_EAL (1 / b)) * ((R_EAL ((1 / (G to_power (1 / b))) to_power b)) * (Integral (M,((abs E) to_power b)))) is ext-real Element of ExtREAL
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
EE is Element of S
((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE)) is ext-real Element of ExtREAL
R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE)))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE))))) - (integral+ (M,(max- (R_EAL (((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE))))) is ext-real Element of ExtREAL
((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE)) is ext-real Element of ExtREAL
R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE))) is ext-real Element of ExtREAL
max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE)))) is ext-real Element of ExtREAL
max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE))))) - (integral+ (M,(max- (R_EAL (((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE))))) is ext-real Element of ExtREAL
(Integral (M,(((1 / a) (#) (((1 / (E2 to_power (1 / a))) (#) (abs ND)) to_power a)) | EE))) + (Integral (M,(((1 / b) (#) (((1 / (G to_power (1 / b))) (#) (abs E)) to_power b)) | EE))) is ext-real Element of ExtREAL
(abs (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b))))) (#) (abs (ND (#) E)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
R_EAL (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) is ext-real Element of ExtREAL
(R_EAL (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b))))) * (Integral (M,(abs (ND (#) E)))) is ext-real Element of ExtREAL
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F2 is Element of S
M . F2 is ext-real Element of ExtREAL
dom E0 is Element of bool X
F2 ` is Element of bool X
X \ F2 is set
EE is ext-real V38() real Element of REAL
R_EAL EE is ext-real Element of ExtREAL
(R_EAL (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b))))) * (R_EAL EE) is ext-real Element of ExtREAL
(1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * EE is ext-real V38() real Element of REAL
((E2 to_power (1 / a)) * (G to_power (1 / b))) * ((1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) * EE) is ext-real V38() real Element of REAL
((E2 to_power (1 / a)) * (G to_power (1 / b))) * 1 is ext-real V38() real Element of REAL
((E2 to_power (1 / a)) * (G to_power (1 / b))) * (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b)))) is ext-real V38() real Element of REAL
(((E2 to_power (1 / a)) * (G to_power (1 / b))) * (1 / ((E2 to_power (1 / a)) * (G to_power (1 / b))))) * EE is ext-real V38() real Element of REAL
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
M . (E0 `) is ext-real set
F2 is Element of S
F2 ` is Element of bool X
X \ F2 is set
M . (F2 `) is ext-real set
KPAbsFMF is Element of S
KPAbsFMF ` is Element of bool X
X \ KPAbsFMF is set
M . (KPAbsFMF `) is ext-real set
PAbsFMF is Element of S
PAbsFMF ` is Element of bool X
X \ PAbsFMF is set
M . (PAbsFMF `) is ext-real set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (x + g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
1 / a is non empty ext-real positive non negative V38() real Element of REAL
a " is non empty ext-real positive non negative V38() real set
1 * (a ") is non empty ext-real positive non negative V38() real set
b is non empty ext-real positive non negative V38() real Element of REAL
1 / b is non empty ext-real positive non negative V38() real Element of REAL
b " is non empty ext-real positive non negative V38() real set
1 * (b ") is non empty ext-real positive non negative V38() real set
(1 / a) + (1 / b) is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power a is_integrable_on M ) } is set
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power a))))) is ext-real Element of ExtREAL
(abs g) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs g) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs g) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs g) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs g) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs g) to_power a))))) is ext-real Element of ExtREAL
(abs (x + g)) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (x + g)) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs (x + g)) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (x + g)) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (x + g)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (x + g)) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (x + g)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (x + g)) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (x + g)) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs (x + g)) to_power a))))) is ext-real Element of ExtREAL
a + b is non empty ext-real positive non negative V38() real Element of REAL
a * b is non empty ext-real positive non negative V38() real Element of REAL
(a * b) " is non empty ext-real positive non negative V38() real Element of REAL
(a + b) * ((a * b) ") is non empty ext-real positive non negative V38() real Element of REAL
((a + b) * ((a * b) ")) * (a * b) is non empty ext-real positive non negative V38() real Element of REAL
1 * (a * b) is non empty ext-real positive non negative V38() real Element of REAL
((a * b) ") * (a * b) is non empty ext-real positive non negative V38() real Element of REAL
(a + b) * (((a * b) ") * (a * b)) is non empty ext-real positive non negative V38() real Element of REAL
(a + b) * 1 is non empty ext-real positive non negative V38() real Element of REAL
a - 1 is ext-real V38() real Element of REAL
- 1 is non empty ext-real non positive negative V38() real set
a + (- 1) is ext-real V38() real set
b * (a - 1) is ext-real V38() real Element of REAL
1 - 1 is ext-real V38() real Element of REAL
1 + (- 1) is ext-real V38() real set
dom x is Element of bool X
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
dom g is Element of bool X
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E is Element of S
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom Ef is Element of bool X
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
ND /\ Ef is Element of S
dom (x + g) is Element of bool X
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
dom EE is Element of bool X
abs EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs EE) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (x + g)) to_power (a - 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((abs (x + g)) to_power (a - 1)) is Element of bool X
dom (abs (x + g)) is Element of bool X
((abs (x + g)) to_power (a - 1)) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs ((abs (x + g)) to_power (a - 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(X,S,M,b) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power b is_integrable_on M ) } is set
G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Gp is Element of S
Gp ` is Element of bool X
X \ Gp is set
M . (Gp `) is ext-real set
dom G is Element of bool X
abs G is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs G) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ((abs (x + g)) to_power (a - 1))) to_power b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs ((abs (x + g)) to_power (a - 1))) to_power b)) is ext-real Element of ExtREAL
R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b))))) - (integral+ (M,(max- (R_EAL ((abs ((abs (x + g)) to_power (a - 1))) to_power b))))) is ext-real Element of ExtREAL
((abs (x + g)) to_power (a - 1)) (#) x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs (x + g)) to_power (a - 1)) (#) g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (((abs (x + g)) to_power (a - 1)) (#) x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (((abs (x + g)) to_power (a - 1)) (#) x))) is ext-real Element of ExtREAL
R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x)))))) - (integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) x)))))) is ext-real Element of ExtREAL
abs (((abs (x + g)) to_power (a - 1)) (#) g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (((abs (x + g)) to_power (a - 1)) (#) g))) is ext-real Element of ExtREAL
R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g)))))) - (integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) g)))))) is ext-real Element of ExtREAL
dom (abs x) is Element of bool X
dom (abs g) is Element of bool X
((abs (x + g)) to_power (a - 1)) (#) (abs x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (((abs (x + g)) to_power (a - 1)) (#) (abs x)) is Element of bool X
(dom ((abs (x + g)) to_power (a - 1))) /\ (dom (abs x)) is Element of bool X
((abs (x + g)) to_power (a - 1)) (#) (abs g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (((abs (x + g)) to_power (a - 1)) (#) (abs g)) is Element of bool X
(dom ((abs (x + g)) to_power (a - 1))) /\ (dom (abs g)) is Element of bool X
((abs (x + g)) to_power (a - 1)) (#) (x + g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (((abs (x + g)) to_power (a - 1)) (#) (x + g)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))) is Element of bool X
(ND /\ Ef) /\ (ND /\ Ef) is Element of S
dom ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))) is Element of bool X
R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)))) is Element of bool X
Ir is Element of S
Ir is Element of X
(((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))) . Ir is ext-real V38() real Element of REAL
abs ((((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))) . Ir) is ext-real V38() real Element of REAL
((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))) . Ir is ext-real V38() real Element of REAL
x . Ir is ext-real V38() real Element of REAL
g . Ir is ext-real V38() real Element of REAL
(x . Ir) + (g . Ir) is ext-real V38() real Element of REAL
abs ((x . Ir) + (g . Ir)) is ext-real V38() real Element of REAL
(x + g) . Ir is ext-real V38() real Element of REAL
abs ((x + g) . Ir) is ext-real V38() real Element of REAL
(abs (x + g)) . Ir is ext-real V38() real Element of REAL
abs (x . Ir) is ext-real V38() real Element of REAL
(abs x) . Ir is ext-real V38() real Element of REAL
abs (g . Ir) is ext-real V38() real Element of REAL
(abs g) . Ir is ext-real V38() real Element of REAL
R_EAL (x . Ir) is ext-real Element of ExtREAL
R_EAL (g . Ir) is ext-real Element of ExtREAL
(R_EAL (x . Ir)) + (R_EAL (g . Ir)) is ext-real Element of ExtREAL
|.(R_EAL (x . Ir)).| is ext-real Element of ExtREAL
|.(R_EAL (g . Ir)).| is ext-real Element of ExtREAL
|.((R_EAL (x . Ir)) + (R_EAL (g . Ir))).| is ext-real Element of ExtREAL
((abs (x + g)) to_power (a - 1)) . Ir is ext-real V38() real Element of REAL
|.(R_EAL (x . Ir)).| + |.(R_EAL (g . Ir)).| is ext-real Element of ExtREAL
(abs (x . Ir)) + (abs (g . Ir)) is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) . Ir) * ((abs (x + g)) . Ir) is ext-real V38() real Element of REAL
((abs x) . Ir) + ((abs g) . Ir) is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) . Ir) * (((abs x) . Ir) + ((abs g) . Ir)) is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) . Ir) * ((abs x) . Ir) is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) (#) (abs x)) . Ir is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) . Ir) * ((abs g) . Ir) is ext-real V38() real Element of REAL
(((abs (x + g)) to_power (a - 1)) (#) (abs g)) . Ir is ext-real V38() real Element of REAL
((((abs (x + g)) to_power (a - 1)) (#) (abs x)) . Ir) + ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) . Ir) is ext-real V38() real Element of REAL
abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))))) is ext-real Element of ExtREAL
R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)))))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g))))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)))))))) - (integral+ (M,(max- (R_EAL (abs (((abs (x + g)) to_power (a - 1)) (#) (abs (x + g)))))))) is ext-real Element of ExtREAL
Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)))) is ext-real Element of ExtREAL
R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))))) is ext-real Element of ExtREAL
max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)))))) is ext-real Element of ExtREAL
max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))))))) - (integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) + (((abs (x + g)) to_power (a - 1)) (#) (abs g))))))) is ext-real Element of ExtREAL
(((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef))) is ext-real Element of ExtREAL
R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef)))) is ext-real Element of ExtREAL
max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef))))) is ext-real Element of ExtREAL
max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef)))))) - (integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | (ND /\ Ef)))))) is ext-real Element of ExtREAL
Integral (M,(((abs (x + g)) to_power (a - 1)) (#) (abs x))) is ext-real Element of ExtREAL
R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x)))) is ext-real Element of ExtREAL
max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x))))) is ext-real Element of ExtREAL
max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x)))))) - (integral+ (M,(max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs x)))))) is ext-real Element of ExtREAL
(((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef))) is ext-real Element of ExtREAL
R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef)))) is ext-real Element of ExtREAL
max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef))))) is ext-real Element of ExtREAL
max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef)))))) - (integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | (ND /\ Ef)))))) is ext-real Element of ExtREAL
Integral (M,(((abs (x + g)) to_power (a - 1)) (#) (abs g))) is ext-real Element of ExtREAL
R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g)))) is ext-real Element of ExtREAL
max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g))))) is ext-real Element of ExtREAL
max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g)))))) - (integral+ (M,(max- (R_EAL (((abs (x + g)) to_power (a - 1)) (#) (abs g)))))) is ext-real Element of ExtREAL
Gp is ext-real V38() real Element of REAL
ExtGp is ext-real V38() real Element of REAL
Gp + ExtGp is ext-real V38() real Element of REAL
Ir is Element of S
(((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir)) is ext-real Element of ExtREAL
R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir))) is ext-real Element of ExtREAL
max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir)))) is ext-real Element of ExtREAL
max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir))))) - (integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir))))) is ext-real Element of ExtREAL
(((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir)) is ext-real Element of ExtREAL
R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir))) is ext-real Element of ExtREAL
max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir)))) is ext-real Element of ExtREAL
max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir))))) - (integral+ (M,(max- (R_EAL ((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir))))) is ext-real Element of ExtREAL
(Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs x)) | Ir))) + (Integral (M,((((abs (x + g)) to_power (a - 1)) (#) (abs g)) | Ir))) is ext-real Element of ExtREAL
G is ext-real V38() real Element of REAL
G to_power (1 / b) is ext-real V38() real Element of REAL
EE is ext-real V38() real Element of REAL
EE to_power (1 / a) is ext-real V38() real Element of REAL
(G to_power (1 / b)) * (EE to_power (1 / a)) is ext-real V38() real Element of REAL
E1 is ext-real V38() real Element of REAL
E1 to_power (1 / a) is ext-real V38() real Element of REAL
(G to_power (1 / b)) * (E1 to_power (1 / a)) is ext-real V38() real Element of REAL
((G to_power (1 / b)) * (EE to_power (1 / a))) + ((G to_power (1 / b)) * (E1 to_power (1 / a))) is ext-real V38() real Element of REAL
PAbsFMF is ext-real V38() real Element of REAL
RF0 is ext-real V38() real Element of REAL
PAbsFMF to_power (1 / b) is ext-real V38() real Element of REAL
RF0 to_power (1 / a) is ext-real V38() real Element of REAL
(PAbsFMF to_power (1 / b)) * (RF0 to_power (1 / a)) is ext-real V38() real Element of REAL
QAbsFMF is ext-real V38() real Element of REAL
Br is ext-real V38() real Element of REAL
QAbsFMF to_power (1 / b) is ext-real V38() real Element of REAL
Br to_power (1 / a) is ext-real V38() real Element of REAL
(QAbsFMF to_power (1 / b)) * (Br to_power (1 / a)) is ext-real V38() real Element of REAL
(abs (x + g)) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs (x + g)) to_power (a - 1)) (#) ((abs (x + g)) to_power 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(a - 1) + 1 is ext-real V38() real Element of REAL
(abs (x + g)) to_power ((a - 1) + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is ext-real V38() real Element of REAL
E0 to_power (1 / b) is ext-real V38() real Element of REAL
(E0 to_power (1 / b)) * (EE to_power (1 / a)) is ext-real V38() real Element of REAL
(E0 to_power (1 / b)) * (E1 to_power (1 / a)) is ext-real V38() real Element of REAL
((E0 to_power (1 / b)) * (EE to_power (1 / a))) + ((E0 to_power (1 / b)) * (E1 to_power (1 / a))) is ext-real V38() real Element of REAL
E0 to_power (1 / a) is ext-real V38() real Element of REAL
(EE to_power (1 / a)) + (E1 to_power (1 / a)) is ext-real V38() real Element of REAL
1 / (E0 to_power (1 / b)) is ext-real V38() real Element of REAL
(E0 to_power (1 / b)) " is ext-real V38() real set
1 * ((E0 to_power (1 / b)) ") is ext-real V38() real set
(1 / (E0 to_power (1 / b))) * (((E0 to_power (1 / b)) * (EE to_power (1 / a))) + ((E0 to_power (1 / b)) * (E1 to_power (1 / a)))) is ext-real V38() real Element of REAL
(1 / (E0 to_power (1 / b))) * (E0 to_power (1 / b)) is ext-real V38() real Element of REAL
(EE to_power (1 / a)) + (E1 to_power (1 / a)) is ext-real V38() real Element of REAL
((1 / (E0 to_power (1 / b))) * (E0 to_power (1 / b))) * ((EE to_power (1 / a)) + (E1 to_power (1 / a))) is ext-real V38() real Element of REAL
1 * ((EE to_power (1 / a)) + (E1 to_power (1 / a))) is ext-real V38() real Element of REAL
(1 / (E0 to_power (1 / b))) * E0 is ext-real V38() real Element of REAL
- (1 / b) is non empty ext-real non positive negative V38() real Element of REAL
E0 to_power (- (1 / b)) is ext-real V38() real Element of REAL
(E0 to_power (- (1 / b))) * E0 is ext-real V38() real Element of REAL
E0 to_power 1 is ext-real V38() real Element of REAL
(E0 to_power (- (1 / b))) * (E0 to_power 1) is ext-real V38() real Element of REAL
(- (1 / b)) + 1 is ext-real V38() real Element of REAL
E0 to_power ((- (1 / b)) + 1) is ext-real V38() real Element of REAL
E0 to_power (1 / a) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
x + g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (x + g) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,a) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power a is_integrable_on M ) } is set
(abs x) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs x) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs x) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs x) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs x) to_power a))))) is ext-real Element of ExtREAL
(abs g) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs g) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs g) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs g) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs g) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs g) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs g) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs g) to_power a))))) is ext-real Element of ExtREAL
(abs (x + g)) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (x + g)) to_power a)) is ext-real Element of ExtREAL
R_EAL ((abs (x + g)) to_power a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (x + g)) to_power a))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (x + g)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (x + g)) to_power a)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (x + g)) to_power a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (x + g)) to_power a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (x + g)) to_power a))))) - (integral+ (M,(max- (R_EAL ((abs (x + g)) to_power a))))) is ext-real Element of ExtREAL
1 / a is non empty ext-real positive non negative V38() real Element of REAL
a " is non empty ext-real positive non negative V38() real set
1 * (a ") is non empty ext-real positive non negative V38() real set
b is ext-real V38() real Element of REAL
ND is ext-real V38() real Element of REAL
Ef is ext-real V38() real Element of REAL
Ef to_power (1 / a) is ext-real V38() real Element of REAL
b to_power (1 / a) is ext-real V38() real Element of REAL
ND to_power (1 / a) is ext-real V38() real Element of REAL
(b to_power (1 / a)) + (ND to_power (1 / a)) is ext-real V38() real Element of REAL
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs x))))) - (integral+ (M,(max- (R_EAL (abs x))))) is ext-real Element of ExtREAL
Integral (M,(abs g)) is ext-real Element of ExtREAL
R_EAL (abs g) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs g))) is ext-real Element of ExtREAL
max+ (R_EAL (abs g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs g)))) is ext-real Element of ExtREAL
max- (R_EAL (abs g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs g)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs g))))) - (integral+ (M,(max- (R_EAL (abs g))))) is ext-real Element of ExtREAL
Integral (M,(abs (x + g))) is ext-real Element of ExtREAL
