:: LPSPACE1 semantic presentation

REAL is non empty V29() V166() V167() V168() V172() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() Element of bool REAL
bool REAL is non empty set
ExtREAL is non empty V167() set
[:NAT,ExtREAL:] is non empty V174() set
bool [:NAT,ExtREAL:] is non empty set
bool ExtREAL is non empty set
COMPLEX is non empty V29() V166() V172() set
omega is non empty epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty V173() V174() V175() set
bool [:NAT,REAL:] is non empty set
bool (bool REAL) is non empty set
RAT is non empty V29() V166() V167() V168() V169() V172() set
INT is non empty V29() V166() V167() V168() V169() V170() V172() set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V166() V167() V168() V169() V170() V171() V172() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V108() V166() V167() V168() V169() V170() V171() V204() Element of NAT
{{},1} is non empty V166() V167() V168() V169() V170() V171() set
bool RAT is non empty set
[:COMPLEX,COMPLEX:] is non empty V173() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V173() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty V173() V174() V175() set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V173() V174() V175() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty V17( RAT ) V173() V174() V175() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty V17( RAT ) V173() V174() V175() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
[:[:NAT,NAT:],NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:[:NAT,NAT:],NAT:] is non empty set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V108() V166() V167() V168() V169() V170() V171() V172() V204() Element of NAT
+infty is non empty non real ext-real positive non negative Element of ExtREAL
-infty is non empty non real ext-real non positive negative Element of ExtREAL
0. is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V166() V167() V168() V169() V170() V171() V172() Element of ExtREAL
+infty is non empty non real ext-real positive non negative set
-infty is non empty non real ext-real non positive negative set
{-infty} is non empty V167() set
{+infty} is non empty V167() set
X is non empty RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
S is Element of bool the carrier of X
M is Element of the carrier of X
x is Element of the carrier of X
M + x is Element of the carrier of X
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 . (M,x) is Element of the carrier of X
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
the addF of X . [M,x] is set
a is Element of the carrier of X
y is V11() real ext-real Element of REAL
y * a is Element of the carrier of X
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (y,a) is set
[y,a] is set
{y,a} is non empty set
{y} is non empty V166() V167() V168() set
{{y,a},{y}} is non empty set
the Mult of X . [y,a] is set
M is Element of the carrier of X
x is Element of the carrier of X
M + x is Element of the carrier of X
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 . (M,x) is Element of the carrier of X
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
the addF of X . [M,x] is set
a is Element of the carrier of X
y is V11() real ext-real Element of REAL
y * a is Element of the carrier of X
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (y,a) is set
[y,a] is set
{y,a} is non empty set
{y} is non empty V166() V167() V168() set
{{y,a},{y}} is non empty set
the Mult of X . [y,a] is set
X is non empty RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
M is set
M is Element of bool the carrier of X
x is non empty Element of bool the carrier of X
a is Element of the carrier of X
y is V11() real ext-real Element of REAL
y * a is Element of the carrier of X
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (y,a) is set
[y,a] is set
{y,a} is non empty set
{y} is non empty V166() V167() V168() set
{{y,a},{y}} is non empty set
the Mult of X . [y,a] is set
f is Element of the carrier of X
g is Element of the carrier of X
f + g is Element of the carrier of X
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 . (f,g) is Element of the carrier of X
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of X . [f,g] is set
X is non empty RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
[:REAL, the carrier of X:] is non empty set
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
S is non empty (X) Element of bool the carrier of X
[:REAL,S:] is non empty set
the Mult of X | [:REAL,S:] is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL,S:],S:] is non empty set
bool [:[:REAL,S:],S:] is non empty set
dom the Mult of X is V13() V16( REAL ) V17( the carrier of X) Element of bool [:REAL, the carrier of X:]
bool [:REAL, the carrier of X:] is non empty set
M is set
x is set
y is set
[x,y] is set
{x,y} is non empty set
{x} is non empty set
{{x,y},{x}} is non empty set
a is V11() real ext-real Element of REAL
[a,y] is set
{a,y} is non empty set
{a} is non empty V166() V167() V168() set
{{a,y},{a}} is non empty set
dom ( the Mult of X | [:REAL,S:]) is V13() V16( REAL ) V17( the carrier of X) Element of bool [:REAL, the carrier of X:]
( the Mult of X | [:REAL,S:]) . M is set
f is Element of the carrier of X
a * f is Element of the carrier of X
the Mult of X . (a,f) is set
[a,f] is set
{a,f} is non empty set
{{a,f},{a}} is non empty set
the Mult of X . [a,f] is set
X is non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
0. X is V55(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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
[:REAL, the carrier of X:] is non empty set
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
S is non empty Element of bool the carrier of X
[:S,S:] is non empty set
[:[:S,S:],S:] is non empty set
bool [:[:S,S:],S:] is non empty set
[:REAL,S:] is non empty set
[:[:REAL,S:],S:] is non empty set
bool [:[:REAL,S:],S:] is non empty set
the addF of X || S is set
K65( the addF of X,[:S,S:]) is V13() set
the Mult of X | [:REAL,S:] is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
M is Element of S
x is V13() V16([:S,S:]) V17(S) Function-like V41([:S,S:],S) Element of bool [:[:S,S:],S:]
y is V13() V16([:REAL,S:]) V17(S) Function-like V41([:REAL,S:],S) Element of bool [:[:REAL,S:],S:]
RLSStruct(# S,M,x,y #) is non empty strict RLSStruct
the carrier of RLSStruct(# S,M,x,y #) is non empty set
f is Element of the carrier of RLSStruct(# S,M,x,y #)
g is V11() real ext-real Element of REAL
g * f is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) is V13() V16([:REAL, the carrier of RLSStruct(# S,M,x,y #):]) V17( the carrier of RLSStruct(# S,M,x,y #)) Function-like V41([:REAL, the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #)) Element of bool [:[:REAL, the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):]
[:REAL, the carrier of RLSStruct(# S,M,x,y #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):] is non empty set
the Mult of RLSStruct(# S,M,x,y #) . (g,f) is set
[g,f] is set
{g,f} is non empty set
{g} is non empty V166() V167() V168() set
{{g,f},{g}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [g,f] is set
[g,f] is Element of [:REAL, the carrier of RLSStruct(# S,M,x,y #):]
the Mult of X . [g,f] is set
the Mult of X . (g,f) is set
the Mult of X . [g,f] is set
g is Element of the carrier of RLSStruct(# S,M,x,y #)
f is Element of the carrier of RLSStruct(# S,M,x,y #)
g + f is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) is V13() V16([: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):]) V17( the carrier of RLSStruct(# S,M,x,y #)) Function-like V41([: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #)) Element of bool [:[: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):]
[: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):] is non empty set
[:[: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):] is non empty set
bool [:[: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):], the carrier of RLSStruct(# S,M,x,y #):] is non empty set
the addF of RLSStruct(# S,M,x,y #) . (g,f) is Element of the carrier of RLSStruct(# S,M,x,y #)
[g,f] is set
{g,f} is non empty set
{g} is non empty set
{{g,f},{g}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [g,f] is set
[g,f] is Element of [: the carrier of RLSStruct(# S,M,x,y #), the carrier of RLSStruct(# S,M,x,y #):]
the addF of X . [g,f] is set
the addF of X . (g,f) is set
the addF of X . [g,f] is set
f is Element of the carrier of RLSStruct(# S,M,x,y #)
A is Element of the carrier of RLSStruct(# S,M,x,y #)
f + A is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (f,A) is Element of the carrier of RLSStruct(# S,M,x,y #)
[f,A] is set
{f,A} is non empty set
{f} is non empty set
{{f,A},{f}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [f,A] is set
n is Element of the carrier of X
n is Element of the carrier of X
n + n is Element of the carrier of X
the addF of X . (n,n) is Element of the carrier of X
[n,n] is set
{n,n} is non empty set
{n} is non empty set
{{n,n},{n}} is non empty set
the addF of X . [n,n] is set
n + n is Element of the carrier of X
the addF of X . (n,n) is Element of the carrier of X
[n,n] is set
{n,n} is non empty set
{n} is non empty set
{{n,n},{n}} is non empty set
the addF of X . [n,n] is set
A + f is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (A,f) is Element of the carrier of RLSStruct(# S,M,x,y #)
[A,f] is set
{A,f} is non empty set
{A} is non empty set
{{A,f},{A}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [A,f] is set
f is Element of the carrier of RLSStruct(# S,M,x,y #)
A is Element of the carrier of RLSStruct(# S,M,x,y #)
n is Element of the carrier of RLSStruct(# S,M,x,y #)
f + A is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (f,A) is Element of the carrier of RLSStruct(# S,M,x,y #)
[f,A] is set
{f,A} is non empty set
{f} is non empty set
{{f,A},{f}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [f,A] is set
(f + A) + n is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . ((f + A),n) is Element of the carrier of RLSStruct(# S,M,x,y #)
[(f + A),n] is set
{(f + A),n} is non empty set
{(f + A)} is non empty set
{{(f + A),n},{(f + A)}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [(f + A),n] is set
a1 is Element of the carrier of X
the addF of X . ((f + A),a1) is set
[(f + A),a1] is set
{(f + A),a1} is non empty set
{{(f + A),a1},{(f + A)}} is non empty set
the addF of X . [(f + A),a1] is set
n is Element of the carrier of X
c is Element of the carrier of X
n + c is Element of the carrier of X
the addF of X . (n,c) is Element of the carrier of X
[n,c] is set
{n,c} is non empty set
{n} is non empty set
{{n,c},{n}} is non empty set
the addF of X . [n,c] is set
(n + c) + a1 is Element of the carrier of X
the addF of X . ((n + c),a1) is Element of the carrier of X
[(n + c),a1] is set
{(n + c),a1} is non empty set
{(n + c)} is non empty set
{{(n + c),a1},{(n + c)}} is non empty set
the addF of X . [(n + c),a1] is set
c + a1 is Element of the carrier of X
the addF of X . (c,a1) is Element of the carrier of X
[c,a1] is set
{c,a1} is non empty set
{c} is non empty set
{{c,a1},{c}} is non empty set
the addF of X . [c,a1] is set
n + (c + a1) is Element of the carrier of X
the addF of X . (n,(c + a1)) is Element of the carrier of X
[n,(c + a1)] is set
{n,(c + a1)} is non empty set
{{n,(c + a1)},{n}} is non empty set
the addF of X . [n,(c + a1)] is set
A + n is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (A,n) is Element of the carrier of RLSStruct(# S,M,x,y #)
[A,n] is set
{A,n} is non empty set
{A} is non empty set
{{A,n},{A}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [A,n] is set
the addF of X . (n,(A + n)) is set
[n,(A + n)] is set
{n,(A + n)} is non empty set
{{n,(A + n)},{n}} is non empty set
the addF of X . [n,(A + n)] is set
f + (A + n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (f,(A + n)) is Element of the carrier of RLSStruct(# S,M,x,y #)
[f,(A + n)] is set
{f,(A + n)} is non empty set
{{f,(A + n)},{f}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [f,(A + n)] is set
f is Element of the carrier of RLSStruct(# S,M,x,y #)
0. RLSStruct(# S,M,x,y #) is V55( RLSStruct(# S,M,x,y #)) Element of the carrier of RLSStruct(# S,M,x,y #)
the ZeroF of RLSStruct(# S,M,x,y #) is Element of the carrier of RLSStruct(# S,M,x,y #)
f + (0. RLSStruct(# S,M,x,y #)) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (f,(0. RLSStruct(# S,M,x,y #))) is Element of the carrier of RLSStruct(# S,M,x,y #)
[f,(0. RLSStruct(# S,M,x,y #))] is set
{f,(0. RLSStruct(# S,M,x,y #))} is non empty set
{f} is non empty set
{{f,(0. RLSStruct(# S,M,x,y #))},{f}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [f,(0. RLSStruct(# S,M,x,y #))] is set
A is Element of the carrier of X
A + (0. X) is Element of the carrier of X
the addF of X . (A,(0. X)) is Element of the carrier of X
[A,(0. X)] is set
{A,(0. X)} is non empty set
{A} is non empty set
{{A,(0. X)},{A}} is non empty set
the addF of X . [A,(0. X)] is set
f is V11() real ext-real set
A is Element of the carrier of RLSStruct(# S,M,x,y #)
n is Element of the carrier of RLSStruct(# S,M,x,y #)
A + n is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . (A,n) is Element of the carrier of RLSStruct(# S,M,x,y #)
[A,n] is set
{A,n} is non empty set
{A} is non empty set
{{A,n},{A}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [A,n] is set
n is V11() real ext-real Element of REAL
n * (A + n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (n,(A + n)) is set
[n,(A + n)] is set
{n,(A + n)} is non empty set
{n} is non empty V166() V167() V168() set
{{n,(A + n)},{n}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [n,(A + n)] is set
the Mult of X . (n,(A + n)) is set
the Mult of X . [n,(A + n)] is set
c is Element of the carrier of X
a1 is Element of the carrier of X
c + a1 is Element of the carrier of X
the addF of X . (c,a1) is Element of the carrier of X
[c,a1] is set
{c,a1} is non empty set
{c} is non empty set
{{c,a1},{c}} is non empty set
the addF of X . [c,a1] is set
n * (c + a1) is Element of the carrier of X
the Mult of X . (n,(c + a1)) is set
[n,(c + a1)] is set
{n,(c + a1)} is non empty set
{{n,(c + a1)},{n}} is non empty set
the Mult of X . [n,(c + a1)] is set
n * c is Element of the carrier of X
the Mult of X . (n,c) is set
[n,c] is set
{n,c} is non empty set
{{n,c},{n}} is non empty set
the Mult of X . [n,c] is set
n * a1 is Element of the carrier of X
the Mult of X . (n,a1) is set
[n,a1] is set
{n,a1} is non empty set
{{n,a1},{n}} is non empty set
the Mult of X . [n,a1] is set
(n * c) + (n * a1) is Element of the carrier of X
the addF of X . ((n * c),(n * a1)) is Element of the carrier of X
[(n * c),(n * a1)] is set
{(n * c),(n * a1)} is non empty set
{(n * c)} is non empty set
{{(n * c),(n * a1)},{(n * c)}} is non empty set
the addF of X . [(n * c),(n * a1)] is set
the Mult of X . (n,c) is Element of the carrier of X
n * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (n,n) is set
[n,n] is set
{n,n} is non empty set
{{n,n},{n}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [n,n] is set
the addF of X . (( the Mult of X . (n,c)),(n * n)) is set
[( the Mult of X . (n,c)),(n * n)] is set
{( the Mult of X . (n,c)),(n * n)} is non empty set
{( the Mult of X . (n,c))} is non empty set
{{( the Mult of X . (n,c)),(n * n)},{( the Mult of X . (n,c))}} is non empty set
the addF of X . [( the Mult of X . (n,c)),(n * n)] is set
n * A is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (n,A) is set
[n,A] is set
{n,A} is non empty set
{{n,A},{n}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [n,A] is set
the addF of X . ((n * A),(n * n)) is set
[(n * A),(n * n)] is set
{(n * A),(n * n)} is non empty set
{(n * A)} is non empty set
{{(n * A),(n * n)},{(n * A)}} is non empty set
the addF of X . [(n * A),(n * n)] is set
(n * A) + (n * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . ((n * A),(n * n)) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . [(n * A),(n * n)] is set
f * (A + n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (f,(A + n)) is set
[f,(A + n)] is set
{f,(A + n)} is non empty set
{f} is non empty V166() V167() V168() set
{{f,(A + n)},{f}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [f,(A + n)] is set
f * A is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (f,A) is set
[f,A] is set
{f,A} is non empty set
{{f,A},{f}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [f,A] is set
f * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (f,n) is set
[f,n] is set
{f,n} is non empty set
{{f,n},{f}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [f,n] is set
(f * A) + (f * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . ((f * A),(f * n)) is Element of the carrier of RLSStruct(# S,M,x,y #)
[(f * A),(f * n)] is set
{(f * A),(f * n)} is non empty set
{(f * A)} is non empty set
{{(f * A),(f * n)},{(f * A)}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [(f * A),(f * n)] is set
f is V11() real ext-real set
A is V11() real ext-real set
n is Element of the carrier of RLSStruct(# S,M,x,y #)
n is V11() real ext-real Element of REAL
c is V11() real ext-real Element of REAL
n + c is V11() real ext-real Element of REAL
(n + c) * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . ((n + c),n) is set
[(n + c),n] is set
{(n + c),n} is non empty set
{(n + c)} is non empty V166() V167() V168() set
{{(n + c),n},{(n + c)}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [(n + c),n] is set
a1 is Element of the carrier of X
(n + c) * a1 is Element of the carrier of X
the Mult of X . ((n + c),a1) is set
[(n + c),a1] is set
{(n + c),a1} is non empty set
{{(n + c),a1},{(n + c)}} is non empty set
the Mult of X . [(n + c),a1] is set
n * a1 is Element of the carrier of X
the Mult of X . (n,a1) is set
[n,a1] is set
{n,a1} is non empty set
{n} is non empty V166() V167() V168() set
{{n,a1},{n}} is non empty set
the Mult of X . [n,a1] is set
c * a1 is Element of the carrier of X
the Mult of X . (c,a1) is set
[c,a1] is set
{c,a1} is non empty set
{c} is non empty V166() V167() V168() set
{{c,a1},{c}} is non empty set
the Mult of X . [c,a1] is set
(n * a1) + (c * a1) is Element of the carrier of X
the addF of X . ((n * a1),(c * a1)) is Element of the carrier of X
[(n * a1),(c * a1)] is set
{(n * a1),(c * a1)} is non empty set
{(n * a1)} is non empty set
{{(n * a1),(c * a1)},{(n * a1)}} is non empty set
the addF of X . [(n * a1),(c * a1)] is set
the Mult of X . (n,a1) is Element of the carrier of X
c * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (c,n) is set
[c,n] is set
{c,n} is non empty set
{{c,n},{c}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [c,n] is set
the addF of X . (( the Mult of X . (n,a1)),(c * n)) is set
[( the Mult of X . (n,a1)),(c * n)] is set
{( the Mult of X . (n,a1)),(c * n)} is non empty set
{( the Mult of X . (n,a1))} is non empty set
{{( the Mult of X . (n,a1)),(c * n)},{( the Mult of X . (n,a1))}} is non empty set
the addF of X . [( the Mult of X . (n,a1)),(c * n)] is set
n * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (n,n) is set
[n,n] is set
{n,n} is non empty set
{{n,n},{n}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [n,n] is set
the addF of X . ((n * n),(c * n)) is set
[(n * n),(c * n)] is set
{(n * n),(c * n)} is non empty set
{(n * n)} is non empty set
{{(n * n),(c * n)},{(n * n)}} is non empty set
the addF of X . [(n * n),(c * n)] is set
(n * n) + (c * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . ((n * n),(c * n)) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . [(n * n),(c * n)] is set
f + A is V11() real ext-real set
(f + A) * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . ((f + A),n) is set
[(f + A),n] is set
{(f + A),n} is non empty set
{(f + A)} is non empty V166() V167() V168() set
{{(f + A),n},{(f + A)}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [(f + A),n] is set
f * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (f,n) is set
[f,n] is set
{f,n} is non empty set
{f} is non empty V166() V167() V168() set
{{f,n},{f}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [f,n] is set
A * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (A,n) is set
[A,n] is set
{A,n} is non empty set
{A} is non empty V166() V167() V168() set
{{A,n},{A}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [A,n] is set
(f * n) + (A * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the addF of RLSStruct(# S,M,x,y #) . ((f * n),(A * n)) is Element of the carrier of RLSStruct(# S,M,x,y #)
[(f * n),(A * n)] is set
{(f * n),(A * n)} is non empty set
{(f * n)} is non empty set
{{(f * n),(A * n)},{(f * n)}} is non empty set
the addF of RLSStruct(# S,M,x,y #) . [(f * n),(A * n)] is set
f is V11() real ext-real set
A is V11() real ext-real set
n is Element of the carrier of RLSStruct(# S,M,x,y #)
n is V11() real ext-real Element of REAL
c is V11() real ext-real Element of REAL
n * c is V11() real ext-real Element of REAL
(n * c) * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . ((n * c),n) is set
[(n * c),n] is set
{(n * c),n} is non empty set
{(n * c)} is non empty V166() V167() V168() set
{{(n * c),n},{(n * c)}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [(n * c),n] is set
a1 is Element of the carrier of X
(n * c) * a1 is Element of the carrier of X
the Mult of X . ((n * c),a1) is set
[(n * c),a1] is set
{(n * c),a1} is non empty set
{{(n * c),a1},{(n * c)}} is non empty set
the Mult of X . [(n * c),a1] is set
c * a1 is Element of the carrier of X
the Mult of X . (c,a1) is set
[c,a1] is set
{c,a1} is non empty set
{c} is non empty V166() V167() V168() set
{{c,a1},{c}} is non empty set
the Mult of X . [c,a1] is set
n * (c * a1) is Element of the carrier of X
the Mult of X . (n,(c * a1)) is set
[n,(c * a1)] is set
{n,(c * a1)} is non empty set
{n} is non empty V166() V167() V168() set
{{n,(c * a1)},{n}} is non empty set
the Mult of X . [n,(c * a1)] is set
c * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (c,n) is set
[c,n] is set
{c,n} is non empty set
{{c,n},{c}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [c,n] is set
the Mult of X . (n,(c * n)) is set
[n,(c * n)] is set
{n,(c * n)} is non empty set
{{n,(c * n)},{n}} is non empty set
the Mult of X . [n,(c * n)] is set
n * (c * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (n,(c * n)) is set
the Mult of RLSStruct(# S,M,x,y #) . [n,(c * n)] is set
f * A is V11() real ext-real set
(f * A) * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . ((f * A),n) is set
[(f * A),n] is set
{(f * A),n} is non empty set
{(f * A)} is non empty V166() V167() V168() set
{{(f * A),n},{(f * A)}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [(f * A),n] is set
A * n is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (A,n) is set
[A,n] is set
{A,n} is non empty set
{A} is non empty V166() V167() V168() set
{{A,n},{A}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [A,n] is set
f * (A * n) is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (f,(A * n)) is set
[f,(A * n)] is set
{f,(A * n)} is non empty set
{f} is non empty V166() V167() V168() set
{{f,(A * n)},{f}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [f,(A * n)] is set
f is Element of the carrier of RLSStruct(# S,M,x,y #)
1 * f is Element of the carrier of RLSStruct(# S,M,x,y #)
the Mult of RLSStruct(# S,M,x,y #) . (1,f) is set
[1,f] is set
{1,f} is non empty set
{1} is non empty V166() V167() V168() V169() V170() V171() set
{{1,f},{1}} is non empty set
the Mult of RLSStruct(# S,M,x,y #) . [1,f] is set
A is Element of the carrier of X
1 * A is Element of the carrier of X
the Mult of X . (1,A) is set
[1,A] is set
{1,A} is non empty set
{{1,A},{1}} is non empty set
the Mult of X . [1,A] is set
X is non empty Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
0. X is V55(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
S is non empty add-closed (X) Element of bool the carrier of X
In ((0. X),S) is Element of S
add| (S,X) is V13() V16([:S,S:]) V17(S) Function-like V41([:S,S:],S) Element of bool [:[:S,S:],S:]
[:S,S:] is non empty set
[:[:S,S:],S:] is non empty set
bool [:[:S,S:],S:] is non empty set
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 is set
K65( the addF of X,[:S,S:]) is V13() set
(X,S) is V13() V16([:REAL,S:]) V17(S) Function-like V41([:REAL,S:],S) Element of bool [:[:REAL,S:],S:]
[:REAL,S:] is non empty set
[:[:REAL,S:],S:] is non empty set
bool [:[:REAL,S:],S:] is non empty set
[:REAL, the carrier of X:] is non empty set
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:REAL,S:] is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is non empty strict RLSStruct
X is non empty RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
0. X is V55(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
S is non empty add-closed (X) Element of bool the carrier of X
In ((0. X),S) is Element of S
add| (S,X) is V13() V16([:S,S:]) V17(S) Function-like V41([:S,S:],S) Element of bool [:[:S,S:],S:]
[:S,S:] is non empty set
[:[:S,S:],S:] is non empty set
bool [:[:S,S:],S:] is non empty set
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 is set
K65( the addF of X,[:S,S:]) is V13() set
(X,S) is V13() V16([:REAL,S:]) V17(S) Function-like V41([:REAL,S:],S) Element of bool [:[:REAL,S:],S:]
[:REAL,S:] is non empty set
[:[:REAL,S:],S:] is non empty set
bool [:[:REAL,S:],S:] is non empty set
[:REAL, the carrier of X:] is non empty set
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:REAL,S:] is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is non empty strict RLSStruct
the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is non empty set
M is Element of the carrier of X
x is Element of the carrier of X
M + x is Element of the carrier of X
the addF of X . (M,x) is Element of the carrier of X
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
the addF of X . [M,x] is set
y is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
a is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
y + a is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
the addF of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is V13() V16([: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):]) V17( the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)) Function-like V41([: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)) Element of bool [:[: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):]
[: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
[:[: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
bool [:[: the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #), the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
the addF of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) . (y,a) is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
the addF of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) . [y,a] is set
[M,x] is Element of [: the carrier of X, the carrier of X:]
X is non empty RLSStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
0. X is V55(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
S is non empty add-closed (X) Element of bool the carrier of X
In ((0. X),S) is Element of S
add| (S,X) is V13() V16([:S,S:]) V17(S) Function-like V41([:S,S:],S) Element of bool [:[:S,S:],S:]
[:S,S:] is non empty set
[:[:S,S:],S:] is non empty set
bool [:[:S,S:],S:] is non empty set
the addF of X is V13() V16([: the carrier of X, the carrier of X:]) V17( the carrier of X) Function-like 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 is set
K65( the addF of X,[:S,S:]) is V13() set
(X,S) is V13() V16([:REAL,S:]) V17(S) Function-like V41([:REAL,S:],S) Element of bool [:[:REAL,S:],S:]
[:REAL,S:] is non empty set
[:[:REAL,S:],S:] is non empty set
bool [:[:REAL,S:],S:] is non empty set
[:REAL, the carrier of X:] is non empty set
the Mult of X is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like V41([:REAL, the carrier of X:], the carrier of X) Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:REAL,S:] is V13() V16([:REAL, the carrier of X:]) V17( the carrier of X) Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is non empty strict RLSStruct
the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is non empty set
M is V11() real ext-real Element of REAL
x is Element of the carrier of X
M * x is Element of the carrier of X
the Mult of X . (M,x) is set
[M,x] is set
{M,x} is non empty set
{M} is non empty V166() V167() V168() set
{{M,x},{M}} is non empty set
the Mult of X . [M,x] is set
y is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
M * y is Element of the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)
the Mult of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) is V13() V16([:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):]) V17( the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)) Function-like V41([:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #)) Element of bool [:[:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):]
[:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):], the carrier of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #):] is non empty set
the Mult of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) . (M,y) is set
[M,y] is set
{M,y} is non empty set
{{M,y},{M}} is non empty set
the Mult of RLSStruct(# S,(In ((0. X),S)),(add| (S,X)),(X,S) #) . [M,y] is set
[M,x] is Element of [:REAL, the carrier of X:]
X is non empty set
S is non empty set
PFuncs (X,S) is non empty functional PartFunc-set of X,S
[:(PFuncs (X,S)),(PFuncs (X,S)):] is non empty set
[:[:(PFuncs (X,S)),(PFuncs (X,S)):],(PFuncs (X,S)):] is non empty set
bool [:[:(PFuncs (X,S)),(PFuncs (X,S)):],(PFuncs (X,S)):] is non empty set
M is V13() V16([:(PFuncs (X,S)),(PFuncs (X,S)):]) V17( PFuncs (X,S)) Function-like V41([:(PFuncs (X,S)),(PFuncs (X,S)):], PFuncs (X,S)) Element of bool [:[:(PFuncs (X,S)),(PFuncs (X,S)):],(PFuncs (X,S)):]
x is V13() V16(X) V17(S) Function-like Element of PFuncs (X,S)
y is V13() V16(X) V17(S) Function-like Element of PFuncs (X,S)
M . (x,y) is set
[x,y] is set
{x,y} is non empty set
{x} is non empty set
{{x,y},{x}} is non empty set
M . [x,y] is set
a is V13() V16(X) V17(S) Function-like Element of PFuncs (X,S)
f is V13() V16(X) V17(S) Function-like Element of PFuncs (X,S)
M . (a,f) is Element of PFuncs (X,S)
[a,f] is set
{a,f} is non empty set
{a} is non empty set
{{a,f},{a}} is non empty set
M . [a,f] is set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of 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
S is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
S is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
M is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,S,x,y) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[x,y] is set
{x,y} is non empty set
{x} is non empty set
{{x,y},{x}} is non empty set
S . [x,y] is set
x (#) y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,M,x,y) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M . [x,y] is set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of 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
S is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
S is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
M is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:REAL,(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:REAL,(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
x is V11() real ext-real Element of REAL
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
S . (x,y) is Element of PFuncs (X,REAL)
[x,y] is set
{x,y} is non empty set
{x} is non empty V166() V167() V168() set
{{x,y},{x}} is non empty set
S . [x,y] is set
x (#) y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M . (x,y) is Element of PFuncs (X,REAL)
M . [x,y] is set
X is non empty set
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
X is non empty set
X --> 1 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom S is Element of bool X
bool X is non empty set
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom M is Element of bool X
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(addpfunc X) . [M,x] is set
dom x is Element of bool X
(dom M) /\ (dom x) is Element of bool X
M + x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (M + x) is Element of bool X
y is Element of X
S . y is V11() real ext-real set
(M + x) . y is V11() real ext-real set
M . y is V11() real ext-real set
x . y is V11() real ext-real set
(M . y) + (x . y) is V11() real ext-real set
a is Element of X
(X,REAL,(addpfunc X),M,x) . a is V11() real ext-real set
(M + x) . a is V11() real ext-real set
M . a is V11() real ext-real set
x . a is V11() real ext-real set
(M . a) + (x . a) is V11() real ext-real set
S . a is V11() real ext-real set
dom (X,REAL,(addpfunc X),M,x) is Element of bool X
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom S is Element of bool X
bool X is non empty set
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom M is Element of bool X
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(X) . [M,x] is set
dom x is Element of bool X
(dom M) /\ (dom x) is Element of bool X
M (#) x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (M (#) x) is Element of bool X
a is Element of X
S . a is V11() real ext-real set
(M (#) x) . a is V11() real ext-real set
M . a is V11() real ext-real set
x . a is V11() real ext-real set
(M . a) * (x . a) is V11() real ext-real set
a is Element of X
(X,REAL,(X),M,x) . a is V11() real ext-real set
(M (#) x) . a is V11() real ext-real set
M . a is V11() real ext-real set
x . a is V11() real ext-real set
(M . a) * (x . a) is V11() real ext-real set
S . a is V11() real ext-real set
dom (X,REAL,(X),M,x) is Element of bool X
X is non empty set
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 1 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
the Element of X is Element of X
(X) . the Element of X is V11() real ext-real set
X is V11() real ext-real Element of REAL
S is non empty set
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
M is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
dom M is Element of bool S
bool S is non empty set
x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S) . (X,x) is Element of PFuncs (S,REAL)
[X,x] is set
{X,x} is non empty set
{X} is non empty V166() V167() V168() set
{{X,x},{X}} is non empty set
(S) . [X,x] is set
dom x is Element of bool S
X (#) x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
dom (X (#) x) is Element of bool S
y is Element of S
(X (#) x) . y is V11() real ext-real set
x . y is V11() real ext-real set
X * (x . y) is V11() real ext-real Element of REAL
M . y is V11() real ext-real set
a is Element of S
(X (#) x) . a is V11() real ext-real set
x . a is V11() real ext-real set
X * (x . a) is V11() real ext-real Element of REAL
M . a is V11() real ext-real set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
y . a is V11() real ext-real set
dom y is Element of bool S
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{S} is non empty set
{{S,M},{S}} is non empty set
(addpfunc X) . [S,M] is set
(X,REAL,(addpfunc X),M,S) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,S] is set
{M,S} is non empty set
{M} is non empty set
{{M,S},{M}} is non empty set
(addpfunc X) . [M,S] is set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{S} is non empty set
{{S,M},{S}} is non empty set
(addpfunc X) . [S,M] is set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(addpfunc X) . [M,x] is set
(X,REAL,(addpfunc X),S,(X,REAL,(addpfunc X),M,x)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,(X,REAL,(addpfunc X),M,x)] is set
{S,(X,REAL,(addpfunc X),M,x)} is non empty set
{{S,(X,REAL,(addpfunc X),M,x)},{S}} is non empty set
(addpfunc X) . [S,(X,REAL,(addpfunc X),M,x)] is set
(X,REAL,(addpfunc X),(X,REAL,(addpfunc X),S,M),x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X,REAL,(addpfunc X),S,M),x] is set
{(X,REAL,(addpfunc X),S,M),x} is non empty set
{(X,REAL,(addpfunc X),S,M)} is non empty set
{{(X,REAL,(addpfunc X),S,M),x},{(X,REAL,(addpfunc X),S,M)}} is non empty set
(addpfunc X) . [(X,REAL,(addpfunc X),S,M),x] is set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{S} is non empty set
{{S,M},{S}} is non empty set
(X) . [S,M] is set
(X,REAL,(X),M,S) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,S] is set
{M,S} is non empty set
{M} is non empty set
{{M,S},{M}} is non empty set
(X) . [M,S] is set
dom (X,REAL,(X),M,S) is Element of bool X
bool X is non empty set
dom M is Element of bool X
dom S is Element of bool X
(dom M) /\ (dom S) is Element of bool X
dom (X,REAL,(X),S,M) is Element of bool X
(dom S) /\ (dom M) is Element of bool X
x is Element of X
(X,REAL,(X),S,M) . x is V11() real ext-real set
M . x is V11() real ext-real set
S . x is V11() real ext-real set
(M . x) * (S . x) is V11() real ext-real set
(X,REAL,(X),M,S) . x is V11() real ext-real set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{S} is non empty set
{{S,M},{S}} is non empty set
(X) . [S,M] is set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(X) . [M,x] is set
(X,REAL,(X),S,(X,REAL,(X),M,x)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,(X,REAL,(X),M,x)] is set
{S,(X,REAL,(X),M,x)} is non empty set
{{S,(X,REAL,(X),M,x)},{S}} is non empty set
(X) . [S,(X,REAL,(X),M,x)] is set
(X,REAL,(X),(X,REAL,(X),S,M),x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X,REAL,(X),S,M),x] is set
{(X,REAL,(X),S,M),x} is non empty set
{(X,REAL,(X),S,M)} is non empty set
{{(X,REAL,(X),S,M),x},{(X,REAL,(X),S,M)}} is non empty set
(X) . [(X,REAL,(X),S,M),x] is set
dom (X,REAL,(X),(X,REAL,(X),S,M),x) is Element of bool X
bool X is non empty set
dom (X,REAL,(X),S,M) is Element of bool X
dom x is Element of bool X
(dom (X,REAL,(X),S,M)) /\ (dom x) is Element of bool X
dom S is Element of bool X
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
((dom S) /\ (dom M)) /\ (dom x) is Element of bool X
dom (X,REAL,(X),S,(X,REAL,(X),M,x)) is Element of bool X
dom (X,REAL,(X),M,x) is Element of bool X
(dom S) /\ (dom (X,REAL,(X),M,x)) is Element of bool X
(dom M) /\ (dom x) is Element of bool X
(dom S) /\ ((dom M) /\ (dom x)) is Element of bool X
f is Element of X
(X,REAL,(X),S,(X,REAL,(X),M,x)) . f is V11() real ext-real set
S . f is V11() real ext-real set
(X,REAL,(X),M,x) . f is V11() real ext-real set
(S . f) * ((X,REAL,(X),M,x) . f) is V11() real ext-real set
M . f is V11() real ext-real set
x . f is V11() real ext-real set
(M . f) * (x . f) is V11() real ext-real set
(S . f) * ((M . f) * (x . f)) is V11() real ext-real set
(S . f) * (M . f) is V11() real ext-real set
((S . f) * (M . f)) * (x . f) is V11() real ext-real set
(X,REAL,(X),S,M) . f is V11() real ext-real set
((X,REAL,(X),S,M) . f) * (x . f) is V11() real ext-real set
(X,REAL,(X),(X,REAL,(X),S,M),x) . f is V11() real ext-real set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 1 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),(X),S) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X),S] is set
{(X),S} is non empty set
{(X)} is non empty set
{{(X),S},{(X)}} is non empty set
(X) . [(X),S] is set
dom (X,REAL,(X),(X),S) is Element of bool X
bool X is non empty set
dom (X) is Element of bool X
dom S is Element of bool X
(dom (X)) /\ (dom S) is Element of bool X
X /\ (dom S) is Element of bool X
x is Element of X
(X,REAL,(X),(X),S) . x is V11() real ext-real set
(X) . x is V11() real ext-real set
S . x is V11() real ext-real set
((X) . x) * (S . x) is V11() real ext-real set
1 * (S . x) is V11() real ext-real Element of REAL
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),(X),S) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X),S] is set
{(X),S} is non empty set
{(X)} is non empty set
{{(X),S},{(X)}} is non empty set
(addpfunc X) . [(X),S] is set
dom (X,REAL,(addpfunc X),(X),S) is Element of bool X
bool X is non empty set
dom (X) is Element of bool X
dom S is Element of bool X
(dom (X)) /\ (dom S) is Element of bool X
X /\ (dom S) is Element of bool X
x is Element of X
(X) . x is V11() real ext-real set