R_EAL (abs (x + g)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (x + g)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (x + g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (x + g))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (x + g))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (x + g))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (x + g)))))) - (integral+ (M,(max- (R_EAL (abs (x + g)))))) is ext-real Element of ExtREAL
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
M . (E0 `) is ext-real set
dom E1 is Element of bool X
abs E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E1) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is Element of S
EE ` is Element of bool X
X \ EE is set
M . (EE `) is ext-real set
dom E is Element of bool X
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
M . (E0 `) is ext-real set
dom E1 is Element of bool X
abs E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E1) to_power a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(Integral (M,(abs x))) + (Integral (M,(abs g))) is ext-real Element of ExtREAL
b + ND is ext-real V38() real Element of REAL
1 - (1 / a) is ext-real V38() real Element of REAL
- (1 / a) is non empty ext-real non positive negative V38() real set
1 + (- (1 / a)) is ext-real V38() real set
1 / (1 - (1 / a)) is ext-real V38() real Element of REAL
(1 - (1 / a)) " is ext-real V38() real set
1 * ((1 - (1 / a)) ") is ext-real V38() real set
EE is non empty ext-real positive non negative V38() real Element of REAL
1 / EE is non empty ext-real positive non negative V38() real Element of REAL
EE " is non empty ext-real positive non negative V38() real set
1 * (EE ") is non empty ext-real positive non negative V38() real set
(1 / a) + (1 / EE) is non empty ext-real positive non negative V38() real Element of REAL
E1 is ext-real V38() real Element of REAL
E0 is ext-real V38() real Element of REAL
E2 is ext-real V38() real Element of REAL
E2 to_power (1 / a) is ext-real V38() real Element of REAL
E1 to_power (1 / a) is ext-real V38() real Element of REAL
E0 to_power (1 / a) is ext-real V38() real Element of REAL
(E1 to_power (1 / a)) + (E0 to_power (1 / a)) is ext-real V38() real Element of REAL
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real () Element of REAL
(X,S,M,x) is non empty NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
0. (X,S,M,x) is V52((X,S,M,x)) Element of the carrier of (X,S,M,x)
(X,S,M,x) is Element of (X,S,M,x)
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
bool (X,S,M,x) is non empty set
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b<sub>1</sub>,x) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,x) } is set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),x) is Element of bool (X,S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & X --> 0 a.e.= b<sub>1</sub>,M ) } is set
abs (X --> 0) is V1() V4(X) V5( REAL ) V5( RAT ) Function-like V163() V164() V165() V166() nonnegative Element of bool [:X,REAL:]
(abs (X --> 0)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (X --> 0)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (X --> 0)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs (X --> 0)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (X --> 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (X --> 0)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (X --> 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (X --> 0)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (X --> 0)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (X --> 0)) to_power x))))) is ext-real Element of ExtREAL
||.(0. (X,S,M,x)).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (0. (X,S,M,x)) is ext-real V38() real Element of REAL
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
(X,S,M,x) . (0. (X,S,M,x)) is ext-real V38() real Element of REAL
a is ext-real V38() real Element of REAL
a to_power (1 / x) is ext-real V38() real Element of REAL
a is ext-real V38() real Element of REAL
a to_power (1 / x) is ext-real V38() real Element of REAL
b is Element of the carrier of (X,S,M,x)
||.b.|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) . b is ext-real V38() real Element of REAL
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs E) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs E) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs E) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs E) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs E) to_power x))))) is ext-real Element of ExtREAL
EE is ext-real V38() real Element of REAL
EE to_power (1 / x) is ext-real V38() real Element of REAL
(X,S,M,x) . b is ext-real V38() real Element of REAL
EE is ext-real V38() real Element of REAL
EE to_power (1 / x) is ext-real V38() real Element of REAL
EE is ext-real V38() real Element of REAL
EE to_power (1 / x) is ext-real V38() real Element of REAL
(X,S,M,E,x) is Element of bool (X,S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & E a.e.= b<sub>1</sub>,M ) } is set
E1 is ext-real V38() real Element of REAL
E1 to_power (1 / x) is ext-real V38() real Element of REAL
E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs E) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs E) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs E) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs E) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs E) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs E) to_power x))))) is ext-real Element of ExtREAL
EE is ext-real V38() real Element of REAL
EE to_power (1 / x) is ext-real V38() real Element of REAL
ND is Element of the carrier of (X,S,M,x)
||.ND.|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) . ND is ext-real V38() real Element of REAL
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs EE) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs EE) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs EE) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs EE) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs EE) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs EE) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs EE) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs EE) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs EE) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs EE) to_power x))))) is ext-real Element of ExtREAL
E1 is ext-real V38() real Element of REAL
E1 to_power (1 / x) is ext-real V38() real Element of REAL
E1 is ext-real V38() real Element of REAL
E1 to_power (1 / x) is ext-real V38() real Element of REAL
E1 is ext-real V38() real Element of REAL
E1 to_power (1 / x) is ext-real V38() real Element of REAL
E + EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b + ND is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (b,ND) is Element of the carrier of (X,S,M,x)
G is ext-real V38() real Element of REAL
G to_power (1 / x) is ext-real V38() real Element of REAL
G is ext-real V38() real Element of REAL
G to_power (1 / x) is ext-real V38() real Element of REAL
Gp is ext-real V38() real Element of REAL
Gp to_power (1 / x) is ext-real V38() real Element of REAL
ExtGp is ext-real V38() real Element of REAL
ExtGp to_power (1 / x) is ext-real V38() real Element of REAL
abs (E + EE) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E + EE)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (E + EE)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E + EE)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E + EE)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E + EE)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E + EE)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E + EE)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E + EE)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E + EE)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E + EE)) to_power x))))) is ext-real Element of ExtREAL
||.(b + ND).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) . (b + ND) is ext-real V38() real Element of REAL
||.b.|| + ||.ND.|| is ext-real V38() real Element of REAL
Gp is ext-real V38() real Element of REAL
Gp to_power (1 / x) is ext-real V38() real Element of REAL
Ef is ext-real V38() real Element of REAL
Ef (#) E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef * b is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . (Ef,b) is set
ExtGp is ext-real V38() real Element of REAL
ExtGp to_power (1 / x) is ext-real V38() real Element of REAL
abs (Ef (#) E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (Ef (#) E)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (Ef (#) E)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (Ef (#) E)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (Ef (#) E)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (Ef (#) E)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (Ef (#) E)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (Ef (#) E)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (Ef (#) E)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (Ef (#) E)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (Ef (#) E)) to_power x))))) is ext-real Element of ExtREAL
||.(Ef * b).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) . (Ef * b) is ext-real V38() real Element of REAL
ExtGp is ext-real V38() real Element of REAL
ExtGp to_power (1 / x) is ext-real V38() real Element of REAL
ExtGp is ext-real V38() real Element of REAL
ExtGp to_power (1 / x) is ext-real V38() real Element of REAL
abs Ef is ext-real V38() real Element of REAL
(abs Ef) to_power x is ext-real V38() real Element of REAL
((abs Ef) to_power x) (#) ((abs E) to_power x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(((abs Ef) to_power x) (#) ((abs E) to_power x))) is ext-real Element of ExtREAL
R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x)))) is ext-real Element of ExtREAL
max+ (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x))))) is ext-real Element of ExtREAL
max- (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x)))))) - (integral+ (M,(max- (R_EAL (((abs Ef) to_power x) (#) ((abs E) to_power x)))))) is ext-real Element of ExtREAL
R_EAL ((abs Ef) to_power x) is ext-real Element of ExtREAL
R_EAL E1 is ext-real Element of ExtREAL
(R_EAL ((abs Ef) to_power x)) * (R_EAL E1) is ext-real Element of ExtREAL
((abs Ef) to_power x) * E1 is ext-real V38() real Element of REAL
((abs Ef) to_power x) to_power (1 / x) is ext-real V38() real Element of REAL
(((abs Ef) to_power x) to_power (1 / x)) * (E1 to_power (1 / x)) is ext-real V38() real Element of REAL
x * (1 / x) is non empty ext-real positive non negative V38() real Element of REAL
(abs Ef) to_power (x * (1 / x)) is ext-real V38() real Element of REAL
((abs Ef) to_power (x * (1 / x))) * (E1 to_power (1 / x)) is ext-real V38() real Element of REAL
(abs Ef) to_power 1 is ext-real V38() real Element of REAL
((abs Ef) to_power 1) * (E1 to_power (1 / x)) is ext-real V38() real Element of REAL
(abs Ef) * ||.b.|| is ext-real V38() real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
M is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V164() set
bool [:M,ExtREAL:] is non empty set
x is V1() V4(M) V5( ExtREAL ) Function-like non empty total V41(M, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
X is non empty ext-real positive non negative V38() real () Element of REAL
(S,M,x,X) is non empty NORMSTR
(S,M,x,X) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (S,M,x,X) is non empty set
the ZeroF of (S,M,x,X) is Element of the carrier of (S,M,x,X)
the addF of (S,M,x,X) is V1() V4([: the carrier of (S,M,x,X), the carrier of (S,M,x,X):]) V5( the carrier of (S,M,x,X)) Function-like non empty total V41([: the carrier of (S,M,x,X), the carrier of (S,M,x,X):], the carrier of (S,M,x,X)) Element of bool [:[: the carrier of (S,M,x,X), the carrier of (S,M,x,X):], the carrier of (S,M,x,X):]
[: the carrier of (S,M,x,X), the carrier of (S,M,x,X):] is non empty set
[:[: the carrier of (S,M,x,X), the carrier of (S,M,x,X):], the carrier of (S,M,x,X):] is non empty set
bool [:[: the carrier of (S,M,x,X), the carrier of (S,M,x,X):], the carrier of (S,M,x,X):] is non empty set
the Mult of (S,M,x,X) is V1() V4([:REAL, the carrier of (S,M,x,X):]) V5( the carrier of (S,M,x,X)) Function-like non empty total V41([:REAL, the carrier of (S,M,x,X):], the carrier of (S,M,x,X)) Element of bool [:[:REAL, the carrier of (S,M,x,X):], the carrier of (S,M,x,X):]
[:REAL, the carrier of (S,M,x,X):] is non empty set
[:[:REAL, the carrier of (S,M,x,X):], the carrier of (S,M,x,X):] is non empty set
bool [:[:REAL, the carrier of (S,M,x,X):], the carrier of (S,M,x,X):] is non empty set
(S,M,x,X) is V1() V4( the carrier of (S,M,x,X)) V5( REAL ) Function-like non empty total V41( the carrier of (S,M,x,X), REAL ) V163() V164() V165() Element of bool [: the carrier of (S,M,x,X),REAL:]
[: the carrier of (S,M,x,X),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (S,M,x,X),REAL:] is non empty set
NORMSTR(# the carrier of (S,M,x,X), the ZeroF of (S,M,x,X), the addF of (S,M,x,X), the Mult of (S,M,x,X),(S,M,x,X) #) is strict NORMSTR
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
the carrier of (X,S,M,x) is non empty set
[:NAT, the carrier of (X,S,M,x):] is non empty set
bool [:NAT, the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
g is V1() V4( NAT ) V5( the carrier of (X,S,M,x)) Function-like non empty total V41( NAT , the carrier of (X,S,M,x)) Element of bool [:NAT, the carrier of (X,S,M,x):]
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . a is Element of the carrier of (X,S,M,x)
||.(g . a).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (g . a) is ext-real V38() real Element of REAL
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,b,x) is Element of bool (X,S,M,x)
bool (X,S,M,x) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b a.e.= b₁,M ) } is set
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs b) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs b) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs b) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs b) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs b) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs b) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs b) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs b) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs b) to_power x))))) is ext-real Element of ExtREAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / x) is ext-real V38() real Element of REAL
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
a is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
b is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
ND is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
b . ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g . ND is Element of the carrier of (X,S,M,x)
||.(g . ND).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (g . ND) is ext-real V38() real Element of REAL
(X,S,M,(b . ND),x) is Element of bool (X,S,M,x)
bool (X,S,M,x) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & b . ND a.e.= b₁,M ) } is set
abs (b . ND) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (b . ND)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (b . ND)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (b . ND)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL ((abs (b . ND)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (b . ND)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (b . ND)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (b . ND)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (b . ND)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (b . ND)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (b . ND)) to_power x))))) is ext-real Element of ExtREAL
Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,Ef,x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & Ef a.e.= b₁,M ) } is set
E is ext-real V38() real Element of REAL
abs Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs Ef) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs Ef) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs Ef) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs Ef) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs Ef) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs Ef) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs Ef) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs Ef) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs Ef) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs Ef) to_power x))))) is ext-real Element of ExtREAL
E to_power (1 / x) is ext-real V38() real Element of REAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
the carrier of (X,S,M,x) is non empty set
[:NAT, the carrier of (X,S,M,x):] is non empty set
bool [:NAT, the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
g is V1() V4( NAT ) V5( the carrier of (X,S,M,x)) Function-like non empty total V41( NAT , the carrier of (X,S,M,x)) Element of bool [:NAT, the carrier of (X,S,M,x):]
a is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a . b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
[:NAT,S:] is non empty set
bool [:NAT,S:] is non empty set
b is V1() V4( NAT ) V5(S) Function-like non empty total V41( NAT ,S) Element of bool [:NAT,S:]
rng b is non empty V146() N_Measure_fam of S
meet (rng b) is Element of S
ND is Element of S
Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
b . Ef is Element of S
X \ (b . Ef) is Element of bool X
M . (X \ (b . Ef)) is ext-real set
a . Ef is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (a . Ef) is Element of bool X
E is Element of S
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E ` is Element of bool X
X \ E is set
M . (E `) is ext-real set
dom EE is Element of bool X
abs EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs EE) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
COMPLEMENT (rng b) is non empty Element of bool (bool X)
Ef is set
E is Element of bool X
E ` is Element of bool X
X \ E is set
EE is set
b . EE is set
(E `) ` is Element of bool X
X \ (E `) is set
E1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
b . E1 is Element of S
X \ (b . E1) is Element of bool X
M . E is ext-real set
Ef is set
E is set
union (COMPLEMENT (rng b)) is Element of bool X
ND ` is Element of bool X
X \ ND is set
X \ (union (COMPLEMENT (rng b))) is Element of bool X
X \ (X \ (union (COMPLEMENT (rng b)))) is Element of bool X
X /\ (union (COMPLEMENT (rng b))) is Element of bool X
M . (ND `) is ext-real set
a || ND is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(a || ND) . E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((a || ND) . E) is Element of bool X
a . E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(a . E) | ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((a . E) | ND) is Element of bool X
dom (a . E) is Element of bool X
(dom (a . E)) /\ ND is Element of bool X
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(a || ND) . E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((a || ND) . E) is Element of bool X
(a || ND) . EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((a || ND) . EE) is Element of bool X
E is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a . EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (a . EE) is Element of bool X
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 is Element of S
E2 ` is Element of bool X
X \ E2 is set
M . (E2 `) is ext-real set
dom E0 is Element of bool X
abs E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E1 is Element of S
E1 /\ ND is Element of S
R_EAL (a . EE) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 is Element of S
E2 ` is Element of bool X
X \ E2 is set
M . (E2 `) is ext-real set
dom E0 is Element of bool X
abs E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(a . EE) | ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E . EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E . EE) is Element of bool X
abs (a . EE) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (a . EE)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom ((abs (a . EE)) to_power x) is Element of bool X
dom (abs (a . EE)) is Element of bool X
abs (E . EE) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E . EE)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom ((abs (E . EE)) to_power x) is Element of bool X
dom (abs (E . EE)) is Element of bool X
E0 is set
((abs (E . EE)) to_power x) . E0 is ext-real V38() real Element of REAL
((abs (a . EE)) to_power x) . E0 is ext-real V38() real Element of REAL
E2 is Element of X
(abs (E . EE)) . E2 is ext-real V38() real Element of REAL
((abs (E . EE)) . E2) to_power x is ext-real V38() real Element of REAL
(E . EE) . E2 is ext-real V38() real Element of REAL
abs ((E . EE) . E2) is ext-real V38() real Element of REAL
(abs ((E . EE) . E2)) to_power x is ext-real V38() real Element of REAL
((a . EE) | ND) . E2 is ext-real V38() real Element of REAL
abs (((a . EE) | ND) . E2) is ext-real V38() real Element of REAL
(abs (((a . EE) | ND) . E2)) to_power x is ext-real V38() real Element of REAL
(a . EE) . E2 is ext-real V38() real Element of REAL
abs ((a . EE) . E2) is ext-real V38() real Element of REAL
(abs ((a . EE) . E2)) to_power x is ext-real V38() real Element of REAL
(abs (a . EE)) . E2 is ext-real V38() real Element of REAL
((abs (a . EE)) . E2) to_power x is ext-real V38() real Element of REAL
((abs (a . EE)) to_power x) | ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 is Element of S
E2 ` is Element of bool X
X \ E2 is set
M . (E2 `) is ext-real set
dom E0 is Element of bool X
abs E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs E0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g . EE is Element of the carrier of (X,S,M,x)
(X,S,M,(a . EE),x) is Element of bool (X,S,M,x)
bool (X,S,M,x) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,x) & a . EE a.e.= b<sub>1</sub>,M ) } is set
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
(E . EE) | (E0 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(a . EE) | (E0 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,(E . EE),x) is Element of bool (X,S,M,x)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E . EE a.e.= b₁,M ) } is set
Integral (M,((abs (E . EE)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E . EE)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E . EE)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E . EE)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E . EE)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E . EE)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E . EE)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E . EE)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E . EE)) to_power x))))) is ext-real Element of ExtREAL
||.(g . EE).|| is ext-real V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (g . EE) is ext-real V38() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
S is V1() V4( NAT ) V5( the carrier of X) Function-like non empty total V41( NAT , the carrier of X) Element of bool [:NAT, the carrier of X:]
lim S is Element of the carrier of X
M is Element of the carrier of X
S - M is V1() V4( NAT ) V5( the carrier of X) Function-like non empty total V41( NAT , the carrier of X) Element of bool [:NAT, the carrier of X:]
||.(S - M).|| is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim ||.(S - M).|| is ext-real V38() real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim x is ext-real V38() real Element of REAL
g is V1() V4( NAT ) V5( NAT ) Function-like non empty total V41( NAT , NAT ) V163() V164() V165() V166() V167() V169() Element of bool [:NAT,NAT:]
a is ext-real V38() real Element of REAL
a / 2 is ext-real V38() real Element of REAL
2 " is non empty ext-real positive non negative V38() real set
a * (2 ") is ext-real V38() real set
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
max (ND,b) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . EE is Element of the carrier of X
(S . EE) - M is Element of the carrier of X
- M is Element of the carrier of X
(S . EE) + (- M) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . EE),(- M)) is Element of the carrier of X
||.((S . EE) - M).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . EE) - M) is ext-real V38() real Element of REAL
x . EE is ext-real V38() real Element of REAL
(x . EE) - 0 is ext-real V38() real Element of REAL
- 0 is ext-real non positive V38() real set
(x . EE) + (- 0) is ext-real V38() real set
abs ((x . EE) - 0) is ext-real V38() real Element of REAL
g . EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . (g . EE) is Element of the carrier of X
M - (S . (g . EE)) is Element of the carrier of X
- (S . (g . EE)) is Element of the carrier of X
M + (- (S . (g . EE))) is Element of the carrier of X
the addF of X . (M,(- (S . (g . EE)))) is Element of the carrier of X
||.(M - (S . (g . EE))).|| is ext-real non negative V38() real Element of REAL
the U9 of X . (M - (S . (g . EE))) is ext-real V38() real Element of REAL
abs ||.(M - (S . (g . EE))).|| is ext-real V38() real Element of REAL
(S . (g . EE)) - (S . EE) is Element of the carrier of X
- (S . EE) is Element of the carrier of X
(S . (g . EE)) + (- (S . EE)) is Element of the carrier of X
the addF of X . ((S . (g . EE)),(- (S . EE))) is Element of the carrier of X
||.((S . (g . EE)) - (S . EE)).|| is ext-real non negative V38() real Element of REAL
the U9 of X . ((S . (g . EE)) - (S . EE)) is ext-real V38() real Element of REAL
(a / 2) + (a / 2) is ext-real V38() real Element of REAL
||.(M - (S . (g . EE))).|| + ||.((S . (g . EE)) - (S . EE)).|| is ext-real non negative V38() real Element of REAL
M - (S . EE) is Element of the carrier of X
M + (- (S . EE)) is Element of the carrier of X
the addF of X . (M,(- (S . EE))) is Element of the carrier of X
||.(M - (S . EE)).|| is ext-real non negative V38() real Element of REAL
the U9 of X . (M - (S . EE)) is ext-real V38() real Element of REAL
(M - (S . (g . EE))) + ((S . (g . EE)) - (S . EE)) is Element of the carrier of X
the addF of X . ((M - (S . (g . EE))),((S . (g . EE)) - (S . EE))) is Element of the carrier of X
||.((M - (S . (g . EE))) + ((S . (g . EE)) - (S . EE))).|| is ext-real non negative V38() real Element of REAL
the U9 of X . ((M - (S . (g . EE))) + ((S . (g . EE)) - (S . EE))) is ext-real V38() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
S is V1() V4( NAT ) V5( the carrier of X) Function-like non empty total V41( NAT , the carrier of X) Element of bool [:NAT, the carrier of X:]
lim S is Element of the carrier of X
M is Element of the carrier of X
S - M is V1() V4( NAT ) V5( the carrier of X) Function-like non empty total V41( NAT , the carrier of X) Element of bool [:NAT, the carrier of X:]
||.(S - M).|| is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim ||.(S - M).|| is ext-real V38() real Element of REAL
x is ext-real V38() real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
||.(S - M).|| . a is ext-real V38() real Element of REAL
(||.(S - M).|| . a) - 0 is ext-real V38() real Element of REAL
- 0 is ext-real non positive V38() real set
(||.(S - M).|| . a) + (- 0) is ext-real V38() real set
abs ((||.(S - M).|| . a) - 0) is ext-real V38() real Element of REAL
(S - M) . a is Element of the carrier of X
||.((S - M) . a).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S - M) . a) is ext-real V38() real Element of REAL
abs ||.((S - M) . a).|| is ext-real V38() real Element of REAL
S . a is Element of the carrier of X
(S . a) - M is Element of the carrier of X
- M is Element of the carrier of X
(S . a) + (- M) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . a),(- M)) is Element of the carrier of X
||.((S . a) - M).|| is ext-real non negative V38() real Element of REAL
the U9 of X . ((S . a) - M) is ext-real V38() real Element of REAL
abs ||.((S . a) - M).|| is ext-real V38() real Element of REAL
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
[:NAT, the carrier of X:] is non empty set
bool [:NAT, the carrier of X:] is non empty set
S is V1() V4( NAT ) V5( the carrier of X) Function-like non empty total V41( NAT , the carrier of X) Element of bool [:NAT, the carrier of X:]
- 0 is ext-real non positive V38() real Element of REAL
2 to_power (- 0) is ext-real V38() real Element of REAL
M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- x is ext-real non positive V38() real Element of REAL
2 to_power (- x) is ext-real V38() real Element of REAL
S . g is Element of the carrier of X
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- (x + 1) is non empty ext-real non positive negative V38() real Element of REAL
2 to_power (- (x + 1)) is ext-real V38() real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
max (g,a) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(max (g,a)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . ND is Element of the carrier of X
Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . Ef is Element of the carrier of X
(S . Ef) - (S . ND) is Element of the carrier of X
- (S . ND) is Element of the carrier of X
(S . Ef) + (- (S . ND)) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . Ef),(- (S . ND))) is Element of the carrier of X
||.((S . Ef) - (S . ND)).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . Ef) - (S . ND)) is ext-real V38() real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . a is Element of the carrier of X
(S . a) - (S . g) is Element of the carrier of X
- (S . g) is Element of the carrier of X
(S . a) + (- (S . g)) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . a),(- (S . g))) is Element of the carrier of X
||.((S . a) - (S . g)).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . a) - (S . g)) is ext-real V38() real Element of REAL
b is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . b is Element of the carrier of X
x is V1() V4( NAT ) V5( NAT ) Function-like non empty total V41( NAT , NAT ) V163() V164() V165() V166() Element of bool [:NAT,NAT:]
x . 0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . (x . 0) is Element of the carrier of X
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . g is Element of the carrier of X
(S . g) - (S . (x . 0)) is Element of the carrier of X
- (S . (x . 0)) is Element of the carrier of X
(S . g) + (- (S . (x . 0))) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . g),(- (S . (x . 0)))) is Element of the carrier of X
||.((S . g) - (S . (x . 0))).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . g) - (S . (x . 0))) is ext-real V38() real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x . g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- g is ext-real non positive V38() real Element of REAL
2 to_power (- g) is ext-real V38() real Element of REAL
S . (x . g) is Element of the carrier of X
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x . (g + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- (g + 1) is non empty ext-real non positive negative V38() real Element of REAL
2 to_power (- (g + 1)) is ext-real V38() real Element of REAL
S . (x . (g + 1)) is Element of the carrier of X
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . a is Element of the carrier of X
(S . a) - (S . (x . (g + 1))) is Element of the carrier of X
- (S . (x . (g + 1))) is Element of the carrier of X
(S . a) + (- (S . (x . (g + 1)))) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . a),(- (S . (x . (g + 1))))) is Element of the carrier of X
||.((S . a) - (S . (x . (g + 1)))).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . a) - (S . (x . (g + 1)))) is ext-real V38() real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- b is ext-real non positive V38() real Element of REAL
2 to_power (- b) is ext-real V38() real Element of REAL
S . E is Element of the carrier of X
S . ND is Element of the carrier of X
(S . E) - (S . ND) is Element of the carrier of X
- (S . ND) is Element of the carrier of X
(S . E) + (- (S . ND)) is Element of the carrier of X
the addF of X . ((S . E),(- (S . ND))) is Element of the carrier of X
||.((S . E) - (S . ND)).|| is ext-real non negative V38() real Element of REAL
the U9 of X . ((S . E) - (S . ND)) is ext-real V38() real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- (b + 1) is non empty ext-real non positive negative V38() real Element of REAL
2 to_power (- (b + 1)) is ext-real V38() real Element of REAL
S . Ef is Element of the carrier of X
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x . g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
x . (g + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ef is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- a is ext-real non positive V38() real Element of REAL
2 to_power (- a) is ext-real V38() real Element of REAL
S . Ef is Element of the carrier of X
S . b is Element of the carrier of X
(S . Ef) - (S . b) is Element of the carrier of X
- (S . b) is Element of the carrier of X
(S . Ef) + (- (S . b)) is Element of the carrier of X
the addF of X is V1() V4([: the carrier of X, the carrier of X:]) V5( the carrier of X) Function-like non empty total V41([: the carrier of X, the carrier of X:], the carrier of X) Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((S . Ef),(- (S . b))) is Element of the carrier of X
||.((S . Ef) - (S . b)).|| is ext-real non negative V38() real Element of REAL
the U9 of X is V1() V4( the carrier of X) V5( REAL ) Function-like non empty total V41( the carrier of X, REAL ) V163() V164() V165() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty V163() V164() V165() set
bool [: the carrier of X,REAL:] is non empty set
the U9 of X . ((S . Ef) - (S . b)) is ext-real V38() real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
- (a + 1) is non empty ext-real non positive negative V38() real Element of REAL
2 to_power (- (a + 1)) is ext-real V38() real Element of REAL
S . ND is Element of the carrier of X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power x is_integrable_on M ) } is set
g is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
Partial_Sums g is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
(Partial_Sums g) . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums g) . a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (a + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums g) . (a + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums g) . a) + (g . (a + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums g) . a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
Partial_Sums S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
(Partial_Sums S) . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums S) . M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
S . (M + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums S) . (M + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . M) + (S . (M + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums S) . M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
X is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
S . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
dom (S . 0) is Element of bool X
bool X is non empty set
Partial_Sums S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
M is Element of X
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
g is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums S) . x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . x) . M is ext-real V38() real Element of REAL
(Partial_Sums S) . g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . g) . M is ext-real V38() real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums S) . b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . b) . M is ext-real V38() real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums S) . (b + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . (b + 1)) . M is ext-real V38() real Element of REAL
S . (b + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . b) + (S . (b + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((Partial_Sums S) . (b + 1)) is Element of bool X
(S . (b + 1)) . M is ext-real V38() real Element of REAL
(((Partial_Sums S) . b) . M) + 0 is ext-real V38() real Element of REAL
(((Partial_Sums S) . b) . M) + ((S . (b + 1)) . M) is ext-real V38() real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(Partial_Sums S) . b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . b) . M is ext-real V38() real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
b - 1 is ext-real V38() real Element of REAL
- 1 is non empty ext-real non positive negative V38() real set
b + (- 1) is ext-real V38() real set
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums S) . ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums S) . ND) . M is ext-real V38() real Element of REAL
ND + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
X is non empty set
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
abs S is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
M is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(abs S) . M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
dom ((abs S) . M) is Element of bool X
bool X is non empty set
x is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(abs S) . x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((abs S) . x) is Element of bool X
S . M is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (S . M) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
S . x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (S . x) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (S . M) is Element of bool X
dom (S . x) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real () Element of REAL
(X,S,M,x) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
the carrier of (X,S,M,x) is non empty set
[:NAT, the carrier of (X,S,M,x):] is non empty set
bool [:NAT, the carrier of (X,S,M,x):] is non empty set
g is V1() V4( NAT ) V5( the carrier of (X,S,M,x)) Function-like non empty total V41( NAT , the carrier of (X,S,M,x)) Element of bool [:NAT, the carrier of (X,S,M,x):]
PFuncs (X,REAL) is functional non empty set
[:NAT,(PFuncs (X,REAL)):] is non empty set
bool [:NAT,(PFuncs (X,REAL)):] is non empty set
(X,S,M,x) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power x is_integrable_on M ) } is set
1 / x is non empty ext-real positive non negative V38() real Element of REAL
x " is non empty ext-real positive non negative V38() real set
1 * (x ") is non empty ext-real positive non negative V38() real set
a is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
a . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (a . 0) is Element of bool X
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
ND is V1() V4( NAT ) V5( NAT ) Function-like non empty total V41( NAT , NAT ) V163() V164() V165() V166() V167() V169() Element of bool [:NAT,NAT:]
Ef is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
b is Element of S
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Ef . E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (Ef . E) is Element of bool X
abs (Ef . E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND . E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a . (ND . E) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E1 is Element of S
E1 ` is Element of bool X
X \ E1 is set
M . (E1 `) is ext-real set
dom EE is Element of bool X
abs EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs EE) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Ef . E is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (Ef . E) is Element of bool X
EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Ef . EE is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (Ef . EE) is Element of bool X
E is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
EE is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
E1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
EE . E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (EE . E1) is Element of bool X
E . E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E . E1) is Element of bool X
E1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E . (E1 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E . (E1 + 1)) is Element of bool X
(E . (E1 + 1)) - (E . E1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (E . E1) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (E . E1) is V1() V4(X) Function-like V163() V164() V165() set
(E . (E1 + 1)) + (- (E . E1)) is V1() V4(X) Function-like V163() V164() V165() set
(dom (E . (E1 + 1))) /\ (dom (E . E1)) is Element of bool X
ND . (E1 + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a . (ND . (E1 + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND . E1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
a . (ND . E1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
EE . E1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (EE . E1) is Element of bool X
E0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
EE . E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (EE . E0) is Element of bool X
E1 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
abs E1 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
E2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(abs E1) . E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((abs E1) . E2) is Element of bool X
abs ((abs E1) . E2) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E1 . E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (E1 . E2) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (E1 . E2) is Element of bool X
G is Element of S
G ` is Element of bool X
X \ G is set
M . (G `) is ext-real set
E . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (E . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
E2 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
Partial_Sums E2 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
G is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
Gp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
G . Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . Gp) is Element of bool X
abs (G . Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(Partial_Sums E2) . Gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . Gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (abs (E . 0)) is Element of bool X
dom ((Partial_Sums E2) . Gp) is Element of bool X
(dom (abs (E . 0))) /\ (dom ((Partial_Sums E2) . Gp)) is Element of bool X
dom (E . 0) is Element of bool X
(dom (E . 0)) /\ (dom ((Partial_Sums E2) . Gp)) is Element of bool X
E2 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E2 . 0) is Element of bool X
(dom (E . 0)) /\ (dom (E2 . 0)) is Element of bool X
Gp is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
ExtGp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
G . ExtGp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . ExtGp) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . ExtGp) is Element of bool X
PFuncs (X,ExtREAL) is functional non empty set
[:NAT,(PFuncs (X,ExtREAL)):] is non empty set
bool [:NAT,(PFuncs (X,ExtREAL)):] is non empty set
R_EAL Gp is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
ExtGp is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
ExtGp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
ExtGp . ExtGp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
dom (ExtGp . ExtGp) is Element of bool X
G . ExtGp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . ExtGp) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
R_EAL ((G . ExtGp) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (G . ExtGp) is Element of bool X