(X,REAL,(addpfunc X),(X),S) . x is V11() real ext-real set
S . x is V11() real ext-real set
0 + (S . x) is V11() real ext-real Element of REAL
- 1 is non empty V11() real ext-real non positive negative Element of REAL
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X) . ((- 1),S) is Element of PFuncs (X,REAL)
[(- 1),S] is set
{(- 1),S} is non empty set
{(- 1)} is non empty V166() V167() V168() set
{{(- 1),S},{(- 1)}} is non empty set
(X) . [(- 1),S] is set
(X,REAL,(addpfunc X),S,((X) . ((- 1),S))) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,((X) . ((- 1),S))] is set
{S,((X) . ((- 1),S))} is non empty set
{S} is non empty set
{{S,((X) . ((- 1),S))},{S}} is non empty set
(addpfunc X) . [S,((X) . ((- 1),S))] is set
dom S is Element of bool X
bool X is non empty set
(X) | (dom S) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{{S,M},{S}} is non empty set
(addpfunc X) . [S,M] is set
dom (X) is Element of bool X
dom ((X) | (dom S)) is Element of bool X
X /\ (dom S) is Element of bool X
dom (X,REAL,(addpfunc X),S,M) is Element of bool X
dom M is Element of bool X
(dom M) /\ (dom S) is Element of bool X
(dom S) /\ (dom S) is Element of bool X
y is Element of X
(- 1) (#) S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom ((- 1) (#) S) is Element of bool X
(X,REAL,(addpfunc X),S,M) . y is V11() real ext-real set
S . y is V11() real ext-real set
M . y is V11() real ext-real set
(S . y) + (M . y) is V11() real ext-real set
((- 1) (#) S) . y is V11() real ext-real set
(S . y) + (((- 1) (#) S) . y) is V11() real ext-real set
(- 1) * (S . y) is V11() real ext-real Element of REAL
(S . y) + ((- 1) * (S . y)) is V11() real ext-real Element of REAL
(X) . y is V11() real ext-real set
((X) | (dom S)) . y is V11() real ext-real set
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X) . (1,S) is Element of PFuncs (X,REAL)
[1,S] is set
{1,S} is non empty set
{1} is non empty V166() V167() V168() V169() V170() V171() set
{{1,S},{1}} is non empty set
(X) . [1,S] is set
x is Element of X
dom S is Element of bool X
bool X is non empty set
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M . x is V11() real ext-real set
S . x is V11() real ext-real set
1 * (S . x) is V11() real ext-real Element of REAL
dom M is Element of bool X
X is V11() real ext-real Element of REAL
S is V11() real ext-real Element of REAL
X * S is V11() real ext-real Element of REAL
M is non empty set
PFuncs (M,REAL) is non empty functional PartFunc-set of M, REAL
(M) is V13() V16([:REAL,(PFuncs (M,REAL)):]) V17( PFuncs (M,REAL)) Function-like V41([:REAL,(PFuncs (M,REAL)):], PFuncs (M,REAL)) Element of bool [:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):]
[:REAL,(PFuncs (M,REAL)):] is non empty set
[:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
x is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
(M) . (S,x) is Element of PFuncs (M,REAL)
[S,x] is set
{S,x} is non empty set
{S} is non empty V166() V167() V168() set
{{S,x},{S}} is non empty set
(M) . [S,x] is set
(M) . (X,((M) . (S,x))) is Element of PFuncs (M,REAL)
[X,((M) . (S,x))] is set
{X,((M) . (S,x))} is non empty set
{X} is non empty V166() V167() V168() set
{{X,((M) . (S,x))},{X}} is non empty set
(M) . [X,((M) . (S,x))] is set
(M) . ((X * S),x) is Element of PFuncs (M,REAL)
[(X * S),x] is set
{(X * S),x} is non empty set
{(X * S)} is non empty V166() V167() V168() set
{{(X * S),x},{(X * S)}} is non empty set
(M) . [(X * S),x] is set
y is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
(M) . (X,y) is Element of PFuncs (M,REAL)
[X,y] is set
{X,y} is non empty set
{{X,y},{X}} is non empty set
(M) . [X,y] is set
a is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
dom a is Element of bool M
bool M is non empty set
dom y is Element of bool M
dom x is Element of bool M
g is Element of M
a . g is V11() real ext-real set
y . g is V11() real ext-real set
X * (y . g) is V11() real ext-real Element of REAL
x . g is V11() real ext-real set
S * (x . g) is V11() real ext-real Element of REAL
X * (S * (x . g)) is V11() real ext-real Element of REAL
(X * S) * (x . g) is V11() real ext-real Element of REAL
f is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
f . g is V11() real ext-real set
dom f is Element of bool M
X is V11() real ext-real Element of REAL
S is V11() real ext-real Element of REAL
X + S is V11() real ext-real Element of REAL
M is non empty set
PFuncs (M,REAL) is non empty functional PartFunc-set of M, REAL
addpfunc M is V13() V16([:(PFuncs (M,REAL)),(PFuncs (M,REAL)):]) V17( PFuncs (M,REAL)) Function-like V41([:(PFuncs (M,REAL)),(PFuncs (M,REAL)):], PFuncs (M,REAL)) Element of bool [:[:(PFuncs (M,REAL)),(PFuncs (M,REAL)):],(PFuncs (M,REAL)):]
[:(PFuncs (M,REAL)),(PFuncs (M,REAL)):] is non empty set
[:[:(PFuncs (M,REAL)),(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
bool [:[:(PFuncs (M,REAL)),(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
(M) is V13() V16([:REAL,(PFuncs (M,REAL)):]) V17( PFuncs (M,REAL)) Function-like V41([:REAL,(PFuncs (M,REAL)):], PFuncs (M,REAL)) Element of bool [:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):]
[:REAL,(PFuncs (M,REAL)):] is non empty set
[:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (M,REAL)):],(PFuncs (M,REAL)):] is non empty set
x is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
(M) . (X,x) is Element of PFuncs (M,REAL)
[X,x] is set
{X,x} is non empty set
{X} is non empty V166() V167() V168() set
{{X,x},{X}} is non empty set
(M) . [X,x] is set
(M) . (S,x) is Element of PFuncs (M,REAL)
[S,x] is set
{S,x} is non empty set
{S} is non empty V166() V167() V168() set
{{S,x},{S}} is non empty set
(M) . [S,x] is set
(M,REAL,(addpfunc M),((M) . (X,x)),((M) . (S,x))) is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
[((M) . (X,x)),((M) . (S,x))] is set
{((M) . (X,x)),((M) . (S,x))} is non empty set
{((M) . (X,x))} is non empty set
{{((M) . (X,x)),((M) . (S,x))},{((M) . (X,x))}} is non empty set
(addpfunc M) . [((M) . (X,x)),((M) . (S,x))] is set
(M) . ((X + S),x) is Element of PFuncs (M,REAL)
[(X + S),x] is set
{(X + S),x} is non empty set
{(X + S)} is non empty V166() V167() V168() set
{{(X + S),x},{(X + S)}} is non empty set
(M) . [(X + S),x] is set
y is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
a is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
(M,REAL,(addpfunc M),y,a) is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
(addpfunc M) . [y,a] is set
dom y is Element of bool M
bool M is non empty set
dom x is Element of bool M
dom a is Element of bool M
(dom a) /\ (dom y) is Element of bool M
(dom x) /\ (dom x) is Element of bool M
dom (M,REAL,(addpfunc M),y,a) is Element of bool M
f is Element of M
S (#) x is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
dom (S (#) x) is Element of bool M
(S (#) x) . f is V11() real ext-real set
x . f is V11() real ext-real set
S * (x . f) is V11() real ext-real Element of REAL
a . f is V11() real ext-real set
X (#) x is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
dom (X (#) x) is Element of bool M
(X (#) x) . f is V11() real ext-real set
X * (x . f) is V11() real ext-real Element of REAL
y . f is V11() real ext-real set
(y . f) + (a . f) is V11() real ext-real set
(X + S) * (x . f) is V11() real ext-real Element of REAL
(M,REAL,(addpfunc M),y,a) . f is V11() real ext-real set
f is V13() V16(M) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (M,REAL)
f . f is V11() real ext-real set
dom f is Element of bool M
X is V11() real ext-real Element of REAL
S is non empty set
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
M is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S) . (X,M) is Element of PFuncs (S,REAL)
[X,M] is set
{X,M} is non empty set
{X} is non empty V166() V167() V168() set
{{X,M},{X}} is non empty set
(S) . [X,M] is set
x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S) . (X,x) is Element of PFuncs (S,REAL)
[X,x] is set
{X,x} is non empty set
{{X,x},{X}} is non empty set
(S) . [X,x] is set
(S,REAL,(addpfunc S),((S) . (X,M)),((S) . (X,x))) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[((S) . (X,M)),((S) . (X,x))] is set
{((S) . (X,M)),((S) . (X,x))} is non empty set
{((S) . (X,M))} is non empty set
{{((S) . (X,M)),((S) . (X,x))},{((S) . (X,M))}} is non empty set
(addpfunc S) . [((S) . (X,M)),((S) . (X,x))] is set
(S,REAL,(addpfunc S),M,x) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(addpfunc S) . [M,x] is set
(S) . (X,(S,REAL,(addpfunc S),M,x)) is Element of PFuncs (S,REAL)
[X,(S,REAL,(addpfunc S),M,x)] is set
{X,(S,REAL,(addpfunc S),M,x)} is non empty set
{{X,(S,REAL,(addpfunc S),M,x)},{X}} is non empty set
(S) . [X,(S,REAL,(addpfunc S),M,x)] is set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S,REAL,(addpfunc S),y,a) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
(addpfunc S) . [y,a] is set
dom y is Element of bool S
bool S is non empty set
dom M is Element of bool S
dom a is Element of bool S
dom x is Element of bool S
dom (S,REAL,(addpfunc S),y,a) is Element of bool S
(dom y) /\ (dom a) is Element of bool S
dom (S,REAL,(addpfunc S),M,x) is Element of bool S
(dom M) /\ (dom x) is Element of bool S
f is Element of S
y . f is V11() real ext-real set
X (#) M is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(X (#) M) . f is V11() real ext-real set
M + x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
dom (M + x) is Element of bool S
a . f is V11() real ext-real set
X (#) x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(X (#) x) . f is V11() real ext-real set
dom (X (#) x) is Element of bool S
x . f is V11() real ext-real set
X * (x . f) is V11() real ext-real Element of REAL
dom (X (#) M) is Element of bool S
M . f is V11() real ext-real set
X * (M . f) is V11() real ext-real Element of REAL
(S,REAL,(addpfunc S),y,a) . f is V11() real ext-real set
(y . f) + (a . f) is V11() real ext-real set
(M . f) + (x . f) is V11() real ext-real set
X * ((M . f) + (x . f)) is V11() real ext-real Element of REAL
(M + x) . f is V11() real ext-real set
X * ((M + x) . f) is V11() real ext-real Element of REAL
(S,REAL,(addpfunc S),M,x) . f is V11() real ext-real set
X * ((S,REAL,(addpfunc S),M,x) . f) is V11() real ext-real Element of REAL
g is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
g . f is V11() real ext-real set
dom g is Element of bool S
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like V41([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):], PFuncs (X,REAL)) Element of bool [:[:(PFuncs (X,REAL)),(PFuncs (X,REAL)):],(PFuncs (X,REAL)):]
S is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(X),S,M) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,M] is set
{S,M} is non empty set
{S} is non empty set
{{S,M},{S}} is non empty set
(X) . [S,M] is set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(addpfunc X) . [M,x] is set
(X,REAL,(X),S,(X,REAL,(addpfunc X),M,x)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,(X,REAL,(addpfunc X),M,x)] is set
{S,(X,REAL,(addpfunc X),M,x)} is non empty set
{{S,(X,REAL,(addpfunc X),M,x)},{S}} is non empty set
(X) . [S,(X,REAL,(addpfunc X),M,x)] is set
(X,REAL,(X),S,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[S,x] is set
{S,x} is non empty set
{{S,x},{S}} is non empty set
(X) . [S,x] is set
(X,REAL,(addpfunc X),(X,REAL,(X),S,M),(X,REAL,(X),S,x)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X,REAL,(X),S,M),(X,REAL,(X),S,x)] is set
{(X,REAL,(X),S,M),(X,REAL,(X),S,x)} is non empty set
{(X,REAL,(X),S,M)} is non empty set
{{(X,REAL,(X),S,M),(X,REAL,(X),S,x)},{(X,REAL,(X),S,M)}} is non empty set
(addpfunc X) . [(X,REAL,(X),S,M),(X,REAL,(X),S,x)] is set
dom S is Element of bool X
bool X is non empty set
dom M is Element of bool X
(dom S) /\ (dom M) is Element of bool X
dom x is Element of bool X
((dom S) /\ (dom M)) /\ (dom x) is Element of bool X
(dom M) /\ (dom x) is Element of bool X
(dom S) /\ ((dom M) /\ (dom x)) is Element of bool X
dom (X,REAL,(X),S,x) is Element of bool X
(dom S) /\ (dom x) is Element of bool X
dom (X,REAL,(X),S,M) is Element of bool X
dom (X,REAL,(addpfunc X),(X,REAL,(X),S,M),(X,REAL,(X),S,x)) is Element of bool X
(dom x) /\ (dom S) is Element of bool X
((dom x) /\ (dom S)) /\ ((dom S) /\ (dom M)) is Element of bool X
(dom S) /\ ((dom S) /\ (dom M)) is Element of bool X
(dom x) /\ ((dom S) /\ ((dom S) /\ (dom M))) is Element of bool X
(dom S) /\ (dom S) is Element of bool X
((dom S) /\ (dom S)) /\ (dom M) is Element of bool X
(dom x) /\ (((dom S) /\ (dom S)) /\ (dom M)) is Element of bool X
(dom M) /\ ((dom S) /\ (dom x)) is Element of bool X
f is Element of X
S (#) M is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (S (#) M) is Element of bool X
M + x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (M + x) is Element of bool X
(X,REAL,(X),S,M) . f is V11() real ext-real set
(S (#) M) . f is V11() real ext-real set
S . f is V11() real ext-real set
M . f is V11() real ext-real set
(S . f) * (M . f) is V11() real ext-real set
S (#) x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (S (#) x) is Element of bool X
(X,REAL,(X),S,x) . f is V11() real ext-real set
(S (#) x) . f is V11() real ext-real set
x . f is V11() real ext-real set
(S . f) * (x . f) is V11() real ext-real set
(X,REAL,(addpfunc X),(X,REAL,(X),S,M),(X,REAL,(X),S,x)) . f is V11() real ext-real set
((X,REAL,(X),S,M) . f) + ((X,REAL,(X),S,x) . f) is V11() real ext-real set
(X,REAL,(addpfunc X),M,x) . f is V11() real ext-real set
(M + x) . f is V11() real ext-real set
(M . f) + (x . f) is V11() real ext-real set
(S . f) * ((M . f) + (x . f)) is V11() real ext-real set
(S . f) * ((X,REAL,(addpfunc X),M,x) . f) is V11() real ext-real set
dom (X,REAL,(addpfunc X),M,x) is Element of bool X
(dom S) /\ (dom (X,REAL,(addpfunc X),M,x)) is Element of bool X
S (#) (X,REAL,(addpfunc X),M,x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom (S (#) (X,REAL,(addpfunc X),M,x)) is Element of bool X
(X,REAL,(X),S,(X,REAL,(addpfunc X),M,x)) . f is V11() real ext-real set
(S (#) (X,REAL,(addpfunc X),M,x)) . f is V11() real ext-real set
dom (X,REAL,(X),S,(X,REAL,(addpfunc X),M,x)) is Element of bool X
X is V11() real ext-real Element of REAL
S is non empty set
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
M is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S) . (X,M) is Element of PFuncs (S,REAL)
[X,M] is set
{X,M} is non empty set
{X} is non empty V166() V167() V168() set
{{X,M},{X}} is non empty set
(S) . [X,M] is set
x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S,REAL,(S),((S) . (X,M)),x) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[((S) . (X,M)),x] is set
{((S) . (X,M)),x} is non empty set
{((S) . (X,M))} is non empty set
{{((S) . (X,M)),x},{((S) . (X,M))}} is non empty set
(S) . [((S) . (X,M)),x] is set
(S,REAL,(S),M,x) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
(S) . [M,x] is set
(S) . (X,(S,REAL,(S),M,x)) is Element of PFuncs (S,REAL)
[X,(S,REAL,(S),M,x)] is set
{X,(S,REAL,(S),M,x)} is non empty set
{{X,(S,REAL,(S),M,x)},{X}} is non empty set
(S) . [X,(S,REAL,(S),M,x)] is set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(S,REAL,(S),y,x) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
[y,x] is set
{y,x} is non empty set
{y} is non empty set
{{y,x},{y}} is non empty set
(S) . [y,x] is set
dom y is Element of bool S
bool S is non empty set
dom M is Element of bool S
dom (S,REAL,(S),y,x) is Element of bool S
dom x is Element of bool S
(dom y) /\ (dom x) is Element of bool S
dom (S,REAL,(S),M,x) is Element of bool S
(dom M) /\ (dom x) is Element of bool S
f is Element of S
(S,REAL,(S),M,x) . f is V11() real ext-real set
M (#) x is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(M (#) x) . f is V11() real ext-real set
dom (M (#) x) is Element of bool S
M . f is V11() real ext-real set
x . f is V11() real ext-real set
(M . f) * (x . f) is V11() real ext-real set
y . f is V11() real ext-real set
X (#) M is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(X (#) M) . f is V11() real ext-real set
dom (X (#) M) is Element of bool S
X * (M . f) is V11() real ext-real Element of REAL
g is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
g . f is V11() real ext-real set
X (#) (S,REAL,(S),M,x) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
(X (#) (S,REAL,(S),M,x)) . f is V11() real ext-real set
dom (X (#) (S,REAL,(S),M,x)) is Element of bool S
X * ((M . f) * (x . f)) is V11() real ext-real Element of REAL
(S,REAL,(S),y,x) . f is V11() real ext-real set
(y . f) * (x . f) is V11() real ext-real set
dom g is Element of bool S
X is non empty set
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
X is non empty set
(X) is non empty RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
M is Element of the carrier of (X)
x is Element of the carrier of (X)
M + x is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (M,x) is Element of the carrier of (X)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
the addF of (X) . [M,x] is set
x + M is Element of the carrier of (X)
the addF of (X) . (x,M) is Element of the carrier of (X)
[x,M] is set
{x,M} is non empty set
{x} is non empty set
{{x,M},{x}} is non empty set
the addF of (X) . [x,M] is set
M is Element of the carrier of (X)
x is Element of the carrier of (X)
M + x is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (M,x) is Element of the carrier of (X)
[M,x] is set
{M,x} is non empty set
{M} is non empty set
{{M,x},{M}} is non empty set
the addF of (X) . [M,x] is set
y is Element of the carrier of (X)
(M + x) + y is Element of the carrier of (X)
the addF of (X) . ((M + x),y) is Element of the carrier of (X)
[(M + x),y] is set
{(M + x),y} is non empty set
{(M + x)} is non empty set
{{(M + x),y},{(M + x)}} is non empty set
the addF of (X) . [(M + x),y] is set
x + y is Element of the carrier of (X)
the addF of (X) . (x,y) is Element of the carrier of (X)
[x,y] is set
{x,y} is non empty set
{x} is non empty set
{{x,y},{x}} is non empty set
the addF of (X) . [x,y] is set
M + (x + y) is Element of the carrier of (X)
the addF of (X) . (M,(x + y)) is Element of the carrier of (X)
[M,(x + y)] is set
{M,(x + y)} is non empty set
{{M,(x + y)},{M}} is non empty set
the addF of (X) . [M,(x + y)] is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
M is Element of the carrier of (X)
M + (0. (X)) is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (M,(0. (X))) is Element of the carrier of (X)
[M,(0. (X))] is set
{M,(0. (X))} is non empty set
{M} is non empty set
{{M,(0. (X))},{M}} is non empty set
the addF of (X) . [M,(0. (X))] is set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X,REAL,(addpfunc X),(X),x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
[(X),x] is set
{(X),x} is non empty set
{(X)} is non empty set
{{(X),x},{(X)}} is non empty set
(addpfunc X) . [(X),x] is set
M is V11() real ext-real set
x is V11() real ext-real Element of REAL
y is Element of the carrier of (X)
a is Element of the carrier of (X)
y + a is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (y,a) is Element of the carrier of (X)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
the addF of (X) . [y,a] is set
x * (y + a) is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (x,(y + a)) is set
[x,(y + a)] is set
{x,(y + a)} is non empty set
{x} is non empty V166() V167() V168() set
{{x,(y + a)},{x}} is non empty set
the Mult of (X) . [x,(y + a)] is set
x * y is Element of the carrier of (X)
the Mult of (X) . (x,y) is set
[x,y] is set
{x,y} is non empty set
{{x,y},{x}} is non empty set
the Mult of (X) . [x,y] is set
x * a is Element of the carrier of (X)
the Mult of (X) . (x,a) is set
[x,a] is set
{x,a} is non empty set
{{x,a},{x}} is non empty set
the Mult of (X) . [x,a] is set
(x * y) + (x * a) is Element of the carrier of (X)
the addF of (X) . ((x * y),(x * a)) is Element of the carrier of (X)
[(x * y),(x * a)] is set
{(x * y),(x * a)} is non empty set
{(x * y)} is non empty set
{{(x * y),(x * a)},{(x * y)}} is non empty set
the addF of (X) . [(x * y),(x * a)] is set