ExtGp . 0 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
dom (ExtGp . 0) is Element of bool X
ExtGp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
ExtGp . ExtGp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (ExtGp . ExtGp) is Element of bool X
I is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
ExtGp . I is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (ExtGp . I) is Element of bool X
ExtGp is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
I is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Ir is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
ExtGp . I is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
ExtGp . Ir is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
KPAbsFMF is Element of X
(ExtGp . I) . KPAbsFMF is ext-real set
(ExtGp . Ir) . KPAbsFMF is ext-real set
G . I is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . I) is Element of bool X
G . Ir is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . Ir) is Element of bool X
(G . I) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((G . I) to_power x) is Element of bool X
(G . Ir) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((G . Ir) to_power x) is Element of bool X
(G . I) . KPAbsFMF is ext-real V38() real Element of REAL
((G . I) . KPAbsFMF) to_power x is ext-real V38() real Element of REAL
((G . I) to_power x) . KPAbsFMF is ext-real V38() real Element of REAL
(G . Ir) . KPAbsFMF is ext-real V38() real Element of REAL
((G . Ir) . KPAbsFMF) to_power x is ext-real V38() real Element of REAL
((G . Ir) to_power x) . KPAbsFMF is ext-real V38() real Element of REAL
E2 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E2 . 0) is Element of bool X
(Partial_Sums E2) . I is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums E2) . I) . KPAbsFMF is ext-real V38() real Element of REAL
(Partial_Sums E2) . Ir is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums E2) . Ir) . KPAbsFMF is ext-real V38() real Element of REAL
(abs (E . 0)) . KPAbsFMF is ext-real V38() real Element of REAL
((abs (E . 0)) . KPAbsFMF) + (((Partial_Sums E2) . I) . KPAbsFMF) is ext-real V38() real Element of REAL
((abs (E . 0)) . KPAbsFMF) + (((Partial_Sums E2) . Ir) . KPAbsFMF) is ext-real V38() real Element of REAL
(abs (E . 0)) + ((Partial_Sums E2) . Ir) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . I) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
I is Element of X
ExtGp # I is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
KPAbsFMF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ir is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(ExtGp # I) . KPAbsFMF is ext-real Element of ExtREAL
(ExtGp # I) . Ir is ext-real Element of ExtREAL
ExtGp . KPAbsFMF is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(ExtGp . KPAbsFMF) . I is ext-real set
ExtGp . Ir is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(ExtGp . Ir) . I is ext-real set
I is Element of X
ExtGp # I is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim ExtGp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(lim ExtGp)) is ext-real Element of ExtREAL
max+ (lim ExtGp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (lim ExtGp))) is ext-real Element of ExtREAL
max- (lim ExtGp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (lim ExtGp))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (lim ExtGp)))) - (integral+ (M,(max- (lim ExtGp)))) is ext-real Element of ExtREAL
I is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim I is ext-real Element of ExtREAL
Ir is set
rng I is V157() Element of bool ExtREAL
KPAbsFMF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
I . KPAbsFMF is ext-real Element of ExtREAL
Gp . KPAbsFMF is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(Gp . KPAbsFMF)) is ext-real Element of ExtREAL
R_EAL (Gp . KPAbsFMF) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (Gp . KPAbsFMF))) is ext-real Element of ExtREAL
max+ (R_EAL (Gp . KPAbsFMF)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (Gp . KPAbsFMF)))) is ext-real Element of ExtREAL
max- (R_EAL (Gp . KPAbsFMF)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (Gp . KPAbsFMF)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (Gp . KPAbsFMF))))) - (integral+ (M,(max- (R_EAL (Gp . KPAbsFMF))))) is ext-real Element of ExtREAL
G . KPAbsFMF is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (G . KPAbsFMF) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (G . KPAbsFMF)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
KPAbsFMF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E2 . KPAbsFMF is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E2 . KPAbsFMF) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((E2 . KPAbsFMF) to_power x)) is ext-real Element of ExtREAL
R_EAL ((E2 . KPAbsFMF) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((E2 . KPAbsFMF) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((E2 . KPAbsFMF) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((E2 . KPAbsFMF) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((E2 . KPAbsFMF) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((E2 . KPAbsFMF) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((E2 . KPAbsFMF) to_power x))))) - (integral+ (M,(max- (R_EAL ((E2 . KPAbsFMF) to_power x))))) is ext-real Element of ExtREAL
abs (E2 . KPAbsFMF) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E2 . KPAbsFMF)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (E2 . KPAbsFMF)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E2 . KPAbsFMF)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E2 . KPAbsFMF)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E2 . KPAbsFMF)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E2 . KPAbsFMF)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E2 . KPAbsFMF)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E2 . KPAbsFMF)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E2 . KPAbsFMF)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E2 . KPAbsFMF)) to_power x))))) is ext-real Element of ExtREAL
KPAbsFMF is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
PAbsFMF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
KPAbsFMF . PAbsFMF is ext-real V38() real Element of REAL
(KPAbsFMF . PAbsFMF) to_power (1 / x) is ext-real V38() real Element of REAL
PAbsFMF is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(abs (E . 0)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (E . 0)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E . 0)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E . 0)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E . 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E . 0)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E . 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E . 0)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E . 0)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E . 0)) to_power x))))) is ext-real Element of ExtREAL
RF0 is ext-real V38() real Element of REAL
RF0 to_power (1 / x) is ext-real V38() real Element of REAL
Partial_Sums PAbsFMF is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
QAbsFMF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums PAbsFMF) . QAbsFMF is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . QAbsFMF) is ext-real V38() real Element of REAL
QAbsFMF is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Ir is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E2 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
G . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums E2) . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + (E2 . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ir . 0 is ext-real V38() real Element of REAL
Gp . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(Gp . 0)) is ext-real Element of ExtREAL
R_EAL (Gp . 0) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (Gp . 0))) is ext-real Element of ExtREAL
max+ (R_EAL (Gp . 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (Gp . 0)))) is ext-real Element of ExtREAL
max- (R_EAL (Gp . 0)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (Gp . 0)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (Gp . 0))))) - (integral+ (M,(max- (R_EAL (Gp . 0))))) is ext-real Element of ExtREAL
(G . 0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((G . 0) to_power x)) is ext-real Element of ExtREAL
R_EAL ((G . 0) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((G . 0) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((G . 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((G . 0) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((G . 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((G . 0) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((G . 0) to_power x))))) - (integral+ (M,(max- (R_EAL ((G . 0) to_power x))))) is ext-real Element of ExtREAL
abs ((abs (E . 0)) + (E2 . 0)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ((abs (E . 0)) + (E2 . 0))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs ((abs (E . 0)) + (E2 . 0))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs ((abs (E . 0)) + (E2 . 0))) to_power x))))) is ext-real Element of ExtREAL
KPAbsFMF . 0 is ext-real V38() real Element of REAL
(E2 . 0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((E2 . 0) to_power x)) is ext-real Element of ExtREAL
R_EAL ((E2 . 0) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((E2 . 0) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((E2 . 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((E2 . 0) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((E2 . 0) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((E2 . 0) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((E2 . 0) to_power x))))) - (integral+ (M,(max- (R_EAL ((E2 . 0) to_power x))))) is ext-real Element of ExtREAL
abs (E2 . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E2 . 0)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (E2 . 0)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E2 . 0)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E2 . 0)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E2 . 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E2 . 0)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E2 . 0)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E2 . 0)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E2 . 0)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E2 . 0)) to_power x))))) is ext-real Element of ExtREAL
abs (abs (E . 0)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (abs (E . 0))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (abs (E . 0))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (abs (E . 0))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (abs (E . 0))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (abs (E . 0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (abs (E . 0))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (abs (E . 0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (abs (E . 0))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (abs (E . 0))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (abs (E . 0))) to_power x))))) is ext-real Element of ExtREAL
QAbsFMF . 0 is ext-real V38() real Element of REAL
(Partial_Sums PAbsFMF) . 0 is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . 0) is ext-real V38() real Element of REAL
PAbsFMF . 0 is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + (PAbsFMF . 0) is ext-real V38() real Element of REAL
(KPAbsFMF . 0) to_power (1 / x) is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((KPAbsFMF . 0) to_power (1 / x)) is ext-real V38() real Element of REAL
(Ir . 0) to_power (1 / x) is ext-real V38() real Element of REAL
Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ir . Br is ext-real V38() real Element of REAL
(Ir . Br) to_power (1 / x) is ext-real V38() real Element of REAL
QAbsFMF . Br is ext-real V38() real Element of REAL
Br + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
PAbsFMF . (Br + 1) is ext-real V38() real Element of REAL
((Ir . Br) to_power (1 / x)) + (PAbsFMF . (Br + 1)) is ext-real V38() real Element of REAL
(QAbsFMF . Br) + (PAbsFMF . (Br + 1)) is ext-real V38() real Element of REAL
G . (Br + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums E2) . (Br + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums E2) . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 . (Br + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums E2) . Br) + (E2 . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + (((Partial_Sums E2) . Br) + (E2 . (Br + 1))) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . Br) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((abs (E . 0)) + ((Partial_Sums E2) . Br)) + (E2 . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
G . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . Br) + (E2 . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
KPAbsFMF . (Br + 1) is ext-real V38() real Element of REAL
(E2 . (Br + 1)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((E2 . (Br + 1)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((E2 . (Br + 1)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((E2 . (Br + 1)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((E2 . (Br + 1)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((E2 . (Br + 1)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((E2 . (Br + 1)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((E2 . (Br + 1)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((E2 . (Br + 1)) to_power x))))) - (integral+ (M,(max- (R_EAL ((E2 . (Br + 1)) to_power x))))) is ext-real Element of ExtREAL
abs (E2 . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E2 . (Br + 1))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (E2 . (Br + 1))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (E2 . (Br + 1))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (E2 . (Br + 1))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (E2 . (Br + 1))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (E2 . (Br + 1))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (E2 . (Br + 1))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (E2 . (Br + 1))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (E2 . (Br + 1))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (E2 . (Br + 1))) to_power x))))) is ext-real Element of ExtREAL
(KPAbsFMF . (Br + 1)) to_power (1 / x) is ext-real V38() real Element of REAL
Gp . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(Gp . Br)) is ext-real Element of ExtREAL
R_EAL (Gp . Br) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (Gp . Br))) is ext-real Element of ExtREAL
max+ (R_EAL (Gp . Br)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (Gp . Br)))) is ext-real Element of ExtREAL
max- (R_EAL (Gp . Br)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (Gp . Br)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (Gp . Br))))) - (integral+ (M,(max- (R_EAL (Gp . Br))))) is ext-real Element of ExtREAL
Ir . (Br + 1) is ext-real V38() real Element of REAL
Gp . (Br + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(Gp . (Br + 1))) is ext-real Element of ExtREAL
R_EAL (Gp . (Br + 1)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (Gp . (Br + 1)))) is ext-real Element of ExtREAL
max+ (R_EAL (Gp . (Br + 1))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (Gp . (Br + 1))))) is ext-real Element of ExtREAL
max- (R_EAL (Gp . (Br + 1))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (Gp . (Br + 1))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (Gp . (Br + 1)))))) - (integral+ (M,(max- (R_EAL (Gp . (Br + 1)))))) is ext-real Element of ExtREAL
(G . Br) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((G . Br) to_power x)) is ext-real Element of ExtREAL
R_EAL ((G . Br) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((G . Br) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((G . Br) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((G . Br) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((G . Br) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((G . Br) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((G . Br) to_power x))))) - (integral+ (M,(max- (R_EAL ((G . Br) to_power x))))) is ext-real Element of ExtREAL
(G . (Br + 1)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((G . (Br + 1)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((G . (Br + 1)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((G . (Br + 1)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((G . (Br + 1)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((G . (Br + 1)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((G . (Br + 1)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((G . (Br + 1)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((G . (Br + 1)) to_power x))))) - (integral+ (M,(max- (R_EAL ((G . (Br + 1)) to_power x))))) is ext-real Element of ExtREAL
abs (G . Br) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (G . Br)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (G . Br)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (G . Br)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (G . Br)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (G . Br)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (G . Br)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (G . Br)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (G . Br)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (G . Br)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (G . Br)) to_power x))))) is ext-real Element of ExtREAL
abs ((G . Br) + (E2 . (Br + 1))) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ((G . Br) + (E2 . (Br + 1)))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs ((G . Br) + (E2 . (Br + 1)))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs ((G . Br) + (E2 . (Br + 1)))) to_power x))))) is ext-real Element of ExtREAL
(Ir . (Br + 1)) to_power (1 / x) is ext-real V38() real Element of REAL
(Partial_Sums PAbsFMF) . Br is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . Br) is ext-real V38() real Element of REAL
((RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . Br)) + (PAbsFMF . (Br + 1)) is ext-real V38() real Element of REAL
((Partial_Sums PAbsFMF) . Br) + (PAbsFMF . (Br + 1)) is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + (((Partial_Sums PAbsFMF) . Br) + (PAbsFMF . (Br + 1))) is ext-real V38() real Element of REAL
(Partial_Sums PAbsFMF) . (Br + 1) is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . (Br + 1)) is ext-real V38() real Element of REAL
QAbsFMF . (Br + 1) is ext-real V38() real Element of REAL
Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ir . Br is ext-real V38() real Element of REAL
(Ir . Br) to_power (1 / x) is ext-real V38() real Element of REAL
QAbsFMF . Br is ext-real V38() real Element of REAL
Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
PAbsFMF . Br is ext-real V38() real Element of REAL
Br + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND . (Br + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . (Br + 1)) is Element of the carrier of (X,S,M,x)
ND . Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . Br) is Element of the carrier of (X,S,M,x)
(g . (ND . (Br + 1))) - (g . (ND . Br)) is Element of the carrier of (X,S,M,x)
- (g . (ND . Br)) is Element of the carrier of (X,S,M,x)
(g . (ND . (Br + 1))) + (- (g . (ND . Br))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . ((g . (ND . (Br + 1))),(- (g . (ND . Br)))) is Element of the carrier of (X,S,M,x)
||.((g . (ND . (Br + 1))) - (g . (ND . Br))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . ((g . (ND . (Br + 1))) - (g . (ND . Br))) is ext-real V38() real Element of REAL
E . (Br + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a . (ND . (Br + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a . (ND . Br) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E1 . Br is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (E1 . Br) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(a . (ND . (Br + 1))) - (a . (ND . Br)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (a . (ND . Br)) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (a . (ND . Br)) is V1() V4(X) Function-like V163() V164() V165() set
(a . (ND . (Br + 1))) + (- (a . (ND . Br))) is V1() V4(X) Function-like V163() V164() V165() set
abs ((a . (ND . (Br + 1))) - (a . (ND . Br))) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(- 1) (#) (a . (ND . Br)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(- 1) * (g . (ND . Br)) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . ((- 1),(g . (ND . Br))) is set
(g . (ND . (Br + 1))) + ((- 1) * (g . (ND . Br))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . ((g . (ND . (Br + 1))),((- 1) * (g . (ND . Br)))) is Element of the carrier of (X,S,M,x)
(abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs ((a . (ND . (Br + 1))) - (a . (ND . Br)))) to_power x))))) is ext-real Element of ExtREAL
KPAbsFMF . Br is ext-real V38() real Element of REAL
(KPAbsFMF . Br) to_power (1 / x) is ext-real V38() real Element of REAL
E0 is ext-real V38() real Element of REAL
E0 to_power (1 / x) is ext-real V38() real Element of REAL
1 / 2 is non empty ext-real positive non negative V38() real Element of REAL
2 " is non empty ext-real positive non negative V38() real set
1 * (2 ") is non empty ext-real positive non negative V38() real set
abs (1 / 2) is ext-real V38() real Element of REAL
(1 / 2) GeoSeq is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Sum ((1 / 2) GeoSeq) is ext-real V38() real Element of REAL
1 - (1 / 2) is ext-real V38() real Element of REAL
- (1 / 2) is non empty ext-real non positive negative V38() real set