y is Element of the carrier of (X)
a is Element of the carrier of (X)
y + a is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (y,a) is Element of the carrier of (X)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
the addF of (X) . [y,a] is set
M * (y + a) is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (M,(y + a)) is set
[M,(y + a)] is set
{M,(y + a)} is non empty set
{M} is non empty V166() V167() V168() set
{{M,(y + a)},{M}} is non empty set
the Mult of (X) . [M,(y + a)] is set
M * y is Element of the carrier of (X)
the Mult of (X) . (M,y) is set
[M,y] is set
{M,y} is non empty set
{{M,y},{M}} is non empty set
the Mult of (X) . [M,y] is set
M * a is Element of the carrier of (X)
the Mult of (X) . (M,a) is set
[M,a] is set
{M,a} is non empty set
{{M,a},{M}} is non empty set
the Mult of (X) . [M,a] is set
(M * y) + (M * a) is Element of the carrier of (X)
the addF of (X) . ((M * y),(M * a)) is Element of the carrier of (X)
[(M * y),(M * a)] is set
{(M * y),(M * a)} is non empty set
{(M * y)} is non empty set
{{(M * y),(M * a)},{(M * y)}} is non empty set
the addF of (X) . [(M * y),(M * a)] is set
M is V11() real ext-real set
x is V11() real ext-real set
M + x is V11() real ext-real set
y is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
y + a is V11() real ext-real Element of REAL
f is Element of the carrier of (X)
(y + a) * f is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . ((y + a),f) is set
[(y + a),f] is set
{(y + a),f} is non empty set
{(y + a)} is non empty V166() V167() V168() set
{{(y + a),f},{(y + a)}} is non empty set
the Mult of (X) . [(y + a),f] is set
y * f is Element of the carrier of (X)
the Mult of (X) . (y,f) is set
[y,f] is set
{y,f} is non empty set
{y} is non empty V166() V167() V168() set
{{y,f},{y}} is non empty set
the Mult of (X) . [y,f] is set
a * f is Element of the carrier of (X)
the Mult of (X) . (a,f) is set
[a,f] is set
{a,f} is non empty set
{a} is non empty V166() V167() V168() set
{{a,f},{a}} is non empty set
the Mult of (X) . [a,f] is set
(y * f) + (a * f) is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . ((y * f),(a * f)) is Element of the carrier of (X)
[(y * f),(a * f)] is set
{(y * f),(a * f)} is non empty set
{(y * f)} is non empty set
{{(y * f),(a * f)},{(y * f)}} is non empty set
the addF of (X) . [(y * f),(a * f)] is set
f is Element of the carrier of (X)
(M + x) * f is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . ((M + x),f) is set
[(M + x),f] is set
{(M + x),f} is non empty set
{(M + x)} is non empty V166() V167() V168() set
{{(M + x),f},{(M + x)}} is non empty set
the Mult of (X) . [(M + x),f] is set
M * f is Element of the carrier of (X)
the Mult of (X) . (M,f) is set
[M,f] is set
{M,f} is non empty set
{M} is non empty V166() V167() V168() set
{{M,f},{M}} is non empty set
the Mult of (X) . [M,f] is set
x * f is Element of the carrier of (X)
the Mult of (X) . (x,f) is set
[x,f] is set
{x,f} is non empty set
{x} is non empty V166() V167() V168() set
{{x,f},{x}} is non empty set
the Mult of (X) . [x,f] is set
(M * f) + (x * f) is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . ((M * f),(x * f)) is Element of the carrier of (X)
[(M * f),(x * f)] is set
{(M * f),(x * f)} is non empty set
{(M * f)} is non empty set
{{(M * f),(x * f)},{(M * f)}} is non empty set
the addF of (X) . [(M * f),(x * f)] is set
M is V11() real ext-real set
x is V11() real ext-real set
M * x is V11() real ext-real set
y is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
y * a is V11() real ext-real Element of REAL
f is Element of the carrier of (X)
(y * a) * f is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . ((y * a),f) is set
[(y * a),f] is set
{(y * a),f} is non empty set
{(y * a)} is non empty V166() V167() V168() set
{{(y * a),f},{(y * a)}} is non empty set
the Mult of (X) . [(y * a),f] is set
a * f is Element of the carrier of (X)
the Mult of (X) . (a,f) is set
[a,f] is set
{a,f} is non empty set
{a} is non empty V166() V167() V168() set
{{a,f},{a}} is non empty set
the Mult of (X) . [a,f] is set
y * (a * f) is Element of the carrier of (X)
the Mult of (X) . (y,(a * f)) is set
[y,(a * f)] is set
{y,(a * f)} is non empty set
{y} is non empty V166() V167() V168() set
{{y,(a * f)},{y}} is non empty set
the Mult of (X) . [y,(a * f)] is set
f is Element of the carrier of (X)
(M * x) * f is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . ((M * x),f) is set
[(M * x),f] is set
{(M * x),f} is non empty set
{(M * x)} is non empty V166() V167() V168() set
{{(M * x),f},{(M * x)}} is non empty set
the Mult of (X) . [(M * x),f] is set
x * f is Element of the carrier of (X)
the Mult of (X) . (x,f) is set
[x,f] is set
{x,f} is non empty set
{x} is non empty V166() V167() V168() set
{{x,f},{x}} is non empty set
the Mult of (X) . [x,f] is set
M * (x * f) is Element of the carrier of (X)
the Mult of (X) . (M,(x * f)) is set
[M,(x * f)] is set
{M,(x * f)} is non empty set
{M} is non empty V166() V167() V168() set
{{M,(x * f)},{M}} is non empty set
the Mult of (X) . [M,(x * f)] is set
M is Element of the carrier of (X)
1 * M is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (1,M) is set
[1,M] is set
{1,M} is non empty set
{1} is non empty V166() V167() V168() V169() V170() V171() set
{{1,M},{1}} is non empty set
the Mult of (X) . [1,M] is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
dom x is Element of bool X
Integral (M,x) is ext-real Element of ExtREAL
max+ x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ x)) is ext-real Element of ExtREAL
max- x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- x)) is ext-real Element of ExtREAL
(integral+ (M,(max+ x))) - (integral+ (M,(max- x))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- x)))) is ext-real set
K118((integral+ (M,(max+ x))),K119((integral+ (M,(max- x))))) is ext-real set
y is Element of S
a is set
x . a is ext-real set
integral+ (M,x) is ext-real Element of ExtREAL
dom (max+ x) is Element of bool X
a is Element of X
x . a is ext-real set
max (0,0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
max ((x . a),0) is ext-real set
(max+ x) . a is ext-real set
a is ext-real Element of ExtREAL
rng x is V167() set
rng x is V167() Element of bool ExtREAL
f is set
x . f is ext-real set
(integral+ (M,x)) - (integral+ (M,(max- x))) is ext-real Element of ExtREAL
K118((integral+ (M,x)),K119((integral+ (M,(max- x))))) is ext-real set
- (integral+ (M,(max- x))) is ext-real Element of ExtREAL
0. + (- (integral+ (M,(max- x)))) is ext-real Element of ExtREAL
- (- (integral+ (M,(max- x)))) is ext-real Element of ExtREAL
- 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V166() V167() V168() V169() V170() V171() V172() Element of REAL
X is non empty set
S is V11() real ext-real Element of REAL
X --> S is V173() V174() V175() set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
X --> S is V13() V16(X) V17( REAL ) Function-like V41(X, REAL ) V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL x)) is ext-real Element of ExtREAL
max+ (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
max- (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL x))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL x)))),K119((integral+ (M,(max- (R_EAL x)))))) 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
f is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom g is Element of bool X
f ` is Element of bool X
X \ f is set
dom (X,0) is Element of bool X
x is Element of S
x ` is Element of bool X
X \ x is set
f is Element of X
(X,0) . f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
M . x is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V41(X, REAL ) V173() V174() V175() Element of bool [:X,REAL:]
dom x is Element of bool X
y is Element of S
y ` is Element of bool X
X \ y is set
a is Element of X
x . a is V11() real ext-real Element of REAL
M . y 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is Element of S
M . x is ext-real Element of ExtREAL
y is Element of S
M . y is ext-real Element of ExtREAL
x \/ y is Element of S
M . (x \/ y) 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom x is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
dom f is Element of bool X
g ` is Element of bool X
X \ g is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
dom y is Element of bool X
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom g is Element of bool X
f ` is Element of bool X
X \ f 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
dom (x + y) is Element of bool X
(f `) /\ (g `) is Element of bool X
f \/ g is Element of S
(f \/ g) ` is Element of bool X
X \ (f \/ g) is set
M . (f \/ g) is ext-real Element of ExtREAL
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
(S,M,x) is non empty Element of bool the carrier of (S)
(S) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
S --> 0 is V13() V16(S) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(S, NAT ) V173() V174() V175() V176() Element of bool [:S,NAT:]
[:S,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:S,NAT:] is non empty set
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
RLSStruct(# (PFuncs (S,REAL)),(S),(addpfunc S),(S) #) is non empty strict RLSStruct
the carrier of (S) is non empty set
bool the carrier of (S) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ex b<sub>2</sub> being Element of M st
( x . b<sub>2</sub> = 0 & dom b<sub>1</sub> = b<sub>2</sub> ` & b₁ is_integrable_on x ) } is set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
dom y is Element of bool S
f is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
g is Element of M
x . g is ext-real Element of ExtREAL
dom f is Element of bool S
g ` is Element of bool S
S \ g is set
f is Element of M
x . f is ext-real Element of ExtREAL
f ` is Element of bool S
S \ f is set
dom (X (#) y) is Element of bool S
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool the carrier of (X)
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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
y is Element of the carrier of (X)
a is Element of the carrier of (X)
y + a is Element of the carrier of (X)
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) . (y,a) is Element of the carrier of (X)
[y,a] is set
{y,a} is non empty set
{y} is non empty set
{{y,a},{y}} is non empty set
the addF of (X) . [y,a] is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom g is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom f is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom f is Element of bool X
(dom g) /\ (dom f) is Element of bool X
f is set
f . f is V11() real ext-real set
g . f is V11() real ext-real set
f . f is V11() real ext-real set
(g . f) + (f . f) is V11() real ext-real set
g + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is Element of the carrier of (X)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom f is Element of bool X
y is V11() real ext-real Element of REAL
y * a is Element of the carrier of (X)
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (y,a) is set
[y,a] is set
{y,a} is non empty set
{y} is non empty V166() V167() V168() set
{{y,a},{y}} is non empty set
the Mult of (X) . [y,a] is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom g is Element of bool X
f is set
g . f is V11() real ext-real set
f . f is V11() real ext-real set
y * (f . f) is V11() real ext-real Element of REAL
y (#) f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool the carrier of (X)
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 non empty set
bool X is non empty set
bool (bool X) is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is Element of the carrier of (X,S,M)
f is Element of the carrier of (X,S,M)
a + f is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (a,f) is Element of the carrier of (X,S,M)
[a,f] is set
{a,f} is non empty set
{a} is non empty set
{{a,f},{a}} is non empty set
the addF of (X,S,M) . [a,f] is set
g is Element of the carrier of (X)
f is Element of the carrier of (X)
g + f is Element of the carrier of (X)
the addF of (X) . (g,f) is Element of the carrier of (X)
[g,f] is set
{g,f} is non empty set
{g} is non empty set
{{g,f},{g}} is non empty set
the addF of (X) . [g,f] is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom f is Element of bool X
A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom A is Element of bool X
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
dom n is Element of bool X
(dom A) /\ (dom n) is Element of bool X
n is set
f . n is V11() real ext-real set
x . n is V11() real ext-real set
y . n is V11() real ext-real set
(x . n) + (y . n) is V11() real ext-real set
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
(S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(S,M,x) is non empty add-closed ((S)) Element of bool the carrier of (S)
(S) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
S --> 0 is V13() V16(S) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(S, NAT ) V173() V174() V175() V176() Element of bool [:S,NAT:]
[:S,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:S,NAT:] is non empty set
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
RLSStruct(# (PFuncs (S,REAL)),(S),(addpfunc S),(S) #) is non empty strict RLSStruct
the carrier of (S) is non empty set
bool the carrier of (S) is non empty set
{ b₁ where b₁ is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ex b₂ being Element of M st
( x . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on x ) } is set
0. (S) is V55((S)) Element of the carrier of (S)
the ZeroF of (S) is Element of the carrier of (S)
In ((0. (S)),(S,M,x)) is Element of (S,M,x)
add| ((S,M,x),(S)) is V13() V16([:(S,M,x),(S,M,x):]) V17((S,M,x)) Function-like V41([:(S,M,x),(S,M,x):],(S,M,x)) Element of bool [:[:(S,M,x),(S,M,x):],(S,M,x):]
[:(S,M,x),(S,M,x):] is non empty set
[:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
bool [:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
the addF of (S) is V13() V16([: the carrier of (S), the carrier of (S):]) V17( the carrier of (S)) Function-like V41([: the carrier of (S), the carrier of (S):], the carrier of (S)) Element of bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):]
[: the carrier of (S), the carrier of (S):] is non empty set
[:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is non empty set
bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is non empty set
the addF of (S) || (S,M,x) is set
K65( the addF of (S),[:(S,M,x),(S,M,x):]) is V13() set
((S),(S,M,x)) is V13() V16([:REAL,(S,M,x):]) V17((S,M,x)) Function-like V41([:REAL,(S,M,x):],(S,M,x)) Element of bool [:[:REAL,(S,M,x):],(S,M,x):]
[:REAL,(S,M,x):] is non empty set
[:[:REAL,(S,M,x):],(S,M,x):] is non empty set
bool [:[:REAL,(S,M,x):],(S,M,x):] is non empty set
[:REAL, the carrier of (S):] is non empty set
the Mult of (S) is V13() V16([:REAL, the carrier of (S):]) V17( the carrier of (S)) Function-like V41([:REAL, the carrier of (S):], the carrier of (S)) Element of bool [:[:REAL, the carrier of (S):], the carrier of (S):]
[:[:REAL, the carrier of (S):], the carrier of (S):] is non empty set
bool [:[:REAL, the carrier of (S):], the carrier of (S):] is non empty set
the Mult of (S) | [:REAL,(S,M,x):] is V13() V16([:REAL, the carrier of (S):]) V17( the carrier of (S)) Function-like Element of bool [:[:REAL, the carrier of (S):], the carrier of (S):]
RLSStruct(# (S,M,x),(In ((0. (S)),(S,M,x))),(add| ((S,M,x),(S))),((S),(S,M,x)) #) is non empty strict RLSStruct
the carrier of (S,M,x) is non empty set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
a is Element of the carrier of (S,M,x)
X * a is Element of the carrier of (S,M,x)
the Mult of (S,M,x) is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([:REAL, the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
[:REAL, the carrier of (S,M,x):] is non empty set
[:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) . (X,a) is set
[X,a] is set
{X,a} is non empty set
{X} is non empty V166() V167() V168() set
{{X,a},{X}} is non empty set
the Mult of (S,M,x) . [X,a] is set
f is Element of the carrier of (S)
X * f is Element of the carrier of (S)
the Mult of (S) . (X,f) is set
[X,f] is set
{X,f} is non empty set
{{X,f},{X}} is non empty set
the Mult of (S) . [X,f] is set
g is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
dom g is Element of bool S
dom y is Element of bool S
f is set
g . f is V11() real ext-real set
y . f is V11() real ext-real set
X * (y . f) is V11() real ext-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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom x is Element of bool X
(X,0) | (dom x) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
y is Element of the carrier of (X,S,M)
(- 1) * y is Element of the carrier of (X,S,M)
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) . ((- 1),y) is set
[(- 1),y] is set
{(- 1),y} is non empty set
{(- 1)} is non empty V166() V167() V168() set
{{(- 1),y},{(- 1)}} is non empty set
the Mult of (X,S,M) . [(- 1),y] is set
y + ((- 1) * y) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (y,((- 1) * y)) is Element of the carrier of (X,S,M)
[y,((- 1) * y)] is set
{y,((- 1) * y)} is non empty set
{y} is non empty set
{{y,((- 1) * y)},{y}} is non empty set
the addF of (X,S,M) . [y,((- 1) * y)] is set
a is Element of the carrier of (X)
(- 1) * a is Element of the carrier of (X)
the Mult of (X) . ((- 1),a) is set
[(- 1),a] is set
{(- 1),a} is non empty set
{{(- 1),a},{(- 1)}} is non empty set
the Mult of (X) . [(- 1),a] is set
a + ((- 1) * a) is Element of the carrier of (X)
the addF of (X) . (a,((- 1) * a)) is Element of the carrier of (X)
[a,((- 1) * a)] is set
{a,((- 1) * a)} is non empty set
{a} is non empty set
{{a,((- 1) * a)},{a}} is non empty set
the addF of (X) . [a,((- 1) * a)] is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom f is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
(X) | (dom x) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
A is Element of S
M . A is ext-real Element of ExtREAL
dom f is Element of bool X
A ` is Element of bool X
X \ A is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
f | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,0) | (f `) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
((X,0) | (f `)) | (f `) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is Element of S
y ` is Element of bool X
X \ y is set
x | (y `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
M . y 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is Element of S
M . a is ext-real Element of ExtREAL
a ` is Element of bool X
X \ a is set
x | (a `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | (a `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
x | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
g ` is Element of bool X
X \ g is set
y | (g `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a | (g `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f \/ g is Element of S
M . (f \/ g) is ext-real Element of ExtREAL
(f \/ g) ` is Element of bool X
X \ (f \/ g) is set
x | ((f \/ g) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(y | (f `)) | ((f \/ g) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | ((f \/ g) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(a | (g `)) | ((f \/ g) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a | ((f \/ g) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
g ` is Element of bool X
X \ g is set
x | (g `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | (g `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
a | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g \/ f is Element of S
(g \/ f) ` is Element of bool X
X \ (g \/ f) is set
x | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(y | (g `)) | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(f | (f `)) | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(x + a) | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(y | ((g \/ f) `)) + (f | ((g \/ f) `)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(y + f) | ((g \/ f) `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