1 + (- (1 / 2)) is ext-real V38() real set
1 / (1 - (1 / 2)) is ext-real V38() real Element of REAL
(1 - (1 / 2)) " is ext-real V38() real set
1 * ((1 - (1 / 2)) ") is ext-real V38() real set
Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
PAbsFMF . Br is ext-real V38() real Element of REAL
((1 / 2) GeoSeq) . Br is ext-real V38() real Element of REAL
Br + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND . (Br + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . (Br + 1)) is Element of the carrier of (X,S,M,x)
ND . Br is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . Br) is Element of the carrier of (X,S,M,x)
(g . (ND . (Br + 1))) - (g . (ND . Br)) is Element of the carrier of (X,S,M,x)
- (g . (ND . Br)) is Element of the carrier of (X,S,M,x)
(g . (ND . (Br + 1))) + (- (g . (ND . Br))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . ((g . (ND . (Br + 1))),(- (g . (ND . Br)))) is Element of the carrier of (X,S,M,x)
||.((g . (ND . (Br + 1))) - (g . (ND . Br))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . ((g . (ND . (Br + 1))) - (g . (ND . Br))) is ext-real V38() real Element of REAL
(1 / 2) |^ Br is ext-real V38() real Element of REAL
(1 / 2) to_power Br is ext-real V38() real Element of REAL
- Br is ext-real non positive V38() real Element of REAL
2 to_power (- Br) is ext-real V38() real Element of REAL
Sum PAbsFMF is ext-real V38() real Element of REAL
Br is ext-real V38() real set
(RF0 to_power (1 / x)) + Br is ext-real V38() real Element of REAL
((RF0 to_power (1 / x)) + Br) to_power x is ext-real V38() real Element of REAL
LExtGp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ir . LExtGp is ext-real V38() real Element of REAL
(Ir . LExtGp) to_power (1 / x) is ext-real V38() real Element of REAL
QAbsFMF . LExtGp is ext-real V38() real Element of REAL
(Partial_Sums PAbsFMF) . LExtGp is ext-real V38() real Element of REAL
(RF0 to_power (1 / x)) + ((Partial_Sums PAbsFMF) . LExtGp) is ext-real V38() real Element of REAL
Gp . LExtGp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(Gp . LExtGp)) is ext-real Element of ExtREAL
R_EAL (Gp . LExtGp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (Gp . LExtGp))) is ext-real Element of ExtREAL
max+ (R_EAL (Gp . LExtGp)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (Gp . LExtGp)))) is ext-real Element of ExtREAL
max- (R_EAL (Gp . LExtGp)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (Gp . LExtGp)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (Gp . LExtGp))))) - (integral+ (M,(max- (R_EAL (Gp . LExtGp))))) is ext-real Element of ExtREAL
G . LExtGp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . LExtGp) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,((G . LExtGp) to_power x)) is ext-real Element of ExtREAL
R_EAL ((G . LExtGp) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((G . LExtGp) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((G . LExtGp) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((G . LExtGp) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((G . LExtGp) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((G . LExtGp) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((G . LExtGp) to_power x))))) - (integral+ (M,(max- (R_EAL ((G . LExtGp) to_power x))))) is ext-real Element of ExtREAL
abs (G . LExtGp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (G . LExtGp)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs (G . LExtGp)) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (G . LExtGp)) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (G . LExtGp)) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (G . LExtGp)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (G . LExtGp)) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (G . LExtGp)) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (G . LExtGp)) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (G . LExtGp)) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (G . LExtGp)) to_power x))))) is ext-real Element of ExtREAL
((Ir . LExtGp) to_power (1 / x)) to_power x is ext-real V38() real Element of REAL
(1 / x) * x is non empty ext-real positive non negative V38() real Element of REAL
(Ir . LExtGp) to_power ((1 / x) * x) is ext-real V38() real Element of REAL
(Ir . LExtGp) to_power 1 is ext-real V38() real Element of REAL
LExtGp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
EE is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ir . LExtGp is ext-real V38() real Element of REAL
Ir . EE is ext-real V38() real Element of REAL
ExtGp . LExtGp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
ExtGp . EE is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
E0 is Element of X
(ExtGp . LExtGp) . E0 is ext-real set
(ExtGp . EE) . E0 is ext-real set
dom (ExtGp . LExtGp) is Element of bool X
dom (ExtGp . EE) is Element of bool X
(ExtGp . LExtGp) | b is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(ExtGp . EE) | b is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
I . LExtGp is ext-real Element of ExtREAL
Integral (M,(ExtGp . LExtGp)) is ext-real Element of ExtREAL
max+ (ExtGp . LExtGp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (ExtGp . LExtGp))) is ext-real Element of ExtREAL
max- (ExtGp . LExtGp) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (ExtGp . LExtGp))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (ExtGp . LExtGp)))) - (integral+ (M,(max- (ExtGp . LExtGp)))) is ext-real Element of ExtREAL
I . EE is ext-real Element of ExtREAL
Integral (M,(ExtGp . EE)) is ext-real Element of ExtREAL
max+ (ExtGp . EE) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (ExtGp . EE))) is ext-real Element of ExtREAL
max- (ExtGp . EE) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (ExtGp . EE))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (ExtGp . EE)))) - (integral+ (M,(max- (ExtGp . EE)))) is ext-real Element of ExtREAL
lim Ir is ext-real V38() real Element of REAL
LExtGp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom LExtGp is Element of bool X
EE is set
LExtGp . EE is ext-real set
E0 is Element of X
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
ExtGp # E0 is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
(ExtGp # E0) . 0 is ext-real Element of ExtREAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(ExtGp # E0) . F2 is ext-real Element of ExtREAL
ExtGp . 0 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(ExtGp . 0) . E0 is ext-real set
(ExtGp # E0) . F2 is ext-real Element of ExtREAL
lim (ExtGp # E0) is ext-real Element of ExtREAL
eq_dom (LExtGp,+infty) is Element of bool X
{+infty} is V157() set
LExtGp " {+infty} is set
b /\ (eq_dom (LExtGp,+infty)) is Element of bool X
EE is Element of S
b \ EE is Element of S
E0 is Element of S
E0 ` is Element of bool X
X \ E0 is set
X \ b is Element of bool X
X /\ EE is Element of bool X
(X \ b) \/ (X /\ EE) is Element of bool X
b ` is Element of bool X
X \ b is set
(b `) \/ EE is Element of bool X
M . EE is ext-real Element of ExtREAL
F2 is Element of S
F2 ` is Element of bool X
X \ F2 is set
M . (F2 `) is ext-real set
M . (E0 `) is ext-real set
F2 is Element of X
LExtGp . F2 is ext-real set
F2 is Element of X
Gp # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (Gp # F2) is ext-real V38() real Element of REAL
ExtGp # F2 is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim (ExtGp # F2) is ext-real Element of ExtREAL
LExtGp . F2 is ext-real set
F2 is ext-real V38() real set
F2 is Element of X
Gp # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
G # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
G . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . F2) is Element of bool X
(G . F2) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((G . F2) to_power x) is Element of bool X
(Gp # F2) . F2 is ext-real V38() real Element of REAL
Gp . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Gp . F2) . F2 is ext-real V38() real Element of REAL
((G . F2) to_power x) . F2 is ext-real V38() real Element of REAL
(G . F2) . F2 is ext-real V38() real Element of REAL
((G . F2) . F2) to_power x is ext-real V38() real Element of REAL
(G # F2) . F2 is ext-real V38() real Element of REAL
((G # F2) . F2) to_power x is ext-real V38() real Element of REAL
F2 is Element of X
(Partial_Sums E2) # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Gp # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
G . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . F2) . F2 is ext-real V38() real Element of REAL
G # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(G # F2) . F2 is ext-real V38() real Element of REAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Gp # F2) . F2 is ext-real V38() real Element of REAL
(G # F2) . F2 is ext-real V38() real Element of REAL
((G # F2) . F2) to_power x is ext-real V38() real Element of REAL
F2 is ext-real V38() real set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(G # F2) . F is ext-real V38() real Element of REAL
gp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
G . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . F) is Element of bool X
G . gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . gp) is Element of bool X
(Partial_Sums E2) . gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums E2) . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . F) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . gp) . F2 is ext-real V38() real Element of REAL
(abs (E . 0)) . F2 is ext-real V38() real Element of REAL
((Partial_Sums E2) . gp) . F2 is ext-real V38() real Element of REAL
((abs (E . 0)) . F2) + (((Partial_Sums E2) . gp) . F2) is ext-real V38() real Element of REAL
(G . F) . F2 is ext-real V38() real Element of REAL
((Partial_Sums E2) . F) . F2 is ext-real V38() real Element of REAL
((abs (E . 0)) . F2) + (((Partial_Sums E2) . F) . F2) is ext-real V38() real Element of REAL
(G # F2) . gp is ext-real V38() real Element of REAL
((G # F2) . gp) - ((G # F2) . F) is ext-real V38() real Element of REAL
- ((G # F2) . F) is ext-real V38() real set
((G # F2) . gp) + (- ((G # F2) . F)) is ext-real V38() real set
(((Partial_Sums E2) . gp) . F2) - (((Partial_Sums E2) . F) . F2) is ext-real V38() real Element of REAL
- (((Partial_Sums E2) . F) . F2) is ext-real V38() real set
(((Partial_Sums E2) . gp) . F2) + (- (((Partial_Sums E2) . F) . F2)) is ext-real V38() real set
((Partial_Sums E2) # F2) . gp is ext-real V38() real Element of REAL
((Partial_Sums E2) # F2) . F is ext-real V38() real Element of REAL
(((Partial_Sums E2) # F2) . gp) - (((Partial_Sums E2) # F2) . F) is ext-real V38() real Element of REAL
- (((Partial_Sums E2) # F2) . F) is ext-real V38() real set
(((Partial_Sums E2) # F2) . gp) + (- (((Partial_Sums E2) # F2) . F)) is ext-real V38() real set
abs ((((Partial_Sums E2) # F2) . gp) - (((Partial_Sums E2) # F2) . F)) is ext-real V38() real Element of REAL
F2 is Element of X
E1 # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
abs (E1 # F2) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
Partial_Sums (abs (E1 # F2)) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums E2) # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (abs (E1 # F2))) . 0 is ext-real V38() real Element of REAL
(abs (E1 # F2)) . 0 is ext-real V38() real Element of REAL
(E1 # F2) . 0 is ext-real V38() real Element of REAL
abs ((E1 # F2) . 0) is ext-real V38() real Element of REAL
E1 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E1 . 0) . F2 is ext-real V38() real Element of REAL
abs ((E1 . 0) . F2) is ext-real V38() real Element of REAL
abs (E1 . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E1 . 0)) . F2 is ext-real V38() real Element of REAL
E2 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E2 . 0) . F2 is ext-real V38() real Element of REAL
(Partial_Sums E2) . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums E2) . 0) . F2 is ext-real V38() real Element of REAL
((Partial_Sums E2) # F2) . 0 is ext-real V38() real Element of REAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums (abs (E1 # F2))) . F2 is ext-real V38() real Element of REAL
((Partial_Sums E2) # F2) . F2 is ext-real V38() real Element of REAL
F2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums E2) . (F2 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Partial_Sums E2) . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E2 . (F2 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((Partial_Sums E2) . F2) + (E2 . (F2 + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E2 . 0) is Element of bool X
dom ((Partial_Sums E2) . (F2 + 1)) is Element of bool X
(abs (E1 # F2)) . (F2 + 1) is ext-real V38() real Element of REAL
(E1 # F2) . (F2 + 1) is ext-real V38() real Element of REAL
abs ((E1 # F2) . (F2 + 1)) is ext-real V38() real Element of REAL
E1 . (F2 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E1 . (F2 + 1)) . F2 is ext-real V38() real Element of REAL
abs ((E1 . (F2 + 1)) . F2) is ext-real V38() real Element of REAL
abs (E1 . (F2 + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E1 . (F2 + 1))) . F2 is ext-real V38() real Element of REAL
(E2 . (F2 + 1)) . F2 is ext-real V38() real Element of REAL
(Partial_Sums (abs (E1 # F2))) . (F2 + 1) is ext-real V38() real Element of REAL
((Partial_Sums (abs (E1 # F2))) . F2) + ((abs (E1 # F2)) . (F2 + 1)) is ext-real V38() real Element of REAL
((Partial_Sums E2) . F2) . F2 is ext-real V38() real Element of REAL
(((Partial_Sums E2) . F2) . F2) + ((E2 . (F2 + 1)) . F2) is ext-real V38() real Element of REAL
((Partial_Sums E2) . (F2 + 1)) . F2 is ext-real V38() real Element of REAL
((Partial_Sums E2) # F2) . (F2 + 1) is ext-real V38() real Element of REAL
F2 is Element of X
E # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(E # F2) . 0 is ext-real V38() real Element of REAL
E1 # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # F2) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E1 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E1 . 0) is Element of bool X
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E . (0 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . (0 + 1)) - (E . 0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (E . 0) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (E . 0) is V1() V4(X) Function-like V163() V164() V165() set
(E . (0 + 1)) + (- (E . 0)) is V1() V4(X) Function-like V163() V164() V165() set
dom ((E . (0 + 1)) - (E . 0)) is Element of bool X
(Partial_Sums (E1 # F2)) . 0 is ext-real V38() real Element of REAL
(E1 # F2) . 0 is ext-real V38() real Element of REAL
(E1 . 0) . F2 is ext-real V38() real Element of REAL
((E . (0 + 1)) - (E . 0)) . F2 is ext-real V38() real Element of REAL
(E . (0 + 1)) . F2 is ext-real V38() real Element of REAL
(E . 0) . F2 is ext-real V38() real Element of REAL
((E . (0 + 1)) . F2) - ((E . 0) . F2) is ext-real V38() real Element of REAL
- ((E . 0) . F2) is ext-real V38() real set
((E . (0 + 1)) . F2) + (- ((E . 0) . F2)) is ext-real V38() real set
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . 0) is ext-real V38() real Element of REAL
((E . 0) . F2) + ((Partial_Sums (E1 # F2)) . 0) is ext-real V38() real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # F2) . (0 + 1) is ext-real V38() real Element of REAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
F2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # F2) . (F2 + 1) is ext-real V38() real Element of REAL
(Partial_Sums (E1 # F2)) . F2 is ext-real V38() real Element of REAL
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . F2) is ext-real V38() real Element of REAL
F2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E1 . (F2 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (E1 . (F2 + 1)) is Element of bool X
(F2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E . ((F2 + 1) + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
E . (F2 + 1) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . ((F2 + 1) + 1)) - (E . (F2 + 1)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (E . (F2 + 1)) is V1() V4(X) Function-like V163() V164() V165() set
(- 1) (#) (E . (F2 + 1)) is V1() V4(X) Function-like V163() V164() V165() set
(E . ((F2 + 1) + 1)) + (- (E . (F2 + 1))) is V1() V4(X) Function-like V163() V164() V165() set
dom ((E . ((F2 + 1) + 1)) - (E . (F2 + 1))) is Element of bool X
(E1 # F2) . (F2 + 1) is ext-real V38() real Element of REAL
(E1 . (F2 + 1)) . F2 is ext-real V38() real Element of REAL
((E . ((F2 + 1) + 1)) - (E . (F2 + 1))) . F2 is ext-real V38() real Element of REAL
(E . ((F2 + 1) + 1)) . F2 is ext-real V38() real Element of REAL
(E . (F2 + 1)) . F2 is ext-real V38() real Element of REAL
((E . ((F2 + 1) + 1)) . F2) - ((E . (F2 + 1)) . F2) is ext-real V38() real Element of REAL
- ((E . (F2 + 1)) . F2) is ext-real V38() real set
((E . ((F2 + 1) + 1)) . F2) + (- ((E . (F2 + 1)) . F2)) is ext-real V38() real set
(Partial_Sums (E1 # F2)) . (F2 + 1) is ext-real V38() real Element of REAL
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . (F2 + 1)) is ext-real V38() real Element of REAL
((Partial_Sums (E1 # F2)) . F2) + ((E1 # F2) . (F2 + 1)) is ext-real V38() real Element of REAL
((E # F2) . 0) + (((Partial_Sums (E1 # F2)) . F2) + ((E1 # F2) . (F2 + 1))) is ext-real V38() real Element of REAL
(((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . F2)) + ((E1 # F2) . (F2 + 1)) is ext-real V38() real Element of REAL
((E . (F2 + 1)) . F2) + ((E1 # F2) . (F2 + 1)) is ext-real V38() real Element of REAL
(F2 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # F2) . ((F2 + 1) + 1) is ext-real V38() real Element of REAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
F2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # F2) . (F2 + 1) is ext-real V38() real Element of REAL
(Partial_Sums (E1 # F2)) . F2 is ext-real V38() real Element of REAL
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . F2) is ext-real V38() real Element of REAL
F2 is Element of X
E # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E1 # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
abs (E1 # F2) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
Partial_Sums (abs (E1 # F2)) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums E2) # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # F2) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F2 is ext-real V38() real set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums (E1 # F2)) . F is ext-real V38() real Element of REAL
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
FP - 1 is ext-real V38() real Element of REAL
- 1 is non empty ext-real non positive negative V38() real set
FP + (- 1) is ext-real V38() real set
(F + 1) - 1 is ext-real V38() real Element of REAL
(F + 1) + (- 1) is ext-real V38() real set
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums (E1 # F2)) . FP is ext-real V38() real Element of REAL
((Partial_Sums (E1 # F2)) . FP) - ((Partial_Sums (E1 # F2)) . F) is ext-real V38() real Element of REAL
- ((Partial_Sums (E1 # F2)) . F) is ext-real V38() real set
((Partial_Sums (E1 # F2)) . FP) + (- ((Partial_Sums (E1 # F2)) . F)) is ext-real V38() real set
abs (((Partial_Sums (E1 # F2)) . FP) - ((Partial_Sums (E1 # F2)) . F)) is ext-real V38() real Element of REAL
FP + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # F2) . (F + 1) is ext-real V38() real Element of REAL
(E # F2) . 0 is ext-real V38() real Element of REAL
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . F) is ext-real V38() real Element of REAL
(E # F2) . FP is ext-real V38() real Element of REAL
((E # F2) . 0) + ((Partial_Sums (E1 # F2)) . FP) is ext-real V38() real Element of REAL
((E # F2) . FP) - ((E # F2) . (F + 1)) is ext-real V38() real Element of REAL
- ((E # F2) . (F + 1)) is ext-real V38() real set
((E # F2) . FP) + (- ((E # F2) . (F + 1))) is ext-real V38() real set
abs (((E # F2) . FP) - ((E # F2) . (F + 1))) is ext-real V38() real Element of REAL
E || E0 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
F2 is Element of X
(E || E0) # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F2 is Element of X
(E || E0) # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E # F2 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
((E || E0) # F2) . F is ext-real V38() real Element of REAL
(E || E0) . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((E || E0) . F) . F2 is ext-real V38() real Element of REAL
E . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . F) | E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
((E . F) | E0) . F2 is ext-real V38() real Element of REAL
(E . F) . F2 is ext-real V38() real Element of REAL
(E # F2) . F is ext-real V38() real Element of REAL
F2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(E || E0) . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((E || E0) . F2) is Element of bool X
dom (E . 0) is Element of bool X
E . F2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . F2) | E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((E . F2) | E0) is Element of bool X
dom (E . F2) is Element of bool X
(dom (E . F2)) /\ E0 is Element of bool X
b /\ E0 is Element of S
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
E . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F2 is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
F2 . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . F) is Element of the carrier of (X,S,M,x)
E . F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . F) | E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs (F2 . F) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
abs (E . F) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (E . F)) | E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . F)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