M . (g \/ f) is ext-real Element of ExtREAL
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
f is Element of M
x . f is ext-real Element of ExtREAL
f ` is Element of bool S
S \ f is set
y | (f `) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
a | (f `) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(X (#) y) | (f `) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) (a | (f `)) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(X (#) a) | (f `) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } is set
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
bool the carrier of (X,S,M) is non empty set
x is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
X is V11() real ext-real Element of REAL
S is non empty set
(S,0) is V13() V16(S) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:S,REAL:]
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
(S,0) + (S,0) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) (S,0) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
y is Element of S
dom (X (#) (S,0)) is Element of bool S
bool S is non empty set
(X (#) (S,0)) . y is V11() real ext-real set
(S,0) . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
X * ((S,0) . y) is V11() real ext-real Element of REAL
X * 0 is V11() real ext-real Element of REAL
dom ((S,0) + (S,0)) is Element of bool S
dom (S,0) is Element of bool S
(dom (S,0)) /\ (dom (S,0)) is Element of bool S
y is Element of S
((S,0) + (S,0)) . y is V11() real ext-real set
(S,0) . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
((S,0) . y) + ((S,0) . y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
0 + ((S,0) . y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
(X,S,M) is non empty Element of bool the carrier of (X,S,M)
the carrier of (X,S,M) is non empty set
bool the carrier of (X,S,M) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } is set
a is Element of the carrier of (X,S,M)
f is Element of the carrier of (X,S,M)
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a + f is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (a,f) is Element of the carrier of (X,S,M)
[a,f] is set
{a,f} is non empty set
{a} is non empty set
{{a,f},{a}} is non empty set
the addF of (X,S,M) . [a,f] is set
g + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,0) + (X,0) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of the carrier of (X,S,M)
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is V11() real ext-real Element of REAL
a * f is Element of the carrier of (X,S,M)
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) . (a,f) is set
[a,f] is set
{a,f} is non empty set
{a} is non empty V166() V167() V168() set
{{a,f},{a}} is non empty set
the Mult of (X,S,M) . [a,f] is set
a (#) g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a (#) (X,0) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
(X,S,M) is non empty Element of bool the carrier of (X,S,M)
the carrier of (X,S,M) is non empty set
bool the carrier of (X,S,M) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } is set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
(X,S,M) is non empty add-closed ((X,S,M)) Element of bool the carrier of (X,S,M)
bool the carrier of (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X,S,M)) Element of bool the carrier of (X,S,M)
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
bool the carrier of (X,S,M) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } is set
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
In ((0. (X,S,M)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X,S,M)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) || (X,S,M) is set
K65( the addF of (X,S,M),[:(X,S,M),(X,S,M):]) is V13() set
((X,S,M),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
RLSStruct(# (X,S,M),(In ((0. (X,S,M)),(X,S,M))),(add| ((X,S,M),(X,S,M))),((X,S,M),(X,S,M)) #) 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty RLSStruct
(X,S,M) is non empty add-closed ((X,S,M)) Element of bool the carrier of (X,S,M)
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ex b₂ being Element of S st
( M . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on M ) } is set
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
bool the carrier of (X,S,M) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,S,M,b₁,(X,0)) ) } is set
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
In ((0. (X,S,M)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X,S,M)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) || (X,S,M) is set
K65( the addF of (X,S,M),[:(X,S,M),(X,S,M):]) is V13() set
((X,S,M),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
RLSStruct(# (X,S,M),(In ((0. (X,S,M)),(X,S,M))),(add| ((X,S,M),(X,S,M))),((X,S,M),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is Element of the carrier of (X,S,M)
f is Element of the carrier of (X,S,M)
a + f is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (a,f) is Element of the carrier of (X,S,M)
[a,f] is set
{a,f} is non empty set
{a} is non empty set
{{a,f},{a}} is non empty set
the addF of (X,S,M) . [a,f] is set
g is Element of the carrier of (X,S,M)
f is Element of the carrier of (X,S,M)
g + f is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (g,f) is Element of the carrier of (X,S,M)
[g,f] is set
{g,f} is non empty set
{g} is non empty set
{{g,f},{g}} is non empty set
the addF of (X,S,M) . [g,f] is set
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
(S,M,x) is non empty RLSStruct
(S,M,x) is non empty add-closed ((S,M,x)) Element of bool the carrier of (S,M,x)
(S,M,x) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
(S,M,x) is non empty add-closed ((S)) Element of bool the carrier of (S)
(S) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
S --> 0 is V13() V16(S) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(S, NAT ) V173() V174() V175() V176() Element of bool [:S,NAT:]
[:S,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:S,NAT:] is non empty set
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
RLSStruct(# (PFuncs (S,REAL)),(S),(addpfunc S),(S) #) is non empty strict RLSStruct
the carrier of (S) is non empty set
bool the carrier of (S) is non empty set
{ b₁ where b₁ is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ex b₂ being Element of M st
( x . b₂ = 0 & dom b₁ = b₂ ` & b₁ is_integrable_on x ) } is set
0. (S) is V55((S)) Element of the carrier of (S)
the ZeroF of (S) is Element of the carrier of (S)
In ((0. (S)),(S,M,x)) is Element of (S,M,x)
add| ((S,M,x),(S)) is V13() V16([:(S,M,x),(S,M,x):]) V17((S,M,x)) Function-like V41([:(S,M,x),(S,M,x):],(S,M,x)) Element of bool [:[:(S,M,x),(S,M,x):],(S,M,x):]
[:(S,M,x),(S,M,x):] is non empty set
[:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
bool [:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
the addF of (S) is V13() V16([: the carrier of (S), the carrier of (S):]) V17( the carrier of (S)) Function-like V41([: the carrier of (S), the carrier of (S):], the carrier of (S)) Element of bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):]
[: the carrier of (S), the carrier of (S):] is non empty set
[:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is non empty set
bool [:[: the carrier of (S), the carrier of (S):], the carrier of (S):] is non empty set
the addF of (S) || (S,M,x) is set
K65( the addF of (S),[:(S,M,x),(S,M,x):]) is V13() set
((S),(S,M,x)) is V13() V16([:REAL,(S,M,x):]) V17((S,M,x)) Function-like V41([:REAL,(S,M,x):],(S,M,x)) Element of bool [:[:REAL,(S,M,x):],(S,M,x):]
[:REAL,(S,M,x):] is non empty set
[:[:REAL,(S,M,x):],(S,M,x):] is non empty set
bool [:[:REAL,(S,M,x):],(S,M,x):] is non empty set
[:REAL, the carrier of (S):] is non empty set
the Mult of (S) is V13() V16([:REAL, the carrier of (S):]) V17( the carrier of (S)) Function-like V41([:REAL, the carrier of (S):], the carrier of (S)) Element of bool [:[:REAL, the carrier of (S):], the carrier of (S):]
[:[:REAL, the carrier of (S):], the carrier of (S):] is non empty set
bool [:[:REAL, the carrier of (S):], the carrier of (S):] is non empty set
the Mult of (S) | [:REAL,(S,M,x):] is V13() V16([:REAL, the carrier of (S):]) V17( the carrier of (S)) Function-like Element of bool [:[:REAL, the carrier of (S):], the carrier of (S):]
RLSStruct(# (S,M,x),(In ((0. (S)),(S,M,x))),(add| ((S,M,x),(S))),((S),(S,M,x)) #) is non empty strict RLSStruct
the carrier of (S,M,x) is non empty set
bool the carrier of (S,M,x) is non empty set
(S,0) is V13() V16(S) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:S,REAL:]
{ b₁ where b₁ is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b₁ in (S,M,x) & (S,M,x,b₁,(S,0)) ) } is set
0. (S,M,x) is V55((S,M,x)) Element of the carrier of (S,M,x)
the ZeroF of (S,M,x) is Element of the carrier of (S,M,x)
In ((0. (S,M,x)),(S,M,x)) is Element of (S,M,x)
add| ((S,M,x),(S,M,x)) is V13() V16([:(S,M,x),(S,M,x):]) V17((S,M,x)) Function-like V41([:(S,M,x),(S,M,x):],(S,M,x)) Element of bool [:[:(S,M,x),(S,M,x):],(S,M,x):]
[:(S,M,x),(S,M,x):] is non empty set
[:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
bool [:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
the addF of (S,M,x) is V13() V16([: the carrier of (S,M,x), the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):]
[: the carrier of (S,M,x), the carrier of (S,M,x):] is non empty set
[:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the addF of (S,M,x) || (S,M,x) is set
K65( the addF of (S,M,x),[:(S,M,x),(S,M,x):]) is V13() set
((S,M,x),(S,M,x)) is V13() V16([:REAL,(S,M,x):]) V17((S,M,x)) Function-like V41([:REAL,(S,M,x):],(S,M,x)) Element of bool [:[:REAL,(S,M,x):],(S,M,x):]
[:REAL,(S,M,x):] is non empty set
[:[:REAL,(S,M,x):],(S,M,x):] is non empty set
bool [:[:REAL,(S,M,x):],(S,M,x):] is non empty set
[:REAL, the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([:REAL, the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
[:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) | [:REAL,(S,M,x):] is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
RLSStruct(# (S,M,x),(In ((0. (S,M,x)),(S,M,x))),(add| ((S,M,x),(S,M,x))),((S,M,x),(S,M,x)) #) is non empty strict RLSStruct
the carrier of (S,M,x) is non empty set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
a is Element of the carrier of (S,M,x)
X * a is Element of the carrier of (S,M,x)
the Mult of (S,M,x) is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([:REAL, the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
[:REAL, the carrier of (S,M,x):] is non empty set
[:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) . (X,a) is set
[X,a] is set
{X,a} is non empty set
{X} is non empty V166() V167() V168() set
{{X,a},{X}} is non empty set
the Mult of (S,M,x) . [X,a] is set
f is Element of the carrier of (S,M,x)
X * f is Element of the carrier of (S,M,x)
the Mult of (S,M,x) . (X,f) is set
[X,f] is set
{X,f} is non empty set
{{X,f},{X}} is non empty set
the Mult of (S,M,x) . [X,f] 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & x in (X,S,M) & (X,S,M,x,b<sub>1</sub>) ) } is set
bool (X,S,M) is non empty set
y is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & x in (X,S,M) & (X,S,M,x,b₁) ) } is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & x in (X,S,M) & (X,S,M,x,b<sub>1</sub>) ) } 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & x in (X,S,M) & (X,S,M,x,b₁) ) } is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,y) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & y in (X,S,M) & (X,S,M,y,b₁) ) } is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & x in (X,S,M) & (X,S,M,x,b<sub>1</sub>) ) } is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,y) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & y in (X,S,M) & (X,S,M,y,b<sub>1</sub>) ) } 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & x in (X,S,M) & (X,S,M,x,b₁) ) } is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,y) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & y in (X,S,M) & (X,S,M,y,b₁) ) } is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,a) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a in (X,S,M) & (X,S,M,a,b₁) ) } is set
x + a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(x + a)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & x + a in (X,S,M) & (X,S,M,x + a,b₁) ) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
y + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(y + f)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & y + f in (X,S,M) & (X,S,M,y + f,b₁) ) } is set
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
(S,M,x) is non empty add-closed ((S)) Element of bool the carrier of (S)
(S) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
S --> 0 is V13() V16(S) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(S, NAT ) V173() V174() V175() V176() Element of bool [:S,NAT:]
[:S,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:S,NAT:] is non empty set
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
RLSStruct(# (PFuncs (S,REAL)),(S),(addpfunc S),(S) #) is non empty strict RLSStruct
the carrier of (S) is non empty set
bool the carrier of (S) is non empty set
{ b₁ where b₁ is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ex b₂ being Element of M st
( x . b₂ = 0 & dom b₁ = b₂ ` & b<sub>1</sub> is_integrable_on x ) } is set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,y) is Element of bool (S,M,x)
bool (S,M,x) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & y in (S,M,x) & (S,M,x,y,b<sub>1</sub>) ) } is set
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(X (#) y)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & X (#) y in (S,M,x) & (S,M,x,X (#) y,b<sub>1</sub>) ) } is set
a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,a) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & a in (S,M,x) & (S,M,x,a,b<sub>1</sub>) ) } is set
X (#) a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(X (#) a)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & X (#) a in (S,M,x) & (S,M,x,X (#) a,b<sub>1</sub>) ) } 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
bool (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
bool (X,S,M) is non empty Element of bool (bool (X,S,M))
y is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,a) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a in (X,S,M) & (X,S,M,a,b₁) ) } is set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
(X,S,M,(X,0)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,0) in (X,S,M) & (X,S,M,(X,0),b₁) ) } 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool (bool (X,S,M))
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M) } is set
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
y is Element of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
a is Element of (X,S,M)
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,g) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & g in (X,S,M) & (X,S,M,g,b<sub>1</sub>) ) } is set
f + g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f + g)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f + g in (X,S,M) & (X,S,M,f + g,b<sub>1</sub>) ) } is set
f is Element of (X,S,M)
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n + n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
A is Element of (X,S,M)
(X,S,M,(n + n)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & n + n in (X,S,M) & (X,S,M,n + n,b<sub>1</sub>) ) } is set
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n + n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(n + n)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & n + n in (X,S,M) & (X,S,M,n + n,b<sub>1</sub>) ) } is set
y is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
a is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
f is Element of (X,S,M)
g is Element of (X,S,M)
a . (f,g) is Element of (X,S,M)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
a . [f,g] is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f + f)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f + f in (X,S,M) & (X,S,M,f + f,b<sub>1</sub>) ) } is set
y is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
a is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
f is Element of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
g is Element of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
y . (f,g) is Element of (X,S,M)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
y . [f,g] is set
f + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f + f)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f + f in (X,S,M) & (X,S,M,f + f,b<sub>1</sub>) ) } is set
a . (f,g) is Element of (X,S,M)
a . [f,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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
(X,S,M,(X,0)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,0) in (X,S,M) & (X,S,M,(X,0),b₁) ) } is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x is Element of (X,S,M)
(X,S,M,a) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a in (X,S,M) & (X,S,M,a,b₁) ) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is Element of (X,S,M)
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool (bool (X,S,M))
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M) } is set
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
a is Element of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
y is V11() real ext-real Element of REAL
y (#) f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(y (#) f)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & y (#) f in (X,S,M) & (X,S,M,y (#) f,b<sub>1</sub>) ) } is set
f is Element of (X,S,M)
A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y (#) A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of (X,S,M)
(X,S,M,(y (#) A)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & y (#) A in (X,S,M) & (X,S,M,y (#) A,b<sub>1</sub>) ) } is set
A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y (#) A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(y (#) A)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & y (#) A in (X,S,M) & (X,S,M,y (#) A,b<sub>1</sub>) ) } is set
y is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
a is V11() real ext-real Element of REAL
f is Element of (X,S,M)
y . (a,f) is Element of (X,S,M)
[a,f] is set
{a,f} is non empty set
{a} is non empty V166() V167() V168() set
{{a,f},{a}} is non empty set
y . [a,f] is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a (#) g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(a (#) g)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & a (#) g in (X,S,M) & (X,S,M,a (#) g,b<sub>1</sub>) ) } is set
y is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
a is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
g is Element of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
f is V11() real ext-real Element of REAL
y . (f,g) is Element of (X,S,M)
[f,g] is set
{f,g} is non empty set
{f} is non empty V166() V167() V168() set
{{f,g},{f}} is non empty set
y . [f,g] is set
f (#) f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f (#) f)) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f (#) f in (X,S,M) & (X,S,M,f (#) f,b<sub>1</sub>) ) } is set
a . (f,g) is Element of (X,S,M)
a . [f,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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty Element of bool (bool (X,S,M))
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
[:(X,S,M),(X,S,M):] is non empty set
(X,S,M) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
(X,S,M) is Element of (X,S,M)
[:REAL,(X,S,M):] is non empty set
(X,S,M) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is non empty strict RLSStruct
the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is non empty set
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f + f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V13() V16([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]) V17( the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Function-like V41([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]