((abs (E . F)) to_power x) | E0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (F2 . F)) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (F2 . F) is Element of bool X
gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
FP is Element of S
FP ` is Element of bool X
X \ FP is set
M . (FP `) is ext-real set
dom gp is Element of bool X
abs gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs gp) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
gp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND . gp is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
E . gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a . (ND . gp) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
g . (ND . gp) is Element of the carrier of (X,S,M,x)
(X,S,M,(E . gp),x) is Element of bool (X,S,M,x)
bool (X,S,M,x) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & E . gp a.e.= b₁,M ) } is set
F2 . gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
FP is Element of S
FP ` is Element of bool X
X \ FP is set
(F2 . gp) | (FP `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(E . gp) | (FP `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
lim F2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
R_EAL F2 is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (R_EAL F2) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (lim F2) is Element of bool X
F2 . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (F2 . 0) is Element of bool X
F is Element of X
(lim F2) . F is ext-real set
F2 # F is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (F2 # F) is ext-real V38() real Element of REAL
R_EAL (F2 # F) is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim (R_EAL (F2 # F)) is ext-real Element of ExtREAL
F is set
rng (lim F2) is V157() Element of bool ExtREAL
gp is Element of X
(lim F2) . gp is ext-real set
F2 # gp is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (F2 # gp) is ext-real V38() real Element of REAL
LExtGp | E0 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (LExtGp | E0) is Element of bool X
b /\ E0 is Element of S
gp is set
rng (LExtGp | E0) is V157() Element of bool ExtREAL
FP is Element of X
(LExtGp | E0) . FP is ext-real set
LExtGp . FP is ext-real set
gp is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
FP is Element of X
gp . FP is ext-real V38() real Element of REAL
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (Gp # FP) is ext-real V38() real Element of REAL
LExtGp . FP is ext-real set
ExtGp # FP is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim (ExtGp # FP) is ext-real Element of ExtREAL
R_EAL gp is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom F is Element of bool X
abs F is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (abs F) is Element of bool X
(abs F) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom ((abs F) to_power x) is Element of bool X
FP is Element of X
E # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(E # FP) . 0 is ext-real V38() real Element of REAL
abs ((E # FP) . 0) is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
G # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums (E1 # FP)) . FP is ext-real V38() real Element of REAL
abs ((Partial_Sums (E1 # FP)) . FP) is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + (abs ((Partial_Sums (E1 # FP)) . FP)) is ext-real V38() real Element of REAL
(G # FP) . FP is ext-real V38() real Element of REAL
G . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . FP) is Element of bool X
(Partial_Sums E2) . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . FP) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . FP) . FP is ext-real V38() real Element of REAL
(abs (E . 0)) . FP is ext-real V38() real Element of REAL
((Partial_Sums E2) . FP) . FP is ext-real V38() real Element of REAL
((abs (E . 0)) . FP) + (((Partial_Sums E2) . FP) . FP) is ext-real V38() real Element of REAL
(E . 0) . FP is ext-real V38() real Element of REAL
abs ((E . 0) . FP) is ext-real V38() real Element of REAL
(abs ((E . 0) . FP)) + (((Partial_Sums E2) . FP) . FP) is ext-real V38() real Element of REAL
(Partial_Sums E2) # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
((Partial_Sums E2) # FP) . FP is ext-real V38() real Element of REAL
(abs ((E . 0) . FP)) + (((Partial_Sums E2) # FP) . FP) is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + (((Partial_Sums E2) # FP) . FP) is ext-real V38() real Element of REAL
abs (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
Partial_Sums (abs (E1 # FP)) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (abs (E1 # FP))) . FP is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + ((Partial_Sums (abs (E1 # FP))) . FP) is ext-real V38() real Element of REAL
FP is Element of X
E # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(E # FP) . 0 is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Partial_Sums (E1 # FP)) . FP is ext-real V38() real Element of REAL
((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . FP) is ext-real V38() real Element of REAL
abs (((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . FP)) is ext-real V38() real Element of REAL
(abs (((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . FP))) to_power x is ext-real V38() real Element of REAL
(Gp # FP) . FP is ext-real V38() real Element of REAL
G # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(G # FP) . FP is ext-real V38() real Element of REAL
((G # FP) . FP) to_power x is ext-real V38() real Element of REAL
abs ((E # FP) . 0) is ext-real V38() real Element of REAL
abs ((Partial_Sums (E1 # FP)) . FP) is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + (abs ((Partial_Sums (E1 # FP)) . FP)) is ext-real V38() real Element of REAL
FP is Element of X
F2 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(F2 # FP) . FP is ext-real V38() real Element of REAL
abs ((F2 # FP) . FP) is ext-real V38() real Element of REAL
(abs ((F2 # FP) . FP)) to_power x is ext-real V38() real Element of REAL
(Gp # FP) . FP is ext-real V38() real Element of REAL
E # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
G # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(G # FP) . FP is ext-real V38() real Element of REAL
((G # FP) . FP) to_power x is ext-real V38() real Element of REAL
(E # FP) . 0 is ext-real V38() real Element of REAL
abs ((E # FP) . 0) is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (E1 # FP)) . FP is ext-real V38() real Element of REAL
abs ((Partial_Sums (E1 # FP)) . FP) is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + (abs ((Partial_Sums (E1 # FP)) . FP)) is ext-real V38() real Element of REAL
0 + (abs ((E # FP) . 0)) is ext-real V38() real Element of REAL
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Ip + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Sq0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Sq0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # FP) . (Sq0 + 1) is ext-real V38() real Element of REAL
(E # FP) . 0 is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (E1 # FP)) . Sq0 is ext-real V38() real Element of REAL
((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . Sq0) is ext-real V38() real Element of REAL
abs ((E # FP) . (Sq0 + 1)) is ext-real V38() real Element of REAL
(abs ((E # FP) . (Sq0 + 1))) to_power x is ext-real V38() real Element of REAL
(Gp # FP) . Sq0 is ext-real V38() real Element of REAL
ExtGp # FP is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
(ExtGp # FP) . Sq0 is ext-real Element of ExtREAL
(ExtGp # FP) . (Sq0 + 1) is ext-real Element of ExtREAL
FP is Element of X
((abs F) to_power x) . FP is ext-real V38() real Element of REAL
abs (((abs F) to_power x) . FP) is ext-real V38() real Element of REAL
gp . FP is ext-real V38() real Element of REAL
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
F2 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
abs (F2 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (Gp # FP) is ext-real V38() real Element of REAL
(abs F) . FP is ext-real V38() real Element of REAL
((abs F) . FP) to_power x is ext-real V38() real Element of REAL
F . FP is ext-real V38() real Element of REAL
abs (F . FP) is ext-real V38() real Element of REAL
(abs (F . FP)) to_power x is ext-real V38() real Element of REAL
lim (F2 # FP) is ext-real V38() real Element of REAL
abs (lim (F2 # FP)) is ext-real V38() real Element of REAL
(abs (lim (F2 # FP))) to_power x is ext-real V38() real Element of REAL
lim (abs (F2 # FP)) is ext-real V38() real Element of REAL
(lim (abs (F2 # FP))) to_power x is ext-real V38() real Element of REAL
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(F2 # FP) . Ip is ext-real V38() real Element of REAL
abs ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
(abs (F2 # FP)) . Ip is ext-real V38() real Element of REAL
lim FP is ext-real V38() real Element of REAL
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(F2 # FP) . Ip is ext-real V38() real Element of REAL
abs ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
(abs ((F2 # FP) . Ip)) to_power x is ext-real V38() real Element of REAL
(Gp # FP) . Ip is ext-real V38() real Element of REAL
(abs (F2 # FP)) . Ip is ext-real V38() real Element of REAL
((abs (F2 # FP)) . Ip) to_power x is ext-real V38() real Element of REAL
FP . Ip is ext-real V38() real Element of REAL
R_EAL F is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
FP is Element of X
E # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # FP) . FP is ext-real V38() real Element of REAL
(E # FP) . Ip is ext-real V38() real Element of REAL
((E # FP) . FP) - ((E # FP) . Ip) is ext-real V38() real Element of REAL
- ((E # FP) . Ip) is ext-real V38() real set
((E # FP) . FP) + (- ((E # FP) . Ip)) is ext-real V38() real set
|.(((E # FP) . FP) - ((E # FP) . Ip)).| is ext-real V38() real Element of REAL
|.(((E # FP) . FP) - ((E # FP) . Ip)).| to_power x is ext-real V38() real Element of REAL
(Gp # FP) . FP is ext-real V38() real Element of REAL
G . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(G . FP) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Gp . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Gp . FP) . FP is ext-real V38() real Element of REAL
Sq0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Sq0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # FP) . (Iq + 1) is ext-real V38() real Element of REAL
(E # FP) . 0 is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (E1 # FP)) . Iq is ext-real V38() real Element of REAL
((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . Iq) is ext-real V38() real Element of REAL
abs ((E # FP) . 0) is ext-real V38() real Element of REAL
abs ((Partial_Sums (E1 # FP)) . Iq) is ext-real V38() real Element of REAL
(abs ((E # FP) . 0)) + (abs ((Partial_Sums (E1 # FP)) . Iq)) is ext-real V38() real Element of REAL
G # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(G # FP) . Iq is ext-real V38() real Element of REAL
(abs ((Partial_Sums (E1 # FP)) . Iq)) + 0 is ext-real V38() real Element of REAL
|.((Partial_Sums (E1 # FP)) . Iq).| is ext-real V38() real Element of REAL
|.((Partial_Sums (E1 # FP)) . Iq).| to_power x is ext-real V38() real Element of REAL
((G # FP) . Iq) to_power x is ext-real V38() real Element of REAL
(Gp # FP) . Iq is ext-real V38() real Element of REAL
ExtGp # FP is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
(ExtGp # FP) . Iq is ext-real Element of ExtREAL
(ExtGp # FP) . (Iq + 1) is ext-real Element of ExtREAL
Sq0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Sq0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ip - 1 is ext-real V38() real Element of REAL
- 1 is non empty ext-real non positive negative V38() real set
Ip + (- 1) is ext-real V38() real set
FP - 1 is ext-real V38() real Element of REAL
FP + (- 1) is ext-real V38() real set
n0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
G . n0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (G . n0) is Element of bool X
(G . n0) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom ((G . n0) to_power x) is Element of bool X
(Gp # FP) . n0 is ext-real V38() real Element of REAL
Gp . n0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(Gp . n0) . FP is ext-real V38() real Element of REAL
((G . n0) to_power x) . FP is ext-real V38() real Element of REAL
(G . n0) . FP is ext-real V38() real Element of REAL
((G . n0) . FP) to_power x is ext-real V38() real Element of REAL
(Partial_Sums E2) . n0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) + ((Partial_Sums E2) . n0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(abs (E . 0)) . FP is ext-real V38() real Element of REAL
((Partial_Sums E2) . n0) . FP is ext-real V38() real Element of REAL
((abs (E . 0)) . FP) + (((Partial_Sums E2) . n0) . FP) is ext-real V38() real Element of REAL
(E . 0) . FP is ext-real V38() real Element of REAL
abs ((E . 0) . FP) is ext-real V38() real Element of REAL
(abs ((E . 0) . FP)) + (((Partial_Sums E2) . n0) . FP) is ext-real V38() real Element of REAL
(Partial_Sums E2) # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
((Partial_Sums E2) # FP) . n0 is ext-real V38() real Element of REAL
(abs ((E . 0) . FP)) + (((Partial_Sums E2) # FP) . n0) is ext-real V38() real Element of REAL
n0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # FP) . (n0 + 1) is ext-real V38() real Element of REAL
(E # FP) . 0 is ext-real V38() real Element of REAL
E1 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Partial_Sums (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (E1 # FP)) . n0 is ext-real V38() real Element of REAL
((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . n0) is ext-real V38() real Element of REAL
Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural ext-real positive non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(E # FP) . (Iq + 1) is ext-real V38() real Element of REAL
(Partial_Sums (E1 # FP)) . Iq is ext-real V38() real Element of REAL
((E # FP) . 0) + ((Partial_Sums (E1 # FP)) . Iq) is ext-real V38() real Element of REAL
((Partial_Sums (E1 # FP)) . n0) - ((Partial_Sums (E1 # FP)) . Iq) is ext-real V38() real Element of REAL
- ((Partial_Sums (E1 # FP)) . Iq) is ext-real V38() real set
((Partial_Sums (E1 # FP)) . n0) + (- ((Partial_Sums (E1 # FP)) . Iq)) is ext-real V38() real set
|.(((Partial_Sums (E1 # FP)) . n0) - ((Partial_Sums (E1 # FP)) . Iq)).| is ext-real V38() real Element of REAL
abs (E1 # FP) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
Partial_Sums (abs (E1 # FP)) is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(Partial_Sums (abs (E1 # FP))) . n0 is ext-real V38() real Element of REAL
0 + ((Partial_Sums (abs (E1 # FP))) . n0) is ext-real V38() real Element of REAL
((E # FP) . (n0 + 1)) - ((E # FP) . (Iq + 1)) is ext-real V38() real Element of REAL
- ((E # FP) . (Iq + 1)) is ext-real V38() real set
((E # FP) . (n0 + 1)) + (- ((E # FP) . (Iq + 1))) is ext-real V38() real set
|.(((E # FP) . (n0 + 1)) - ((E # FP) . (Iq + 1))).| is ext-real V38() real Element of REAL
|.(((E # FP) . (n0 + 1)) - ((E # FP) . (Iq + 1))).| to_power x is ext-real V38() real Element of REAL
ExtGp # FP is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
(ExtGp # FP) . n0 is ext-real Element of ExtREAL
(ExtGp # FP) . (n0 + 1) is ext-real Element of ExtREAL
FP is Element of X
F . FP is ext-real V38() real Element of REAL
F2 # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
gp . FP is ext-real V38() real Element of REAL
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
(F2 # FP) . FP is ext-real V38() real Element of REAL
(F . FP) - ((F2 # FP) . FP) is ext-real V38() real Element of REAL
- ((F2 # FP) . FP) is ext-real V38() real set
(F . FP) + (- ((F2 # FP) . FP)) is ext-real V38() real set
|.((F . FP) - ((F2 # FP) . FP)).| is ext-real V38() real Element of REAL
|.((F . FP) - ((F2 # FP) . FP)).| to_power x is ext-real V38() real Element of REAL
Gp # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
E # FP is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(F2 # FP) . Ip is ext-real V38() real Element of REAL
Sq0 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Iq is ext-real V38() real set
lim (F2 # FP) is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
n0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(lim (F2 # FP)) - ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
- ((F2 # FP) . Ip) is ext-real V38() real set
(lim (F2 # FP)) + (- ((F2 # FP) . Ip)) is ext-real V38() real set
r is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Sq0 . r is ext-real V38() real Element of REAL
(Sq0 . r) - ((lim (F2 # FP)) - ((F2 # FP) . Ip)) is ext-real V38() real Element of REAL
- ((lim (F2 # FP)) - ((F2 # FP) . Ip)) is ext-real V38() real set
(Sq0 . r) + (- ((lim (F2 # FP)) - ((F2 # FP) . Ip))) is ext-real V38() real set
abs ((Sq0 . r) - ((lim (F2 # FP)) - ((F2 # FP) . Ip))) is ext-real V38() real Element of REAL
(F2 # FP) . r is ext-real V38() real Element of REAL
((F2 # FP) . r) - ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
((F2 # FP) . r) + (- ((F2 # FP) . Ip)) is ext-real V38() real set
(((F2 # FP) . r) - ((F2 # FP) . Ip)) - ((lim (F2 # FP)) - ((F2 # FP) . Ip)) is ext-real V38() real Element of REAL
(((F2 # FP) . r) - ((F2 # FP) . Ip)) + (- ((lim (F2 # FP)) - ((F2 # FP) . Ip))) is ext-real V38() real set
((F2 # FP) . r) - (lim (F2 # FP)) is ext-real V38() real Element of REAL
- (lim (F2 # FP)) is ext-real V38() real set
((F2 # FP) . r) + (- (lim (F2 # FP))) is ext-real V38() real set
lim Sq0 is ext-real V38() real Element of REAL
abs Sq0 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() nonnegative Element of bool [:NAT,REAL:]
lim (abs Sq0) is ext-real V38() real Element of REAL
abs ((lim (F2 # FP)) - ((F2 # FP) . Ip)) is ext-real V38() real Element of REAL
Iq is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq . n is ext-real V38() real Element of REAL
(Gp # FP) . n is ext-real V38() real Element of REAL
(F2 # FP) . n is ext-real V38() real Element of REAL
((F2 # FP) . n) - ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
((F2 # FP) . n) + (- ((F2 # FP) . Ip)) is ext-real V38() real set
|.(((F2 # FP) . n) - ((F2 # FP) . Ip)).| is ext-real V38() real Element of REAL
|.(((F2 # FP) . n) - ((F2 # FP) . Ip)).| to_power x is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Sq0 . n is ext-real V38() real Element of REAL
abs (Sq0 . n) is ext-real V38() real Element of REAL
(abs Sq0) . n is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq . n is ext-real V38() real Element of REAL
(F2 # FP) . n is ext-real V38() real Element of REAL
((F2 # FP) . n) - ((F2 # FP) . Ip) is ext-real V38() real Element of REAL
((F2 # FP) . n) + (- ((F2 # FP) . Ip)) is ext-real V38() real set
|.(((F2 # FP) . n) - ((F2 # FP) . Ip)).| is ext-real V38() real Element of REAL
|.(((F2 # FP) . n) - ((F2 # FP) . Ip)).| to_power x is ext-real V38() real Element of REAL
Sq0 . n is ext-real V38() real Element of REAL
abs (Sq0 . n) is ext-real V38() real Element of REAL
(abs (Sq0 . n)) to_power x is ext-real V38() real Element of REAL
(abs Sq0) . n is ext-real V38() real Element of REAL
((abs Sq0) . n) to_power x is ext-real V38() real Element of REAL
lim Iq is ext-real V38() real Element of REAL
(lim (abs Sq0)) to_power x is ext-real V38() real Element of REAL
Iq ^\ Ip is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (Iq ^\ Ip) is ext-real V38() real Element of REAL
lim (Gp # FP) is ext-real V38() real Element of REAL
(Gp # FP) ^\ Ip is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim ((Gp # FP) ^\ Ip) is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(Iq ^\ Ip) . n is ext-real V38() real Element of REAL
((Gp # FP) ^\ Ip) . n is ext-real V38() real Element of REAL
Ip + n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq . (Ip + n) is ext-real V38() real Element of REAL
(Gp # FP) . (Ip + n) is ext-real V38() real Element of REAL
FP is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) Element of bool [:NAT,(PFuncs (X,REAL)):]
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
FP . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . FP) is Element of bool X
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
F2 . Ip is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (F2 . Ip) is Element of bool X
F - (F2 . Ip) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . Ip) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (F2 . Ip) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . Ip)) is V1() V4(X) Function-like V163() V164() V165() set
abs (F - (F2 . Ip)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (F - (F2 . Ip))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom ((abs (F - (F2 . Ip))) to_power x) is Element of bool X
dom (abs (F - (F2 . Ip))) is Element of bool X
dom (F - (F2 . Ip)) is Element of bool X
E0 /\ E0 is Element of S
FP . 0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . 0) is Element of bool X
lim FP is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
R_EAL FP is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (R_EAL FP) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (lim FP) is Element of bool X
FP is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
FP . FP is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . FP) is Element of bool X
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
FP . Ip is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . Ip) is Element of bool X
FP is V1() V4( NAT ) V5( PFuncs (X,REAL)) Function-like non empty total V41( NAT , PFuncs (X,REAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,REAL)):]
Ip is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
FP . Ip is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Sq0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
F2 . Sq0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (F2 . Sq0) is Element of bool X