[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
[:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,f) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,f] is set
{f,f} is non empty set
{f} is non empty set
{{f,f},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,f] is set
f + f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,f) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,f] is set
{f,f} is non empty set
{f} is non empty set
{{f,f},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,f] is set
A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,A) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & A in (X,S,M) & (X,S,M,A,b₁) ) } is set
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,n) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n in (X,S,M) & (X,S,M,n,b₁) ) } is set
A + n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(A + n)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & A + n in (X,S,M) & (X,S,M,A + n,b₁) ) } is set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V55( RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is non empty set
the ZeroF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f + (0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V13() V16([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]) V17( the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Function-like V41([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]
[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
[:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,(0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #))) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,(0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #))] is set
{f,(0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #))} is non empty set
{f} is non empty set
{{f,(0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #))},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,(0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #))] is set
A is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,A) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & A in (X,S,M) & (X,S,M,A,b₁) ) } is set
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
A + f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n is Element of the carrier of (X,S,M)
n is Element of the carrier of (X,S,M)
n + n is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (n,n) is Element of the carrier of (X,S,M)
[n,n] is set
{n,n} is non empty set
{n} is non empty set
{{n,n},{n}} is non empty set
the addF of (X,S,M) . [n,n] is set
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
n + (0. (X,S,M)) is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (n,(0. (X,S,M))) is Element of the carrier of (X,S,M)
[n,(0. (X,S,M))] is set
{n,(0. (X,S,M))} is non empty set
{{n,(0. (X,S,M))},{n}} is non empty set
the addF of (X,S,M) . [n,(0. (X,S,M))] is set
the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is non empty set
f is V11() real ext-real set
f is V11() real ext-real set
A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,a1) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a1 in (X,S,M) & (X,S,M,a1,b₁) ) } is set
n is V11() real ext-real Element of REAL
(X,S,M) . (n,A) is set
[n,A] is set
{n,A} is non empty set
{n} is non empty V166() V167() V168() set
{{n,A},{n}} is non empty set
(X,S,M) . [n,A] is set
f (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f (#) a1)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f (#) a1 in (X,S,M) & (X,S,M,f (#) a1,b₁) ) } is set
f * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V13() V16([:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]) V17( the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Function-like V41([:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of bool [:[:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]
[:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
[:[:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
bool [:[:REAL, the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,A) is set
[f,A] is set
{f,A} is non empty set
{f} is non empty V166() V167() V168() set
{{f,A},{f}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,A] is set
n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
b1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,b1) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & b1 in (X,S,M) & (X,S,M,b1,b₁) ) } is set
(X,S,M) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
0. (X) is V55((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
In ((0. (X)),(X,S,M)) is Element of (X,S,M)
add| ((X,S,M),(X)) is V13() V16([:(X,S,M),(X,S,M):]) V17((X,S,M)) Function-like V41([:(X,S,M),(X,S,M):],(X,S,M)) Element of bool [:[:(X,S,M),(X,S,M):],(X,S,M):]
[:(X,S,M),(X,S,M):] is non empty set
[:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
bool [:[:(X,S,M),(X,S,M):],(X,S,M):] is non empty set
the addF of (X) is V13() V16([: the carrier of (X), the carrier of (X):]) V17( the carrier of (X)) Function-like 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) || (X,S,M) is set
K65( the addF of (X),[:(X,S,M),(X,S,M):]) is V13() set
((X),(X,S,M)) is V13() V16([:REAL,(X,S,M):]) V17((X,S,M)) Function-like V41([:REAL,(X,S,M):],(X,S,M)) Element of bool [:[:REAL,(X,S,M):],(X,S,M):]
[:REAL,(X,S,M):] is non empty set
[:[:REAL,(X,S,M):],(X,S,M):] is non empty set
bool [:[:REAL,(X,S,M):],(X,S,M):] is non empty set
[:REAL, the carrier of (X):] is non empty set
the Mult of (X) is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like V41([:REAL, the carrier of (X):], the carrier of (X)) Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) | [:REAL,(X,S,M):] is V13() V16([:REAL, the carrier of (X):]) V17( the carrier of (X)) Function-like Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
RLSStruct(# (X,S,M),(In ((0. (X)),(X,S,M))),(add| ((X,S,M),(X))),((X),(X,S,M)) #) is non empty strict RLSStruct
the carrier of (X,S,M) is non empty set
c1 is Element of the carrier of (X,S,M)
n * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) . (n,c1) is set
[n,c1] is set
{n,c1} is non empty set
{{n,c1},{n}} is non empty set
the Mult of (X,S,M) . [n,c1] is set
(X,S,M) . (n,n) is set
[n,n] is set
{n,n} is non empty set
{{n,n},{n}} is non empty set
(X,S,M) . [n,n] is set
f (#) b1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f (#) b1)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f (#) b1 in (X,S,M) & (X,S,M,f (#) b1,b₁) ) } is set
n * n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (n,n) is set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [n,n] is set
a1 + b1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
b1 is Element of the carrier of (X,S,M)
c1 + b1 is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (c1,b1) is Element of the carrier of (X,S,M)
[c1,b1] is set
{c1,b1} is non empty set
{c1} is non empty set
{{c1,b1},{c1}} is non empty set
the addF of (X,S,M) . [c1,b1] is set
f (#) (a1 + b1) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n * (c1 + b1) is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (n,(c1 + b1)) is set
[n,(c1 + b1)] is set
{n,(c1 + b1)} is non empty set
{{n,(c1 + b1)},{n}} is non empty set
the Mult of (X,S,M) . [n,(c1 + b1)] is set
f * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (f,c1) is set
[f,c1] is set
{f,c1} is non empty set
{{f,c1},{f}} is non empty set
the Mult of (X,S,M) . [f,c1] is set
f * b1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (f,b1) is set
[f,b1] is set
{f,b1} is non empty set
{{f,b1},{f}} is non empty set
the Mult of (X,S,M) . [f,b1] is set
(f * c1) + (f * b1) is Element of the carrier of (X,S,M)
the addF of (X,S,M) . ((f * c1),(f * b1)) is Element of the carrier of (X,S,M)
[(f * c1),(f * b1)] is set
{(f * c1),(f * b1)} is non empty set
{(f * c1)} is non empty set
{{(f * c1),(f * b1)},{(f * c1)}} is non empty set
the addF of (X,S,M) . [(f * c1),(f * b1)] is set
n * b1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (n,b1) is set
[n,b1] is set
{n,b1} is non empty set
{{n,b1},{n}} is non empty set
the Mult of (X,S,M) . [n,b1] is set
(f (#) a1) + (f (#) b1) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M) . (A,n) is set
[A,n] is set
{A,n} is non empty set
{A} is non empty set
{{A,n},{A}} is non empty set
(X,S,M) . [A,n] is set
(X,S,M,(a1 + b1)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a1 + b1 in (X,S,M) & (X,S,M,a1 + b1,b₁) ) } is set
A + n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V13() V16([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]) V17( the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Function-like V41([: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):]
[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
[:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
bool [:[: the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #), the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):], the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #):] is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (A,n) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [A,n] is set
n * (A + n) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (n,(A + n)) is set
[n,(A + n)] is set
{n,(A + n)} is non empty set
{{n,(A + n)},{n}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [n,(A + n)] is set
(X,S,M,((f (#) a1) + (f (#) b1))) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (f (#) a1) + (f (#) b1) in (X,S,M) & (X,S,M,(f (#) a1) + (f (#) b1),b₁) ) } is set
f * (A + n) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,(A + n)) is set
[f,(A + n)] is set
{f,(A + n)} is non empty set
{{f,(A + n)},{f}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,(A + n)] is set
f * n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,n) is set
[f,n] is set
{f,n} is non empty set
{{f,n},{f}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,n] is set
(f * A) + (f * n) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((f * A),(f * n)) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[(f * A),(f * n)] is set
{(f * A),(f * n)} is non empty set
{(f * A)} is non empty set
{{(f * A),(f * n)},{(f * A)}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(f * A),(f * n)] is set
f (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
c is V11() real ext-real Element of REAL
c * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (c,c1) is set
[c,c1] is set
{c,c1} is non empty set
{c} is non empty V166() V167() V168() set
{{c,c1},{c}} is non empty set
the Mult of (X,S,M) . [c,c1] is set
(X,S,M) . (c,A) is set
[c,A] is set
{c,A} is non empty set
{{c,A},{c}} is non empty set
(X,S,M) . [c,A] is set
(X,S,M,(f (#) a1)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f (#) a1 in (X,S,M) & (X,S,M,f (#) a1,b₁) ) } is set
c * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (c,A) is set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [c,A] is set
f + f is V11() real ext-real set
(f + f) (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n + c is V11() real ext-real Element of REAL
(n + c) * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . ((n + c),c1) is set
[(n + c),c1] is set
{(n + c),c1} is non empty set
{(n + c)} is non empty V166() V167() V168() set
{{(n + c),c1},{(n + c)}} is non empty set
the Mult of (X,S,M) . [(n + c),c1] is set
f * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (f,c1) is set
[f,c1] is set
{f,c1} is non empty set
{f} is non empty V166() V167() V168() set
{{f,c1},{f}} is non empty set
the Mult of (X,S,M) . [f,c1] is set
(f * c1) + (f * c1) is Element of the carrier of (X,S,M)
the addF of (X,S,M) . ((f * c1),(f * c1)) is Element of the carrier of (X,S,M)
[(f * c1),(f * c1)] is set
{(f * c1),(f * c1)} is non empty set
{{(f * c1),(f * c1)},{(f * c1)}} is non empty set
the addF of (X,S,M) . [(f * c1),(f * c1)] is set
(f (#) a1) + (f (#) a1) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(n + c) * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((n + c),A) is set
[(n + c),A] is set
{(n + c),A} is non empty set
{{(n + c),A},{(n + c)}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(n + c),A] is set
(X,S,M,((f (#) a1) + (f (#) a1))) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (f (#) a1) + (f (#) a1) in (X,S,M) & (X,S,M,(f (#) a1) + (f (#) a1),b₁) ) } is set
(f + f) * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((f + f),A) is set
[(f + f),A] is set
{(f + f),A} is non empty set
{(f + f)} is non empty V166() V167() V168() set
{{(f + f),A},{(f + f)}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(f + f),A] is set
f * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,A) is set
[f,A] is set
{f,A} is non empty set
{{f,A},{f}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,A] is set
(f * A) + (f * A) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((f * A),(f * A)) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[(f * A),(f * A)] is set
{(f * A),(f * A)} is non empty set
{{(f * A),(f * A)},{(f * A)}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(f * A),(f * A)] is set
f (#) (f (#) a1) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f * (f * c1) is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (f,(f * c1)) is set
[f,(f * c1)] is set
{f,(f * c1)} is non empty set
{{f,(f * c1)},{f}} is non empty set
the Mult of (X,S,M) . [f,(f * c1)] is set
n * c is V11() real ext-real Element of REAL
(n * c) * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . ((n * c),c1) is set
[(n * c),c1] is set
{(n * c),c1} is non empty set
{(n * c)} is non empty V166() V167() V168() set
{{(n * c),c1},{(n * c)}} is non empty set
the Mult of (X,S,M) . [(n * c),c1] is set
f * f is V11() real ext-real set
(f * f) (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(n * c) * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((n * c),A) is set
[(n * c),A] is set
{(n * c),A} is non empty set
{{(n * c),A},{(n * c)}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(n * c),A] is set
c (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n (#) (c (#) a1) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(n (#) (c (#) a1))) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n (#) (c (#) a1) in (X,S,M) & (X,S,M,n (#) (c (#) a1),b₁) ) } is set
f * (f * A) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,(f * A)) is set
[f,(f * A)] is set
{f,(f * A)} is non empty set
{{f,(f * A)},{f}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,(f * A)] is set
(f * f) * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((f * f),A) is set
[(f * f),A] is set
{(f * f),A} is non empty set
{(f * f)} is non empty V166() V167() V168() set
{{(f * f),A},{(f * f)}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(f * f),A] is set
1 (#) a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
1 * c1 is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . (1,c1) is set
[1,c1] is set
{1,c1} is non empty set
{1} is non empty V166() V167() V168() V169() V170() V171() set
{{1,c1},{1}} is non empty set
the Mult of (X,S,M) . [1,c1] is set
1 * A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (1,A) is set
[1,A] is set
{1,A} is non empty set
{{1,A},{1}} is non empty set
the Mult of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [1,A] is set
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
(- 1) (#) f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,((- 1) (#) f)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (- 1) (#) f in (X,S,M) & (X,S,M,(- 1) (#) f,b₁) ) } is set
n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f + n is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,n) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,n] is set
{f,n} is non empty set
{f} is non empty set
{{f,n},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,n] is set
0. RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is V55( RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)) Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the ZeroF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
n is Element of the carrier of (X,S,M)
(- 1) * n is Element of the carrier of (X,S,M)
the Mult of (X,S,M) . ((- 1),n) is set
[(- 1),n] is set
{(- 1),n} is non empty set
{(- 1)} is non empty V166() V167() V168() set
{{(- 1),n},{(- 1)}} is non empty set
the Mult of (X,S,M) . [(- 1),n] is set
n + ((- 1) * n) is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (n,((- 1) * n)) is Element of the carrier of (X,S,M)
[n,((- 1) * n)] is set
{n,((- 1) * n)} is non empty set
{n} is non empty set
{{n,((- 1) * n)},{n}} is non empty set
the addF of (X,S,M) . [n,((- 1) * n)] is set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
c is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f + ((- 1) (#) f) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(f + ((- 1) (#) f))) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f + ((- 1) (#) f) in (X,S,M) & (X,S,M,f + ((- 1) (#) f),b₁) ) } is set
b1 is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,b1) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & b1 in (X,S,M) & (X,S,M,b1,b₁) ) } is set
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
f + f is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,f) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,f] is set
{f,f} is non empty set
{f} is non empty set
{{f,f},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,f] is set
A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
(f + f) + A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . ((f + f),A) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[(f + f),A] is set
{(f + f),A} is non empty set
{(f + f)} is non empty set
{{(f + f),A},{(f + f)}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [(f + f),A] is set
f + A is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,A) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,A] is set
{f,A} is non empty set
{f} is non empty set
{{f,A},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,A] is set
f + (f + A) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . (f,(f + A)) is Element of the carrier of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #)
[f,(f + A)] is set
{f,(f + A)} is non empty set
{{f,(f + A)},{f}} is non empty set
the addF of RLSStruct(# (X,S,M),(X,S,M),(X,S,M),(X,S,M) #) . [f,(f + A)] is set
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,n) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n in (X,S,M) & (X,S,M,n,b₁) ) } is set
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,n) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n in (X,S,M) & (X,S,M,n,b₁) ) } is set
c is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,c) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & c in (X,S,M) & (X,S,M,c,b₁) ) } is set
(X,S,M) . (f,A) is set
(X,S,M) . [f,A] is set
n + c is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,(n + c)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n + c in (X,S,M) & (X,S,M,n + c,b₁) ) } is set
b1 is Element of the carrier of (X,S,M)
c1 is Element of the carrier of (X,S,M)
b1 + c1 is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (b1,c1) is Element of the carrier of (X,S,M)
[b1,c1] is set
{b1,c1} is non empty set
{b1} is non empty set
{{b1,c1},{b1}} is non empty set
the addF of (X,S,M) . [b1,c1] is set
n + (n + c) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a1 is Element of the carrier of (X,S,M)
a1 + (b1 + c1) is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (a1,(b1 + c1)) is Element of the carrier of (X,S,M)
[a1,(b1 + c1)] is set
{a1,(b1 + c1)} is non empty set
{a1} is non empty set
{{a1,(b1 + c1)},{a1}} is non empty set
the addF of (X,S,M) . [a1,(b1 + c1)] is set
a1 + b1 is Element of the carrier of (X,S,M)
the addF of (X,S,M) . (a1,b1) is Element of the carrier of (X,S,M)
[a1,b1] is set
{a1,b1} is non empty set
{{a1,b1},{a1}} is non empty set
the addF of (X,S,M) . [a1,b1] is set
(a1 + b1) + c1 is Element of the carrier of (X,S,M)
the addF of (X,S,M) . ((a1 + b1),c1) is Element of the carrier of (X,S,M)
[(a1 + b1),c1] is set
{(a1 + b1),c1} is non empty set
{(a1 + b1)} is non empty set
{{(a1 + b1),c1},{(a1 + b1)}} is non empty set
the addF of (X,S,M) . [(a1 + b1),c1] is set
n + n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(n + n) + c is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M) . (f,f) is set
(X,S,M) . [f,f] is set
(X,S,M,(n + n)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n + n in (X,S,M) & (X,S,M,n + n,b₁) ) } is set
(X,S,M,(n + (n + c))) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & n + (n + c) in (X,S,M) & (X,S,M,n + (n + c),b₁) ) } is set
f is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of f is non empty set
[: the carrier of f, the carrier of f:] is non empty set
the addF of f is V13() V16([: the carrier of f, the carrier of f:]) V17( the carrier of f) Function-like V41([: the carrier of f, the carrier of f:], the carrier of f) Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
0. f is V55(f) Element of the carrier of f
the ZeroF of f is Element of the carrier of f
[:REAL, the carrier of f:] is non empty set
the Mult of f is V13() V16([:REAL, the carrier of f:]) V17( the carrier of f) Function-like V41([:REAL, the carrier of f:], the carrier of f) Element of bool [:[:REAL, the carrier of f:], the carrier of f:]
[:[:REAL, the carrier of f:], the carrier of f:] is non empty set
bool [:[:REAL, the carrier of f:], the carrier of f:] is non empty set
x is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is non empty set
the addF of x is V13() V16([: the carrier of x, the carrier of x:]) V17( the carrier of x) Function-like 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:], 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
0. x is V55(x) Element of the carrier of x
the ZeroF of x is Element of the carrier of x
[:REAL, the carrier of x:] is non empty set
the Mult of x is V13() V16([:REAL, the carrier of x:]) V17( the carrier of x) Function-like V41([:REAL, the carrier of x:], the carrier of x) Element of bool [:[:REAL, the carrier of x:], the carrier of x:]
[:[:REAL, the carrier of x:], the carrier of x:] is non empty set
bool [:[:REAL, the carrier of x:], the carrier of x:] is non empty set
y is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of y is non empty set
[: the carrier of y, the carrier of y:] is non empty set
the addF of y is V13() V16([: the carrier of y, the carrier of y:]) V17( the carrier of y) Function-like V41([: the carrier of y, the carrier of y:], the carrier of y) Element of bool [:[: the carrier of y, the carrier of y:], the carrier of y:]
[:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty set
bool [:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty set
0. y is V55(y) Element of the carrier of y
the ZeroF of y is Element of the carrier of y
[:REAL, the carrier of y:] is non empty set
the Mult of y is V13() V16([:REAL, the carrier of y:]) V17( the carrier of y) Function-like V41([:REAL, the carrier of y:], the carrier of y) Element of bool [:[:REAL, the carrier of y:], the carrier of y:]
[:[:REAL, the carrier of y:], the carrier of y:] is non empty set
bool [:[:REAL, the carrier of y:], the carrier of y:] is non empty set
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,x) is ext-real Element of ExtREAL
R_EAL x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL x)) is ext-real Element of ExtREAL
max+ (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
max- (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL x))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL x)))),K119((integral+ (M,(max- (R_EAL x)))))) is ext-real set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,y) is ext-real Element of ExtREAL
R_EAL y is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL y)) is ext-real Element of ExtREAL
max+ (R_EAL y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL y))) is ext-real Element of ExtREAL
max- (R_EAL y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL y))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL y)))) - (integral+ (M,(max- (R_EAL y)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL y))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL y)))),K119((integral+ (M,(max- (R_EAL y)))))) is ext-real set
a is Element of S
M . a is ext-real Element of ExtREAL
a ` is Element of bool X
X \ a is set
x | (a `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | (a `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom x is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
dom f is Element of bool X
g ` is Element of bool X
X \ g is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
a \/ f is Element of S
M . (a \/ f) is ext-real Element of ExtREAL
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom g is Element of bool X
f ` is Element of bool X
X \ f is set
dom (R_EAL x) is Element of bool X
g is Element of S
dom y is Element of bool X
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom f is Element of bool X
f ` is Element of bool X
X \ f is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
a \/ f is Element of S
M . (a \/ f) is ext-real Element of ExtREAL
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
A is Element of S
M . A is ext-real Element of ExtREAL
dom f is Element of bool X
A ` is Element of bool X
X \ A is set
dom (R_EAL y) is Element of bool X
f is Element of S
f \/ f is Element of S
(a `) \ (f \/ f) is Element of bool X
a \/ (f \/ f) is Element of S
(a \/ (f \/ f)) ` is Element of bool X
X \ (a \/ (f \/ f)) is set
f \/ (a \/ f) is Element of S
(f \/ (a \/ f)) ` is Element of bool X
X \ (f \/ (a \/ f)) is set
(f `) \ (a \/ f) is Element of bool X
f \/ (a \/ f) is Element of S
(f \/ (a \/ f)) ` is Element of bool X
X \ (f \/ (a \/ f)) is set
(f `) \ (a \/ f) is Element of bool X
x | ((a `) \ (f \/ f)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(y | (a `)) | ((a `) \ (f \/ f)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | ((a `) \ (f \/ f)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | ((f `) \ (a \/ f)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(y | ((f `) \ (a \/ f)))) is ext-real Element of ExtREAL
R_EAL (y | ((f `) \ (a \/ f))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (y | ((f `) \ (a \/ f))))) is ext-real Element of ExtREAL
max+ (R_EAL (y | ((f `) \ (a \/ f)))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (y | ((f `) \ (a \/ f)))))) is ext-real Element of ExtREAL
max- (R_EAL (y | ((f `) \ (a \/ f)))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (y | ((f `) \ (a \/ f)))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (y | ((f `) \ (a \/ f))))))) - (integral+ (M,(max- (R_EAL (y | ((f `) \ (a \/ f))))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (y | ((f `) \ (a \/ f)))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (y | ((f `) \ (a \/ f))))))),K119((integral+ (M,(max- (R_EAL (y | ((f `) \ (a \/ f))))))))) 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,x) is ext-real Element of ExtREAL
R_EAL x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL x)) is ext-real Element of ExtREAL
max+ (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
max- (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL x))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL x)))),K119((integral+ (M,(max- (R_EAL x)))))) is ext-real set
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
dom (R_EAL x) is Element of bool X
y 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs y)) is ext-real Element of ExtREAL
R_EAL (abs y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs y))) is ext-real Element of ExtREAL
max+ (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs y)))) is ext-real Element of ExtREAL
max- (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs y)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs y))))) - (integral+ (M,(max- (R_EAL (abs y))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs y)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs y))))),K119((integral+ (M,(max- (R_EAL (abs y))))))) is ext-real set
dom x is Element of bool X
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom a is Element of bool X
f ` is Element of bool X
X \ f is set
a is Element of S
M . a is ext-real Element of ExtREAL
a ` is Element of bool X
X \ a is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
dom f is Element of bool X
g ` is Element of bool X
X \ g is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
x | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(abs x) | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs (y | (f `)) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(abs y) | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom y is Element of bool X
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom g is Element of bool X
f ` is Element of bool X
X \ f 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
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom f is Element of bool X
f ` is Element of bool X
X \ f is set
dom (abs y) is Element of bool X
dom (abs x) is Element of bool X
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
a is Element of the carrier of (X,S,M)
a is Element of the carrier of (X,S,M)
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
y is Element of the carrier of (X,S,M)
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,a) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & a in (X,S,M) & (X,S,M,a,b₁) ) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
g is Element of S
M . g is ext-real Element of ExtREAL
dom f is Element of bool X
g ` is Element of bool X
X \ g 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,x) is ext-real Element of ExtREAL
R_EAL x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL x)) is ext-real Element of ExtREAL
max+ (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL x))) is ext-real Element of ExtREAL
max- (R_EAL x) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL x))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL x)))),K119((integral+ (M,(max- (R_EAL x)))))) is ext-real set
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,y) is ext-real Element of ExtREAL
R_EAL y is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL y)) is ext-real Element of ExtREAL
max+ (R_EAL y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL y))) is ext-real Element of ExtREAL
max- (R_EAL y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL y))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL y)))) - (integral+ (M,(max- (R_EAL y)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL y))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL y)))),K119((integral+ (M,(max- (R_EAL y)))))) is ext-real set
abs y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs y)) is ext-real Element of ExtREAL
R_EAL (abs y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs y))) is ext-real Element of ExtREAL
max+ (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs y)))) is ext-real Element of ExtREAL
max- (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs y)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs y))))) - (integral+ (M,(max- (R_EAL (abs y))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs y)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs y))))),K119((integral+ (M,(max- (R_EAL (abs y))))))) is ext-real set
a is Element of the carrier of (X,S,M)
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 non empty set
bool X is non empty set
bool (bool X) is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
x is Element of the carrier of (X,S,M)
(X,S,M) is non empty Element of bool (bool (X,S,M))
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M) } is set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,y) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & y in (X,S,M) & (X,S,M,y,b<sub>1</sub>) ) } is set
abs y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs y)) is ext-real Element of ExtREAL
R_EAL (abs y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs y))) is ext-real Element of ExtREAL
max+ (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs y)))) is ext-real Element of ExtREAL
max- (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs y)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs y))))) - (integral+ (M,(max- (R_EAL (abs y))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs y)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs y))))),K119((integral+ (M,(max- (R_EAL (abs y))))))) is ext-real set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
f is Element of S
M . f is ext-real Element of ExtREAL
dom a is Element of bool X
f ` is Element of bool X
X \ f is set
x is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
y is Element of the carrier of (X,S,M)
x . y is V11() real ext-real Element of REAL
x is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
y is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
a is Element of the carrier of (X,S,M)
x . a is V11() real ext-real Element of REAL
y . a is V11() real ext-real Element of REAL
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs f)) is ext-real Element of ExtREAL
R_EAL (abs f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs f))) is ext-real Element of ExtREAL
max+ (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs f)))) is ext-real Element of ExtREAL
max- (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs f))))) - (integral+ (M,(max- (R_EAL (abs f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs f))))),K119((integral+ (M,(max- (R_EAL (abs f))))))) is ext-real set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs g)) is ext-real Element of ExtREAL
R_EAL (abs g) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs g))) is ext-real Element of ExtREAL
max+ (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs g)))) is ext-real Element of ExtREAL
max- (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs g)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs g))))),K119((integral+ (M,(max- (R_EAL (abs 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
(X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M), the ZeroF of (X,S,M), the addF of (X,S,M), the Mult of (X,S,M),(X,S,M) #) 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty NORMSTR
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
(X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M), the ZeroF of (X,S,M), the addF of (X,S,M), the Mult of (X,S,M),(X,S,M) #) is strict NORMSTR
the carrier of (X,S,M) is non empty set
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 Element of the carrier of (X,S,M)
||.x.|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
the U9 of (X,S,M) . x is V11() real ext-real Element of REAL
y is Element of the carrier of (X,S,M)
(X,S,M) . y is V11() real ext-real Element of REAL
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs a)) is ext-real Element of ExtREAL
R_EAL (abs a) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() 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 V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs a)))) is ext-real Element of ExtREAL
max- (R_EAL (abs a)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs a)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs a))))),K119((integral+ (M,(max- (R_EAL (abs a))))))) is ext-real set
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (X,S,M) is non empty set
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b<sub>1</sub> in (X,S,M) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & f in (X,S,M) & (X,S,M,f,b<sub>1</sub>) ) } is set
(X,S,M,a) is Element of bool (X,S,M)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b<sub>1</sub> in (X,S,M) & a in (X,S,M) & (X,S,M,a,b<sub>1</sub>) ) } is set
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs g)) is ext-real Element of ExtREAL
R_EAL (abs g) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs g))) is ext-real Element of ExtREAL
max+ (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs g)))) is ext-real Element of ExtREAL
max- (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs g)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs g))))),K119((integral+ (M,(max- (R_EAL (abs g))))))) 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty NORMSTR
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
(X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M), the ZeroF of (X,S,M), the addF of (X,S,M), the Mult of (X,S,M),(X,S,M) #) is strict NORMSTR
the carrier of (X,S,M) is non empty set
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,x) is Element of bool (X,S,M)
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & x in (X,S,M) & (X,S,M,x,b₁) ) } is set
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
y is Element of the carrier of (X,S,M)
||.y.|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
the U9 of (X,S,M) . y is V11() real ext-real Element of REAL
a is Element of the carrier of (X,S,M)
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
[:M,ExtREAL:] is non empty V174() set
bool [:M,ExtREAL:] is non empty set
x is V13() V16(M) V17( ExtREAL ) Function-like V41(M, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:M,ExtREAL:]
(S,M,x) is non empty NORMSTR
(S,M,x) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (S,M,x) is non empty set
the ZeroF of (S,M,x) is Element of the carrier of (S,M,x)
the addF of (S,M,x) is V13() V16([: the carrier of (S,M,x), the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):]
[: the carrier of (S,M,x), the carrier of (S,M,x):] is non empty set
[:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([:REAL, the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
[:REAL, the carrier of (S,M,x):] is non empty set
[:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
(S,M,x) is V13() V16( the carrier of (S,M,x)) V17( REAL ) Function-like V41( the carrier of (S,M,x), REAL ) V173() V174() V175() Element of bool [: the carrier of (S,M,x),REAL:]
[: the carrier of (S,M,x),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (S,M,x),REAL:] is non empty set
NORMSTR(# the carrier of (S,M,x), the ZeroF of (S,M,x), the addF of (S,M,x), the Mult of (S,M,x),(S,M,x) #) is strict NORMSTR
the carrier of (S,M,x) is non empty set
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
X (#) y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
y + a is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
f is Element of the carrier of (S,M,x)
X * f is Element of the carrier of (S,M,x)
the Mult of (S,M,x) is V13() V16([:REAL, the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([:REAL, the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):]
[:REAL, the carrier of (S,M,x):] is non empty set
[:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[:REAL, the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the Mult of (S,M,x) . (X,f) is set
[X,f] is set
{X,f} is non empty set
{X} is non empty V166() V167() V168() set
{{X,f},{X}} is non empty set
the Mult of (S,M,x) . [X,f] is set
g is Element of the carrier of (S,M,x)
f + g is Element of the carrier of (S,M,x)
the addF of (S,M,x) is V13() V16([: the carrier of (S,M,x), the carrier of (S,M,x):]) V17( the carrier of (S,M,x)) Function-like V41([: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x)) Element of bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):]
[: the carrier of (S,M,x), the carrier of (S,M,x):] is non empty set
[:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
bool [:[: the carrier of (S,M,x), the carrier of (S,M,x):], the carrier of (S,M,x):] is non empty set
the addF of (S,M,x) . (f,g) is Element of the carrier of (S,M,x)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of (S,M,x) . [f,g] is set
(S,M,x) is non empty Element of bool (bool (S,M,x))
(S,M,x) is non empty add-closed ((S)) Element of bool the carrier of (S)