F - (F2 . Sq0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . Sq0) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (F2 . Sq0) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . Sq0)) is V1() V4(X) Function-like V163() V164() V165() set
dom (F - (F2 . Sq0)) is Element of bool X
E0 /\ E0 is Element of S
abs (F - (F2 . Sq0)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (abs (F - (F2 . Sq0))) is Element of bool X
(abs (F - (F2 . Sq0))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Ip is Element of X
gp . Ip is ext-real V38() real Element of REAL
Sq0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
FP . Sq0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
|.(FP . Sq0).| is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
|.(FP . Sq0).| . Ip is ext-real V38() real Element of REAL
Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
FP . Iq is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . Iq) is Element of bool X
F2 . Iq is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F - (F2 . Iq) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . Iq) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (F2 . Iq) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . Iq)) is V1() V4(X) Function-like V163() V164() V165() set
|.(F - (F2 . Iq)).| is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
|.(F - (F2 . Iq)).| to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (|.(F - (F2 . Iq)).| to_power x) is Element of bool X
dom |.(F - (F2 . Iq)).| is Element of bool X
dom (F - (F2 . Iq)) is Element of bool X
F2 . Sq0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F - (F2 . Sq0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . Sq0) is V1() V4(X) Function-like V163() V164() V165() set
(- 1) (#) (F2 . Sq0) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . Sq0)) is V1() V4(X) Function-like V163() V164() V165() set
|.(F - (F2 . Sq0)).| is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
|.(F - (F2 . Sq0)).| to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
F . Ip is ext-real V38() real Element of REAL
(F2 . Sq0) . Ip is ext-real V38() real Element of REAL
(F . Ip) - ((F2 . Sq0) . Ip) is ext-real V38() real Element of REAL
- ((F2 . Sq0) . Ip) is ext-real V38() real set
(F . Ip) + (- ((F2 . Sq0) . Ip)) is ext-real V38() real set
|.((F . Ip) - ((F2 . Sq0) . Ip)).| is ext-real V38() real Element of REAL
|.((F . Ip) - ((F2 . Sq0) . Ip)).| to_power x is ext-real V38() real Element of REAL
|.(FP . Iq).| is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
|.(FP . Iq).| . Ip is ext-real V38() real Element of REAL
(FP . Iq) . Ip is ext-real V38() real Element of REAL
|.((FP . Iq) . Ip).| is ext-real V38() real Element of REAL
|.(F - (F2 . Sq0)).| . Ip is ext-real V38() real Element of REAL
(|.(F - (F2 . Sq0)).| . Ip) to_power x is ext-real V38() real Element of REAL
|.((|.(F - (F2 . Sq0)).| . Ip) to_power x).| is ext-real V38() real Element of REAL
(F - (F2 . Sq0)) . Ip is ext-real V38() real Element of REAL
|.((F - (F2 . Sq0)) . Ip).| is ext-real V38() real Element of REAL
|.((F - (F2 . Sq0)) . Ip).| to_power x is ext-real V38() real Element of REAL
|.(|.((F - (F2 . Sq0)) . Ip).| to_power x).| is ext-real V38() real Element of REAL
|.(|.((F . Ip) - ((F2 . Sq0) . Ip)).| to_power x).| is ext-real V38() real Element of REAL
F2 # Ip is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(F2 # Ip) . Iq is ext-real V38() real Element of REAL
(F . Ip) - ((F2 # Ip) . Iq) is ext-real V38() real Element of REAL
- ((F2 # Ip) . Iq) is ext-real V38() real set
(F . Ip) + (- ((F2 # Ip) . Iq)) is ext-real V38() real set
|.((F . Ip) - ((F2 # Ip) . Iq)).| is ext-real V38() real Element of REAL
|.((F . Ip) - ((F2 # Ip) . Iq)).| to_power x is ext-real V38() real Element of REAL
lim FP is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
R_EAL FP is V1() V4( NAT ) V5( PFuncs (X,ExtREAL)) Function-like non empty total V41( NAT , PFuncs (X,ExtREAL)) with_the_same_dom Element of bool [:NAT,(PFuncs (X,ExtREAL)):]
lim (R_EAL FP) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(lim FP)) is ext-real Element of ExtREAL
max+ (lim FP) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (lim FP))) is ext-real Element of ExtREAL
max- (lim FP) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (lim FP))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (lim FP)))) - (integral+ (M,(max- (lim FP)))) is ext-real Element of ExtREAL
Ip is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim Ip is ext-real V38() real Element of REAL
Sq0 is Element of X
FP # Sq0 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (FP # Sq0) is ext-real V38() real Element of REAL
F2 # Sq0 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (F2 # Sq0) is ext-real V38() real Element of REAL
Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
(FP # Sq0) . Iq is ext-real V38() real Element of REAL
(F2 # Sq0) . Iq is ext-real V38() real Element of REAL
(lim (F2 # Sq0)) - ((F2 # Sq0) . Iq) is ext-real V38() real Element of REAL
- ((F2 # Sq0) . Iq) is ext-real V38() real set
(lim (F2 # Sq0)) + (- ((F2 # Sq0) . Iq)) is ext-real V38() real set
|.((lim (F2 # Sq0)) - ((F2 # Sq0) . Iq)).| is ext-real V38() real Element of REAL
|.((lim (F2 # Sq0)) - ((F2 # Sq0) . Iq)).| to_power x is ext-real V38() real Element of REAL
FP . Iq is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom (FP . Iq) is Element of bool X
F2 . Iq is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F - (F2 . Iq) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . Iq) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (F2 . Iq) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . Iq)) is V1() V4(X) Function-like V163() V164() V165() set
|.(F - (F2 . Iq)).| is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
|.(F - (F2 . Iq)).| to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
dom (|.(F - (F2 . Iq)).| to_power x) is Element of bool X
dom |.(F - (F2 . Iq)).| is Element of bool X
dom (F - (F2 . Iq)) is Element of bool X
(FP . Iq) . Sq0 is ext-real V38() real Element of REAL
(|.(F - (F2 . Iq)).| to_power x) . Sq0 is ext-real V38() real Element of REAL
|.(F - (F2 . Iq)).| . Sq0 is ext-real V38() real Element of REAL
(|.(F - (F2 . Iq)).| . Sq0) to_power x is ext-real V38() real Element of REAL
(F - (F2 . Iq)) . Sq0 is ext-real V38() real Element of REAL
|.((F - (F2 . Iq)) . Sq0).| is ext-real V38() real Element of REAL
|.((F - (F2 . Iq)) . Sq0).| to_power x is ext-real V38() real Element of REAL
F . Sq0 is ext-real V38() real Element of REAL
(F2 . Iq) . Sq0 is ext-real V38() real Element of REAL
(F . Sq0) - ((F2 . Iq) . Sq0) is ext-real V38() real Element of REAL
- ((F2 . Iq) . Sq0) is ext-real V38() real set
(F . Sq0) + (- ((F2 . Iq) . Sq0)) is ext-real V38() real set
|.((F . Sq0) - ((F2 . Iq) . Sq0)).| is ext-real V38() real Element of REAL
|.((F . Sq0) - ((F2 . Iq) . Sq0)).| to_power x is ext-real V38() real Element of REAL
(lim (F2 # Sq0)) - ((F2 . Iq) . Sq0) is ext-real V38() real Element of REAL
(lim (F2 # Sq0)) + (- ((F2 . Iq) . Sq0)) is ext-real V38() real set
|.((lim (F2 # Sq0)) - ((F2 . Iq) . Sq0)).| is ext-real V38() real Element of REAL
|.((lim (F2 # Sq0)) - ((F2 . Iq) . Sq0)).| to_power x is ext-real V38() real Element of REAL
dom (lim FP) is Element of bool X
Sq0 is Element of X
(lim FP) . Sq0 is ext-real set
FP # Sq0 is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
lim (FP # Sq0) is ext-real V38() real Element of REAL
R_EAL (FP # Sq0) is V1() V4( NAT ) V5( ExtREAL ) Function-like non empty total V41( NAT , ExtREAL ) V164() Element of bool [:NAT,ExtREAL:]
lim (R_EAL (FP # Sq0)) is ext-real Element of ExtREAL
(X,S,M,F,x) is Element of bool (X,S,M,x)
bool (X,S,M,x) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,x) & F a.e.= b₁,M ) } is set
(X,S,M,x) is non empty Element of bool (bool (X,S,M,x))
bool (bool (X,S,M,x)) is non empty set
{ (X,S,M,b₁,x) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,x) } is set
Sq0 is Element of the carrier of (X,S,M,x)
Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ip . Iq is ext-real V38() real Element of REAL
ND . Iq is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . Iq) is Element of the carrier of (X,S,M,x)
Sq0 - (g . (ND . Iq)) is Element of the carrier of (X,S,M,x)
- (g . (ND . Iq)) is Element of the carrier of (X,S,M,x)
Sq0 + (- (g . (ND . Iq))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (Sq0,(- (g . (ND . Iq)))) is Element of the carrier of (X,S,M,x)
||.(Sq0 - (g . (ND . Iq))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (Sq0 - (g . (ND . Iq))) is ext-real V38() real Element of REAL
||.(Sq0 - (g . (ND . Iq))).|| to_power x is ext-real V38() real Element of REAL
FP . Iq is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
n0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
F2 . n0 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
F - (F2 . n0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
- (F2 . n0) is V1() V4(X) Function-like V163() V164() V165() set
- 1 is non empty ext-real non positive negative V38() real set
(- 1) (#) (F2 . n0) is V1() V4(X) Function-like V163() V164() V165() set
F + (- (F2 . n0)) is V1() V4(X) Function-like V163() V164() V165() set
abs (F - (F2 . n0)) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs (F - (F2 . n0))) to_power x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(- 1) (#) (F2 . n0) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(- 1) * (g . (ND . Iq)) is Element of the carrier of (X,S,M,x)
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) . ((- 1),(g . (ND . Iq))) is set
Sq0 + ((- 1) * (g . (ND . Iq))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Sq0,((- 1) * (g . (ND . Iq)))) is Element of the carrier of (X,S,M,x)
Integral (M,((abs (F - (F2 . n0))) to_power x)) is ext-real Element of ExtREAL
R_EAL ((abs (F - (F2 . n0))) to_power x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs (F - (F2 . n0))) to_power x))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs (F - (F2 . n0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs (F - (F2 . n0))) to_power x)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs (F - (F2 . n0))) to_power x)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs (F - (F2 . n0))) to_power x)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs (F - (F2 . n0))) to_power x))))) - (integral+ (M,(max- (R_EAL ((abs (F - (F2 . n0))) to_power x))))) is ext-real Element of ExtREAL
r is ext-real V38() real Element of REAL
r to_power (1 / x) is ext-real V38() real Element of REAL
(1 / x) * x is non empty ext-real positive non negative V38() real Element of REAL
r to_power ((1 / x) * x) is ext-real V38() real Element of REAL
r to_power 1 is ext-real V38() real Element of REAL
c₄₁ is Element of X
FP # c₄₁ is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
Iq is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real set
Iq . n is ext-real V38() real Element of REAL
n0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
ND . n0 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . n0) is Element of the carrier of (X,S,M,x)
Sq0 - (g . (ND . n0)) is Element of the carrier of (X,S,M,x)
- (g . (ND . n0)) is Element of the carrier of (X,S,M,x)
Sq0 + (- (g . (ND . n0))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the addF of (X,S,M,x) . (Sq0,(- (g . (ND . n0)))) is Element of the carrier of (X,S,M,x)
||.(Sq0 - (g . (ND . n0))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
the U9 of (X,S,M,x) . (Sq0 - (g . (ND . n0))) is ext-real V38() real Element of REAL
ND . n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . n) is Element of the carrier of (X,S,M,x)
Sq0 - (g . (ND . n)) is Element of the carrier of (X,S,M,x)
- (g . (ND . n)) is Element of the carrier of (X,S,M,x)
Sq0 + (- (g . (ND . n))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Sq0,(- (g . (ND . n)))) is Element of the carrier of (X,S,M,x)
||.(Sq0 - (g . (ND . n))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) . (Sq0 - (g . (ND . n))) is ext-real V38() real Element of REAL
lim Iq is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Ip . n is ext-real V38() real Element of REAL
ND . n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . n) is Element of the carrier of (X,S,M,x)
Sq0 - (g . (ND . n)) is Element of the carrier of (X,S,M,x)
- (g . (ND . n)) is Element of the carrier of (X,S,M,x)
Sq0 + (- (g . (ND . n))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Sq0,(- (g . (ND . n)))) is Element of the carrier of (X,S,M,x)
||.(Sq0 - (g . (ND . n))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) . (Sq0 - (g . (ND . n))) is ext-real V38() real Element of REAL
||.(Sq0 - (g . (ND . n))).|| to_power x is ext-real V38() real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
Iq . n is ext-real V38() real Element of REAL
Ip . n is ext-real V38() real Element of REAL
(Ip . n) to_power (1 / x) is ext-real V38() real Element of REAL
ND . n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V38() real V104() V156() V157() V158() V159() V160() V161() V194() Element of NAT
g . (ND . n) is Element of the carrier of (X,S,M,x)
Sq0 - (g . (ND . n)) is Element of the carrier of (X,S,M,x)
- (g . (ND . n)) is Element of the carrier of (X,S,M,x)
Sq0 + (- (g . (ND . n))) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) . (Sq0,(- (g . (ND . n)))) is Element of the carrier of (X,S,M,x)
||.(Sq0 - (g . (ND . n))).|| is ext-real non negative V38() real Element of REAL
the U9 of (X,S,M,x) . (Sq0 - (g . (ND . n))) is ext-real V38() real Element of REAL
||.(Sq0 - (g . (ND . n))).|| to_power x is ext-real V38() real Element of REAL
(||.(Sq0 - (g . (ND . n))).|| to_power x) to_power (1 / x) is ext-real V38() real Element of REAL
x * (1 / x) is non empty ext-real positive non negative V38() real Element of REAL
||.(Sq0 - (g . (ND . n))).|| to_power (x * (1 / x)) is ext-real V38() real Element of REAL
||.(Sq0 - (g . (ND . n))).|| to_power 1 is ext-real V38() real Element of REAL
n is Element of X
FP # n is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
(lim Ip) to_power (1 / x) is ext-real V38() real Element of REAL
n is Element of X
FP # n is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
0 to_power (1 / x) is ext-real V38() real Element of REAL
n is Element of X
FP # n is V1() V4( NAT ) V5( REAL ) Function-like non empty total V41( NAT , REAL ) V163() V164() V165() Element of bool [:NAT,REAL:]
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is non empty ext-real positive non negative V38() real () Element of REAL
(X,S,M,x) is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed discerning reflexive RealNormSpace-like NORMSTR
(X,S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,x) is non empty set
the ZeroF of (X,S,M,x) is Element of the carrier of (X,S,M,x)
the addF of (X,S,M,x) is V1() V4([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):] is non empty set
[:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[: the carrier of (X,S,M,x), the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
the Mult of (X,S,M,x) is V1() V4([:REAL, the carrier of (X,S,M,x):]) V5( the carrier of (X,S,M,x)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x)) Element of bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):]
[:REAL, the carrier of (X,S,M,x):] is non empty set
[:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,x):], the carrier of (X,S,M,x):] is non empty set
(X,S,M,x) is V1() V4( the carrier of (X,S,M,x)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,x), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,x),REAL:]
[: the carrier of (X,S,M,x),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,x), the ZeroF of (X,S,M,x), the addF of (X,S,M,x), the Mult of (X,S,M,x),(X,S,M,x) #) is strict NORMSTR
the carrier of (X,S,M,x) is non empty set
[:NAT, the carrier of (X,S,M,x):] is non empty set
bool [:NAT, the carrier of (X,S,M,x):] is non empty set
g is V1() V4( NAT ) V5( the carrier of (X,S,M,x)) Function-like non empty total V41( NAT , the carrier of (X,S,M,x)) Element of bool [:NAT, the carrier of (X,S,M,x):]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b<sub>1</sub> is_integrable_on M ) } is set
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a is Element of S
M . a is ext-real Element of ExtREAL
dom g is Element of bool X
a ` is Element of bool X
X \ a is set
g is Element of S
M . g is ext-real Element of ExtREAL
g ` is Element of bool X
X \ g is set
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
R_EAL x is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
dom (R_EAL x) is Element of bool X
b is Element of S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
g is non empty ext-real positive non negative V38() real Element of REAL
(X,S,M,g) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power g is_integrable_on M ) } is set
(abs x) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
(X,S,M,x,1) is Element of bool (X,S,M,1)
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & x a.e.= b<sub>1</sub>,M ) } is set
a.e-eq-class (x,M) is Element of bool (L1_Functions M)
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & x in L1_Functions M & x a.e.= b₁,M ) } is set
g is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom a is Element of bool X
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
dom b is Element of bool X
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
dom ((abs a) to_power 1) is Element of bool X
dom (abs a) is Element of bool X
ND is Element of X
((abs a) to_power 1) . ND is ext-real V38() real Element of REAL
(abs a) . ND is ext-real V38() real Element of REAL
((abs a) . ND) to_power 1 is ext-real V38() real Element of REAL
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
dom ND is Element of bool X
abs ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs ND) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is Element of S
M . ND is ext-real Element of ExtREAL
ND ` is Element of bool X
X \ ND is set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
Ef is Element of S
Ef ` is Element of bool X
X \ Ef is set
M . (Ef `) is ext-real set
E is Element of S
M . E is ext-real Element of ExtREAL
E ` is Element of bool X
X \ E is set
EE is Element of S
M . EE is ext-real Element of ExtREAL
EE ` is Element of bool X
X \ EE is set
x | (EE `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a | (EE `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND \/ EE is Element of S
E1 is Element of S
M . E1 is ext-real Element of ExtREAL
E1 ` is Element of bool X
X \ E1 is set
x | (E1 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(x | (EE `)) | (E1 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a | (E1 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(a | (EE `)) | (E1 `) is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
R_EAL x is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
max+ (R_EAL x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (max+ (R_EAL x)) is Element of bool X
max- (R_EAL x) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
dom (max- (R_EAL x)) is Element of bool X
dom (R_EAL x) is Element of bool X
E0 is Element of X
(max+ (R_EAL x)) . E0 is ext-real set
E2 is Element of X
(max- (R_EAL x)) . E2 is ext-real set
Ef /\ E1 is Element of S
Ef \ E1 is Element of S
(Ef /\ E1) \/ (Ef \ E1) is Element of S
Ef /\ (E1 `) is Element of bool X
(max+ (R_EAL x)) | (dom (max+ (R_EAL x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max- (R_EAL x)) | (dom (max- (R_EAL x))) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
E0 is Element of S
(max+ (R_EAL x)) | E0 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL x)) | E0)) is ext-real Element of ExtREAL
E2 is Element of S
(max+ (R_EAL x)) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL x)) | E2)) is ext-real Element of ExtREAL
(integral+ (M,((max+ (R_EAL x)) | E0))) + (integral+ (M,((max+ (R_EAL x)) | E2))) is ext-real Element of ExtREAL
integral+ (M,(max- (R_EAL x))) is ext-real Element of ExtREAL
(max- (R_EAL x)) | E0 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL x)) | E0)) is ext-real Element of ExtREAL
(max- (R_EAL x)) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL x)) | E2)) is ext-real Element of ExtREAL
(integral+ (M,((max- (R_EAL x)) | E0))) + (integral+ (M,((max- (R_EAL x)) | E2))) is ext-real Element of ExtREAL
(max+ (R_EAL x)) | E1 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max+ (R_EAL x)) | E1)) is ext-real Element of ExtREAL
(max- (R_EAL x)) | E1 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,((max- (R_EAL x)) | E1)) is ext-real Element of ExtREAL
x | E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(a | (E1 `)) | E2 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(R_EAL x) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
R_EAL a is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(R_EAL a) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ ((R_EAL x) | E2) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ (R_EAL a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max+ (R_EAL a)) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max+ ((R_EAL a) | E2) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- ((R_EAL x) | E2) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- (R_EAL a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
(max- (R_EAL a)) | E2 is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
max- ((R_EAL a) | E2) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL a))) is ext-real Element of ExtREAL
integral+ (M,(max- (R_EAL a))) is ext-real Element of ExtREAL
integral+ (M,(max+ ((R_EAL a) | E2))) is ext-real Element of ExtREAL
integral+ (M,(max- ((R_EAL a) | E2))) is ext-real Element of ExtREAL
Ef is Element of S
M . Ef is ext-real Element of ExtREAL
Ef ` is Element of bool X
X \ Ef is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (x,M) is Element of bool (L1_Functions M)
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & x in L1_Functions M & x a.e.= b₁,M ) } is set
(X,S,M,x,1) is Element of bool (X,S,M,1)
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & x a.e.= b<sub>1</sub>,M ) } is set
g is set
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
R_EAL a is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is Element of S
M . ND is ext-real Element of ExtREAL
dom b is Element of bool X
ND ` is Element of bool X
X \ ND is set
dom (R_EAL a) is Element of bool X
b is Element of S
dom a is Element of bool X
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
M . Ef is ext-real Element of ExtREAL
dom ND is Element of bool X
Ef ` is Element of bool X
X \ Ef is set
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs a) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Ef is Element of S
M . Ef is ext-real Element of ExtREAL
dom ND is Element of bool X
Ef ` is Element of bool X
X \ Ef is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
dom x is Element of bool X
(X,S,M,x,1) is Element of bool (X,S,M,1)
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & x a.e.= b<sub>1</sub>,M ) } is set
a.e-eq-class (x,M) is Element of bool (L1_Functions M)
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & x in L1_Functions M & x a.e.= b₁,M ) } is set
g is Element of S
g ` is Element of bool X
X \ g is set
M . (g `) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
CosetSet M is non empty Element of bool (bool (L1_Functions M))
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b<sub>1</sub> is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b<sub>1</sub>,M)) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in L1_Functions M } is set
(X,S,M,1) is non empty Element of bool (bool (X,S,M,1))
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . (b<sub>2</sub> `) = 0 & dom b₁ = b₂ & b₁ is_measurable_on b₂ & (abs b₁) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
bool (bool (X,S,M,1)) is non empty set
{ (X,S,M,b₁,1) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in (X,S,M,1) } is set
x is set
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (g,M) is Element of bool (L1_Functions M)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & g in L1_Functions M & g a.e.= b₁,M ) } is set
dom g is Element of bool X
(X,S,M,g,1) is Element of bool (X,S,M,1)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,1) & g a.e.= b₁,M ) } is set
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
x is set
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
(X,S,M,g,1) is Element of bool (X,S,M,1)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in (X,S,M,1) & g a.e.= b₁,M ) } is set
dom g is Element of bool X
a is Element of S
a ` is Element of bool X
X \ a is set
M . (a `) is ext-real set
a.e-eq-class (g,M) is Element of bool (L1_Functions M)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & g in L1_Functions M & g a.e.= b₁,M ) } is set
b is Element of S
M . b is ext-real Element of ExtREAL
b ` is Element of bool X
X \ b is set
abs g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs g) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
CosetSet M is non empty Element of bool (bool (L1_Functions M))
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b₁,M)) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in L1_Functions M } is set
[:(CosetSet M),(CosetSet M):] is non empty set
(X,S,M,1) is non empty Element of bool (bool (X,S,M,1))
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
bool (bool (X,S,M,1)) is non empty set
{ (X,S,M,b<sub>1</sub>,1) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,1) } is set
[:(X,S,M,1),(X,S,M,1):] is non empty set
addCoset M is V1() V4([:(CosetSet M),(CosetSet M):]) V5( CosetSet M) Function-like non empty total V41([:(CosetSet M),(CosetSet M):], CosetSet M) Element of bool [:[:(CosetSet M),(CosetSet M):],(CosetSet M):]
[:[:(CosetSet M),(CosetSet M):],(CosetSet M):] is non empty set
bool [:[:(CosetSet M),(CosetSet M):],(CosetSet M):] is non empty set
(X,S,M,1) is V1() V4([:(X,S,M,1),(X,S,M,1):]) V5((X,S,M,1)) Function-like non empty total V41([:(X,S,M,1),(X,S,M,1):],(X,S,M,1)) Element of bool [:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):]
[:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):] is non empty set
bool [:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):] is non empty set
x is Element of CosetSet M
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (a,M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & a in L1_Functions M & a a.e.= b<sub>1</sub>,M ) } is set
g is Element of CosetSet M
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (b,M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & b in L1_Functions M & b a.e.= b<sub>1</sub>,M ) } is set
(addCoset M) . (x,g) is Element of CosetSet M
a + b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class ((a + b),M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & a + b in L1_Functions M & a + b a.e.= b<sub>1</sub>,M ) } is set
dom (a + b) is Element of bool X
(X,S,M,(a + b),1) is Element of bool (X,S,M,1)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & a + b a.e.= b<sub>1</sub>,M ) } is set
ND is Element of S
ND ` is Element of bool X
X \ ND is set
M . (ND `) is ext-real set
(X,S,M,1) . (x,g) is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
zeroCoset M is Element of CosetSet M
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
CosetSet M is non empty Element of bool (bool (L1_Functions M))
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b₁,M)) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in L1_Functions M } is set
(X,S,M,1) is Element of (X,S,M,1)
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
(X,S,M,1) is non empty Element of bool (bool (X,S,M,1))
bool (bool (X,S,M,1)) is non empty set
{ (X,S,M,b<sub>1</sub>,1) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,1) } is set
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),1) is Element of bool (X,S,M,1)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & X --> 0 a.e.= b<sub>1</sub>,M ) } is set
dom (X --> 0) is Element of bool X
x is Element of CosetSet M
a.e-eq-class ((X --> 0),M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & X --> 0 in L1_Functions M & X --> 0 a.e.= b<sub>1</sub>,M ) } is set
g is Element of S
g ` is Element of bool X
X \ g is set
M . (g `) 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
CosetSet M is non empty Element of bool (bool (L1_Functions M))
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b₁,M)) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in L1_Functions M } is set
[:REAL,(CosetSet M):] is non empty set
(X,S,M,1) is non empty Element of bool (bool (X,S,M,1))
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
bool (bool (X,S,M,1)) is non empty set
{ (X,S,M,b<sub>1</sub>,1) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,1) } is set
[:REAL,(X,S,M,1):] is non empty set
lmultCoset M is V1() V4([:REAL,(CosetSet M):]) V5( CosetSet M) Function-like non empty total V41([:REAL,(CosetSet M):], CosetSet M) Element of bool [:[:REAL,(CosetSet M):],(CosetSet M):]
[:[:REAL,(CosetSet M):],(CosetSet M):] is non empty set
bool [:[:REAL,(CosetSet M):],(CosetSet M):] is non empty set
(X,S,M,1) is V1() V4([:REAL,(X,S,M,1):]) V5((X,S,M,1)) Function-like non empty total V41([:REAL,(X,S,M,1):],(X,S,M,1)) Element of bool [:[:REAL,(X,S,M,1):],(X,S,M,1):]
[:[:REAL,(X,S,M,1):],(X,S,M,1):] is non empty set
bool [:[:REAL,(X,S,M,1):],(X,S,M,1):] is non empty set
g is Element of CosetSet M
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (a,M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & a in L1_Functions M & a a.e.= b<sub>1</sub>,M ) } is set
x is ext-real V38() real Element of REAL
(lmultCoset M) . (x,g) is Element of CosetSet M
x (#) a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class ((x (#) a),M) is Element of bool (L1_Functions M)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in L1_Functions M & x (#) a in L1_Functions M & x (#) a a.e.= b<sub>1</sub>,M ) } is set
dom (x (#) a) is Element of bool X
(X,S,M,(x (#) a),1) is Element of bool (X,S,M,1)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & x (#) a a.e.= b<sub>1</sub>,M ) } is set
b is Element of S
b ` is Element of bool X
X \ b is set
M . (b `) is ext-real set
(X,S,M,1) . (x,g) is 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
Pre-L-Space M is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
(X,S,M,1) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (Pre-L-Space M) is non empty set
CosetSet M is non empty Element of bool (bool (L1_Functions M))
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b₁,M)) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in L1_Functions M } is set
[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):] is non empty set
[:(CosetSet M),(CosetSet M):] is non empty set
the addF of (Pre-L-Space M) is V1() V4([: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):]) V5( the carrier of (Pre-L-Space M)) Function-like non empty total V41([: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M)) Element of bool [:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):]
[:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
bool [:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
addCoset M is V1() V4([:(CosetSet M),(CosetSet M):]) V5( CosetSet M) Function-like non empty total V41([:(CosetSet M),(CosetSet M):], CosetSet M) Element of bool [:[:(CosetSet M),(CosetSet M):],(CosetSet M):]
[:[:(CosetSet M),(CosetSet M):],(CosetSet M):] is non empty set
bool [:[:(CosetSet M),(CosetSet M):],(CosetSet M):] is non empty set
0. (Pre-L-Space M) is V52( Pre-L-Space M) Element of the carrier of (Pre-L-Space M)
the ZeroF of (Pre-L-Space M) is Element of the carrier of (Pre-L-Space M)
zeroCoset M is Element of CosetSet M
[:REAL, the carrier of (Pre-L-Space M):] is non empty set
[:REAL,(CosetSet M):] is non empty set
the Mult of (Pre-L-Space M) is V1() V4([:REAL, the carrier of (Pre-L-Space M):]) V5( the carrier of (Pre-L-Space M)) Function-like non empty total V41([:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M)) Element of bool [:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):]
[:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
bool [:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
lmultCoset M is V1() V4([:REAL,(CosetSet M):]) V5( CosetSet M) Function-like non empty total V41([:REAL,(CosetSet M):], CosetSet M) Element of bool [:[:REAL,(CosetSet M):],(CosetSet M):]
[:[:REAL,(CosetSet M):],(CosetSet M):] is non empty set
bool [:[:REAL,(CosetSet M):],(CosetSet M):] is non empty set
(X,S,M,1) is non empty Element of bool (bool (X,S,M,1))
(X,S,M,1) is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . (b₂ `) = 0 & dom b<sub>1</sub> = b<sub>2</sub> & b<sub>1</sub> is_measurable_on b<sub>2</sub> & (abs b<sub>1</sub>) to_power 1 is_integrable_on M ) } is set
bool (X,S,M,1) is non empty set
bool (bool (X,S,M,1)) is non empty set
{ (X,S,M,b<sub>1</sub>,1) where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M,1) } is set
[:(X,S,M,1),(X,S,M,1):] is non empty set
(X,S,M,1) is V1() V4([:(X,S,M,1),(X,S,M,1):]) V5((X,S,M,1)) Function-like non empty total V41([:(X,S,M,1),(X,S,M,1):],(X,S,M,1)) Element of bool [:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):]
[:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):] is non empty set
bool [:[:(X,S,M,1),(X,S,M,1):],(X,S,M,1):] is non empty set
(X,S,M,1) is Element of (X,S,M,1)
X --> 0 is V1() V4(X) V5( REAL ) V5( RAT ) V5( INT ) Function-like V163() V164() V165() V166() Element of bool [:X,REAL:]
(X,S,M,(X --> 0),1) is Element of bool (X,S,M,1)
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M,1) & X --> 0 a.e.= b<sub>1</sub>,M ) } is set
[:REAL,(X,S,M,1):] is non empty set
(X,S,M,1) is V1() V4([:REAL,(X,S,M,1):]) V5((X,S,M,1)) Function-like non empty total V41([:REAL,(X,S,M,1):],(X,S,M,1)) Element of bool [:[:REAL,(X,S,M,1):],(X,S,M,1):]
[:[:REAL,(X,S,M,1):],(X,S,M,1):] is non empty set
bool [:[:REAL,(X,S,M,1):],(X,S,M,1):] is non empty 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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
Pre-L-Space M is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (Pre-L-Space M) is non empty set
(X,S,M,1) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,1) is non empty set
L-1-Norm M is V1() V4( the carrier of (Pre-L-Space M)) V5( REAL ) Function-like non empty total V41( the carrier of (Pre-L-Space M), REAL ) V163() V164() V165() Element of bool [: the carrier of (Pre-L-Space M),REAL:]
[: the carrier of (Pre-L-Space M),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (Pre-L-Space M),REAL:] is non empty set
(X,S,M,1) is V1() V4( the carrier of (X,S,M,1)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,1), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,1),REAL:]
[: the carrier of (X,S,M,1),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,1),REAL:] is non empty set
x is Element of the carrier of (Pre-L-Space M)
CosetSet M is non empty Element of bool (bool (L1_Functions M))
L1_Functions M is non empty add-closed multi-closed Element of bool the carrier of (RLSp_PFunct X)
RLSp_PFunct X is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
PFuncs (X,REAL) is functional non empty PartFunc-set of X, REAL
RealPFuncZero X is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of PFuncs (X,REAL)
K517(NAT,X,0) is V1() V4(X) V5( NAT ) V5( RAT ) V5( INT ) Function-like non empty total V41(X, NAT ) V163() V164() V165() V166() Element of bool [:X,NAT:]
[:X,NAT:] is V5( RAT ) V5( INT ) non empty V163() V164() V165() V166() set
bool [:X,NAT:] is non empty set
addpfunc X is V1() V4([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):] is non empty set
[:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
multrealpfunc X is V1() V4([:REAL,(PFuncs (X,REAL)):]) V5( PFuncs (X,REAL)) Function-like non empty total V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
[:REAL,(PFuncs (X,REAL)):] is non empty set
[:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):] is non empty set
RLSStruct(# (PFuncs (X,REAL)),(RealPFuncZero X),(addpfunc X),(multrealpfunc X) #) is non empty strict RLSStruct
the carrier of (RLSp_PFunct X) is non empty set
bool the carrier of (RLSp_PFunct X) is non empty set
[:X,REAL:] is non empty V163() V164() V165() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ex b<sub>2</sub> being Element of S st
( M . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on M ) } is set
bool (L1_Functions M) is non empty set
bool (bool (L1_Functions M)) is non empty set
{ (a.e-eq-class (b₁,M)) where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : b₁ in L1_Functions M } is set
g is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
a.e-eq-class (g,M) is Element of bool (L1_Functions M)
{ b₁ where b₁ is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:] : ( b₁ in L1_Functions M & g in L1_Functions M & g a.e.= b₁,M ) } is set
(L-1-Norm M) . x is ext-real V38() real Element of REAL
a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs a is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,(abs a)) is ext-real Element of ExtREAL
R_EAL (abs a) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V164() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs a))) is ext-real Element of ExtREAL
max+ (R_EAL (abs a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs a)))) is ext-real Element of ExtREAL
max- (R_EAL (abs a)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs a)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs a))))) - (integral+ (M,(max- (R_EAL (abs a))))) is ext-real Element of ExtREAL
(X,S,M,1) . x is ext-real V38() real Element of REAL
1 / 1 is non empty ext-real positive non negative V38() real Element of REAL
1 " is non empty ext-real positive non negative V38() real set
1 * (1 ") is non empty ext-real positive non negative V38() real set
b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
abs b is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
(abs b) to_power 1 is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() nonnegative Element of bool [:X,REAL:]
Integral (M,((abs b) to_power 1)) is ext-real Element of ExtREAL
R_EAL ((abs b) to_power 1) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs b) to_power 1))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs b) to_power 1)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs b) to_power 1)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs b) to_power 1)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs b) to_power 1)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs b) to_power 1))))) - (integral+ (M,(max- (R_EAL ((abs b) to_power 1))))) is ext-real Element of ExtREAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / 1) is ext-real V38() real Element of REAL
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
ND is V1() V4(X) V5( REAL ) Function-like V163() V164() V165() Element of bool [:X,REAL:]
Integral (M,(abs b)) is ext-real Element of ExtREAL
R_EAL (abs b) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs b))) is ext-real Element of ExtREAL
max+ (R_EAL (abs b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs b)))) is ext-real Element of ExtREAL
max- (R_EAL (abs b)) is V1() V4(X) V5( ExtREAL ) Function-like V164() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs b)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs b))))) - (integral+ (M,(max- (R_EAL (abs b))))) is ext-real Element of ExtREAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / 1) is ext-real V38() real Element of REAL
ND is ext-real V38() real Element of REAL
ND to_power (1 / 1) is ext-real V38() real Element of REAL
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 V152() V153() V154() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V164() set
bool [:S,ExtREAL:] is non empty set
M is V1() V4(S) V5( ExtREAL ) Function-like non empty total V41(S, ExtREAL ) V164() V172() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
L-1-Space M is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RealNormSpace-like NORMSTR
Pre-L-Space M is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (Pre-L-Space M) is non empty set
the ZeroF of (Pre-L-Space M) is Element of the carrier of (Pre-L-Space M)
the addF of (Pre-L-Space M) is V1() V4([: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):]) V5( the carrier of (Pre-L-Space M)) Function-like non empty total V41([: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M)) Element of bool [:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):]
[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):] is non empty set
[:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
bool [:[: the carrier of (Pre-L-Space M), the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
the Mult of (Pre-L-Space M) is V1() V4([:REAL, the carrier of (Pre-L-Space M):]) V5( the carrier of (Pre-L-Space M)) Function-like non empty total V41([:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M)) Element of bool [:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):]
[:REAL, the carrier of (Pre-L-Space M):] is non empty set
[:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
bool [:[:REAL, the carrier of (Pre-L-Space M):], the carrier of (Pre-L-Space M):] is non empty set
L-1-Norm M is V1() V4( the carrier of (Pre-L-Space M)) V5( REAL ) Function-like non empty total V41( the carrier of (Pre-L-Space M), REAL ) V163() V164() V165() Element of bool [: the carrier of (Pre-L-Space M),REAL:]
[: the carrier of (Pre-L-Space M),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (Pre-L-Space M),REAL:] is non empty set
NORMSTR(# the carrier of (Pre-L-Space M), the ZeroF of (Pre-L-Space M), the addF of (Pre-L-Space M), the Mult of (Pre-L-Space M),(L-1-Norm M) #) is strict NORMSTR
(X,S,M,1) is non empty NORMSTR
(X,S,M,1) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed RLSStruct
the carrier of (X,S,M,1) is non empty set
the ZeroF of (X,S,M,1) is Element of the carrier of (X,S,M,1)
the addF of (X,S,M,1) is V1() V4([: the carrier of (X,S,M,1), the carrier of (X,S,M,1):]) V5( the carrier of (X,S,M,1)) Function-like non empty total V41([: the carrier of (X,S,M,1), the carrier of (X,S,M,1):], the carrier of (X,S,M,1)) Element of bool [:[: the carrier of (X,S,M,1), the carrier of (X,S,M,1):], the carrier of (X,S,M,1):]
[: the carrier of (X,S,M,1), the carrier of (X,S,M,1):] is non empty set
[:[: the carrier of (X,S,M,1), the carrier of (X,S,M,1):], the carrier of (X,S,M,1):] is non empty set
bool [:[: the carrier of (X,S,M,1), the carrier of (X,S,M,1):], the carrier of (X,S,M,1):] is non empty set
the Mult of (X,S,M,1) is V1() V4([:REAL, the carrier of (X,S,M,1):]) V5( the carrier of (X,S,M,1)) Function-like non empty total V41([:REAL, the carrier of (X,S,M,1):], the carrier of (X,S,M,1)) Element of bool [:[:REAL, the carrier of (X,S,M,1):], the carrier of (X,S,M,1):]
[:REAL, the carrier of (X,S,M,1):] is non empty set
[:[:REAL, the carrier of (X,S,M,1):], the carrier of (X,S,M,1):] is non empty set
bool [:[:REAL, the carrier of (X,S,M,1):], the carrier of (X,S,M,1):] is non empty set
(X,S,M,1) is V1() V4( the carrier of (X,S,M,1)) V5( REAL ) Function-like non empty total V41( the carrier of (X,S,M,1), REAL ) V163() V164() V165() Element of bool [: the carrier of (X,S,M,1),REAL:]
[: the carrier of (X,S,M,1),REAL:] is non empty V163() V164() V165() set
bool [: the carrier of (X,S,M,1),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M,1), the ZeroF of (X,S,M,1), the addF of (X,S,M,1), the Mult of (X,S,M,1),(X,S,M,1) #) is strict NORMSTR