(S) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (S,REAL) is non empty functional PartFunc-set of S, REAL
(S) is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (S,REAL)
S --> 0 is V13() V16(S) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(S, NAT ) V173() V174() V175() V176() Element of bool [:S,NAT:]
[:S,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:S,NAT:] is non empty set
addpfunc S is V13() V16([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:(PFuncs (S,REAL)),(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):] is non empty set
[:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:(PFuncs (S,REAL)),(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
(S) is V13() V16([:REAL,(PFuncs (S,REAL)):]) V17( PFuncs (S,REAL)) Function-like V41([:REAL,(PFuncs (S,REAL)):], PFuncs (S,REAL)) Element of bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):]
[:REAL,(PFuncs (S,REAL)):] is non empty set
[:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
bool [:[:REAL,(PFuncs (S,REAL)):],(PFuncs (S,REAL)):] is non empty set
RLSStruct(# (PFuncs (S,REAL)),(S),(addpfunc S),(S) #) is non empty strict RLSStruct
the carrier of (S) is non empty set
bool the carrier of (S) is non empty set
{ b₁ where b₁ is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ex b₂ being Element of M st
( x . b₂ = 0 & dom b₁ = b₂ ` & b<sub>1</sub> is_integrable_on x ) } is set
bool (S,M,x) is non empty set
bool (bool (S,M,x)) is non empty set
{ (S,M,x,b<sub>1</sub>) where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : b<sub>1</sub> in (S,M,x) } is set
A is Element of the carrier of (S,M,x)
n is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,n) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & n in (S,M,x) & (S,M,x,n,b<sub>1</sub>) ) } is set
(S,M,x,a) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & a in (S,M,x) & (S,M,x,a,b<sub>1</sub>) ) } is set
n is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
f is Element of the carrier of (S,M,x)
n is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,n) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & n in (S,M,x) & (S,M,x,n,b<sub>1</sub>) ) } is set
(S,M,x) is V13() V16([:(S,M,x),(S,M,x):]) V17((S,M,x)) Function-like V41([:(S,M,x),(S,M,x):],(S,M,x)) Element of bool [:[:(S,M,x),(S,M,x):],(S,M,x):]
[:(S,M,x),(S,M,x):] is non empty set
[:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
bool [:[:(S,M,x),(S,M,x):],(S,M,x):] is non empty set
(S,M,x) . (f,A) is set
[f,A] is set
{f,A} is non empty set
{f} is non empty set
{{f,A},{f}} is non empty set
(S,M,x) . [f,A] is set
n + n is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(n + n)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & n + n in (S,M,x) & (S,M,x,n + n,b<sub>1</sub>) ) } is set
f + A is Element of the carrier of (S,M,x)
the addF of (S,M,x) . (f,A) is Element of the carrier of (S,M,x)
the addF of (S,M,x) . [f,A] is set
(S,M,x,y) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & y in (S,M,x) & (S,M,x,y,b<sub>1</sub>) ) } is set
c is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(y + a)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & y + a in (S,M,x) & (S,M,x,y + a,b<sub>1</sub>) ) } is set
f is Element of the carrier of (S,M,x)
A is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,A) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & A in (S,M,x) & (S,M,x,A,b<sub>1</sub>) ) } is set
(S,M,x) is V13() V16([:REAL,(S,M,x):]) V17((S,M,x)) Function-like V41([:REAL,(S,M,x):],(S,M,x)) Element of bool [:[:REAL,(S,M,x):],(S,M,x):]
[:REAL,(S,M,x):] is non empty set
[:[:REAL,(S,M,x):],(S,M,x):] is non empty set
bool [:[:REAL,(S,M,x):],(S,M,x):] is non empty set
(S,M,x) . (X,f) is set
[X,f] is set
{X,f} is non empty set
{{X,f},{X}} is non empty set
(S,M,x) . [X,f] is set
X (#) A is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(X (#) A)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & X (#) A in (S,M,x) & (S,M,x,X (#) A,b<sub>1</sub>) ) } is set
X * f is Element of the carrier of (S,M,x)
the Mult of (S,M,x) . (X,f) is set
the Mult of (S,M,x) . [X,f] is set
n is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
(S,M,x,(X (#) y)) is Element of bool (S,M,x)
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:] : ( b<sub>1</sub> in (S,M,x) & X (#) y in (S,M,x) & (S,M,x,X (#) y,b<sub>1</sub>) ) } is set
X is V11() real ext-real Element of REAL
S is non empty set
bool S is non empty set
bool (bool S) is non empty set
[:S,REAL:] is non empty V173() V174() V175() set
bool [:S,REAL:] is non empty set
M is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool S)
x is Element of M
y is V13() V16(S) V17( REAL ) Function-like V173() V174() V175() Element of bool [:S,REAL:]
dom y is Element of bool S
R_EAL y is V13() V16(S) V17( ExtREAL ) Function-like V174() Element of bool [:S,ExtREAL:]
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
a is ext-real Element of ExtREAL
g is V13() V16(S) V17( ExtREAL ) Function-like V174() Element of bool [:S,ExtREAL:]
dom g is Element of bool S
f is Element of S
dom (R_EAL y) is Element of bool S
(R_EAL y) . f is ext-real set
g . f 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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 V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
a is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom a is Element of bool X
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
dom (R_EAL (abs x)) is Element of bool X
f is Element of S
g is set
dom (abs x) is Element of bool X
(abs x) . g is V11() real ext-real set
x . g is V11() real ext-real set
abs (x . g) is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V166() V167() V168() V169() V170() V171() V204() Element of NAT
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative Element of REAL
1 / (g + 1) is non empty V11() real ext-real positive non negative Element of REAL
great_eq_dom ((abs x),(1 / (g + 1))) is set
f /\ (great_eq_dom ((abs x),(1 / (g + 1)))) is Element of bool X
f is Element of S
f --> (1 / (g + 1)) is V13() V16(f) V17( REAL ) Function-like V41(f, REAL ) V173() V174() V175() Element of bool [:f,REAL:]
[:f,REAL:] is V173() V174() V175() set
bool [:f,REAL:] is non empty set
dom (f --> (1 / (g + 1))) is Element of bool f
bool f is non empty set
n is set
(f --> (1 / (g + 1))) . n is V11() real ext-real set
(abs x) | f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom n is Element of bool X
n is set
n . n is V11() real ext-real set
Integral (M,n) is ext-real Element of ExtREAL
R_EAL n is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL n)) is ext-real Element of ExtREAL
max+ (R_EAL n) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL n))) is ext-real Element of ExtREAL
max- (R_EAL n) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL n))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL n)))) - (integral+ (M,(max- (R_EAL n)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL n))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL n)))),K119((integral+ (M,(max- (R_EAL n)))))) is ext-real set
abs n is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom (abs n) is Element of bool X
n is Element of X
(abs n) . n is V11() real ext-real set
n . n is V11() real ext-real set
abs (n . n) is V11() real ext-real Element of REAL
abs (1 / (g + 1)) is V11() real ext-real Element of REAL
dom ((abs x) | f) is Element of bool X
(dom (abs x)) /\ f is Element of bool X
n is Element of X
(abs x) . n is V11() real ext-real set
n . n is V11() real ext-real set
abs (n . n) is V11() real ext-real Element of REAL
((abs x) | f) . n is V11() real ext-real set
c is V11() real ext-real Element of REAL
Integral (M,(abs n)) is ext-real Element of ExtREAL
R_EAL (abs n) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs n))) is ext-real Element of ExtREAL
max+ (R_EAL (abs n)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs n)))) is ext-real Element of ExtREAL
max- (R_EAL (abs n)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs n)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs n))))) - (integral+ (M,(max- (R_EAL (abs n))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs n)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs n))))),K119((integral+ (M,(max- (R_EAL (abs n))))))) is ext-real set
Integral (M,((abs x) | f)) is ext-real Element of ExtREAL
R_EAL ((abs x) | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs x) | f))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) | f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) | f)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) | f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) | f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) | f))))) - (integral+ (M,(max- (R_EAL ((abs x) | f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL ((abs x) | f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL ((abs x) | f))))),K119((integral+ (M,(max- (R_EAL ((abs x) | f))))))) is ext-real set
n is set
n . n is V11() real ext-real set
(abs x) | f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,((abs x) | f)) is ext-real Element of ExtREAL
R_EAL ((abs x) | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs x) | f))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs x) | f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs x) | f)))) is ext-real Element of ExtREAL
max- (R_EAL ((abs x) | f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs x) | f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs x) | f))))) - (integral+ (M,(max- (R_EAL ((abs x) | f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL ((abs x) | f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL ((abs x) | f))))),K119((integral+ (M,(max- (R_EAL ((abs x) | f))))))) is ext-real set
R_EAL (1 / (g + 1)) is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
(R_EAL (1 / (g + 1))) * (M . f) is ext-real Element of ExtREAL
n is set
[:NAT,S:] is non empty set
bool [:NAT,S:] is non empty set
g is V13() V16( NAT ) V17(S) Function-like V41( NAT ,S) Element of bool [:NAT,S:]
f is set
rng g is non empty V156() M25(X,S)
union (rng g) is Element of S
f is set
A is set
g . A is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V166() V167() V168() V169() V170() V171() V204() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative Element of REAL
1 / (n + 1) is non empty V11() real ext-real positive non negative Element of REAL
great_eq_dom ((abs x),(1 / (n + 1))) is set
dom x is Element of bool X
(abs x) . f is V11() real ext-real set
x . f is V11() real ext-real set
abs (x . f) is V11() real ext-real Element of REAL
{ b<sub>1</sub> where b<sub>1</sub> is Element of X : ( b<sub>1</sub> in dom x & not x . b<sub>1</sub> = 0 ) } is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
f is Element of S
M . f is ext-real Element of ExtREAL
f ` is Element of bool X
X \ f is set
(union (rng g)) \/ f is Element of S
f is Element of S
f ` is Element of bool X
X \ f is set
(union (rng g)) ` is Element of bool X
X \ (union (rng g)) is set
(f `) /\ ((union (rng g)) `) is Element of bool X
dom (X,0) is Element of bool X
(X,0) | (f `) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
dom ((X,0) | (f `)) is Element of bool X
x | (f `) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
dom (x | (f `)) is Element of bool X
A is set
n is Element of X
x . n is V11() real ext-real set
(abs x) . n is V11() real ext-real set
abs (x . n) is V11() real ext-real Element of REAL
((abs x) . n) - 0 is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V166() V167() V168() V169() V170() V171() V204() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative Element of REAL
1 / (n + 1) is non empty V11() real ext-real positive non negative Element of REAL
great_eq_dom ((abs x),(1 / (n + 1))) is set
g . n is Element of S
A is set
x . A is V11() real ext-real set
A is set
(x | (f `)) . A is V11() real ext-real set
x . A is V11() real ext-real set
(X,0) . A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
((X,0) | (f `)) . A is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
A is set
n is set
g . n is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V166() V167() V168() V169() V170() V171() V204() Element of NAT
g . n is Element of S
M . (g . n) is ext-real Element of ExtREAL
M . f is ext-real Element of ExtREAL
X is non empty set
bool X is non empty set
bool (bool X) is non empty set
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
abs (X,0) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
Integral (M,(abs (X,0))) is ext-real Element of ExtREAL
R_EAL (abs (X,0)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs (X,0)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (X,0))) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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 V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs (X,0))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs (X,0)))))),K119((integral+ (M,(max- (R_EAL (abs (X,0)))))))) is ext-real set
y is Element of X
(X,0) . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V108() V204() set
abs ((X,0) . y) is V11() real ext-real Element of REAL
dom (abs (X,0)) is Element of bool X
(abs (X,0)) . y is V11() real ext-real set
dom (X,0) 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 V173() V174() V175() set
bool [:X,REAL:] is non empty set
S is non empty compl-closed sigma-multiplicative V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs x is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs x)) is ext-real Element of ExtREAL
R_EAL (abs x) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs x))) is ext-real Element of ExtREAL
max+ (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs x)))) is ext-real Element of ExtREAL
max- (R_EAL (abs x)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs x)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs x))))),K119((integral+ (M,(max- (R_EAL (abs x))))))) is ext-real set
y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs (x + y) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs (x + y))) is ext-real Element of ExtREAL
R_EAL (abs (x + y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (x + y)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (x + y))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (x + y))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (x + y))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (x + y))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (x + y)))))) - (integral+ (M,(max- (R_EAL (abs (x + y)))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs (x + y))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs (x + y)))))),K119((integral+ (M,(max- (R_EAL (abs (x + y)))))))) is ext-real set
abs y is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs y)) is ext-real Element of ExtREAL
R_EAL (abs y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs y))) is ext-real Element of ExtREAL
max+ (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs y)))) is ext-real Element of ExtREAL
max- (R_EAL (abs y)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs y)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs y))))) - (integral+ (M,(max- (R_EAL (abs y))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs y)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs y))))),K119((integral+ (M,(max- (R_EAL (abs y))))))) is ext-real set
(Integral (M,(abs x))) + (Integral (M,(abs y))) is ext-real Element of ExtREAL
R_EAL x is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
R_EAL y is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
dom (R_EAL x) is Element of bool X
g is Element of S
|.(R_EAL x).| is V13() V16(X) V17( ExtREAL ) Function-like V174() nonnegative Element of bool [:X,ExtREAL:]
dom |.(R_EAL x).| is Element of bool X
dom (R_EAL y) is Element of bool X
f is Element of S
|.(R_EAL y).| is V13() V16(X) V17( ExtREAL ) Function-like V174() nonnegative Element of bool [:X,ExtREAL:]
dom |.(R_EAL y).| is Element of bool X
(R_EAL x) + (R_EAL y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
dom ((R_EAL x) + (R_EAL y)) is Element of bool X
|.((R_EAL x) + (R_EAL y)).| is V13() V16(X) V17( ExtREAL ) Function-like V174() nonnegative Element of bool [:X,ExtREAL:]
f is Element of S
|.((R_EAL x) + (R_EAL y)).| | f is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(|.((R_EAL x) + (R_EAL y)).| | f)) is ext-real Element of ExtREAL
max+ (|.((R_EAL x) + (R_EAL y)).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (|.((R_EAL x) + (R_EAL y)).| | f))) is ext-real Element of ExtREAL
max- (|.((R_EAL x) + (R_EAL y)).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (|.((R_EAL x) + (R_EAL y)).| | f))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (|.((R_EAL x) + (R_EAL y)).| | f)))) - (integral+ (M,(max- (|.((R_EAL x) + (R_EAL y)).| | f)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (|.((R_EAL x) + (R_EAL y)).| | f))))) is ext-real set
K118((integral+ (M,(max+ (|.((R_EAL x) + (R_EAL y)).| | f)))),K119((integral+ (M,(max- (|.((R_EAL x) + (R_EAL y)).| | f)))))) is ext-real set
|.(R_EAL x).| | f is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(|.(R_EAL x).| | f)) is ext-real Element of ExtREAL
max+ (|.(R_EAL x).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (|.(R_EAL x).| | f))) is ext-real Element of ExtREAL
max- (|.(R_EAL x).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (|.(R_EAL x).| | f))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (|.(R_EAL x).| | f)))) - (integral+ (M,(max- (|.(R_EAL x).| | f)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (|.(R_EAL x).| | f))))) is ext-real set
K118((integral+ (M,(max+ (|.(R_EAL x).| | f)))),K119((integral+ (M,(max- (|.(R_EAL x).| | f)))))) is ext-real set
|.(R_EAL y).| | f is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(|.(R_EAL y).| | f)) is ext-real Element of ExtREAL
max+ (|.(R_EAL y).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (|.(R_EAL y).| | f))) is ext-real Element of ExtREAL
max- (|.(R_EAL y).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (|.(R_EAL y).| | f))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (|.(R_EAL y).| | f)))) - (integral+ (M,(max- (|.(R_EAL y).| | f)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (|.(R_EAL y).| | f))))) is ext-real set
K118((integral+ (M,(max+ (|.(R_EAL y).| | f)))),K119((integral+ (M,(max- (|.(R_EAL y).| | f)))))) is ext-real set
(Integral (M,(|.(R_EAL x).| | f))) + (Integral (M,(|.(R_EAL y).| | f))) is ext-real Element of ExtREAL
dom |.((R_EAL x) + (R_EAL y)).| is Element of bool X
R_EAL (x + y) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
|.(x + y).| is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,|.(x + y).|) is ext-real Element of ExtREAL
R_EAL |.(x + y).| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL |.(x + y).|)) is ext-real Element of ExtREAL
max+ (R_EAL |.(x + y).|) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL |.(x + y).|))) is ext-real Element of ExtREAL
max- (R_EAL |.(x + y).|) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL |.(x + y).|))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL |.(x + y).|)))) - (integral+ (M,(max- (R_EAL |.(x + y).|)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL |.(x + y).|))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL |.(x + y).|)))),K119((integral+ (M,(max- (R_EAL |.(x + y).|)))))) is ext-real set
(dom (R_EAL x)) /\ (dom (R_EAL y)) is Element of bool X
{-infty} is non empty V167() Element of bool ExtREAL
(R_EAL x) " {-infty} is Element of bool X
{+infty} is non empty V167() Element of bool ExtREAL
(R_EAL y) " {+infty} is Element of bool X
((R_EAL x) " {-infty}) /\ ((R_EAL y) " {+infty}) is Element of bool X
(R_EAL x) " {+infty} is Element of bool X
(R_EAL y) " {-infty} is Element of bool X
((R_EAL x) " {+infty}) /\ ((R_EAL y) " {-infty}) is Element of bool X
(((R_EAL x) " {-infty}) /\ ((R_EAL y) " {+infty})) \/ (((R_EAL x) " {+infty}) /\ ((R_EAL y) " {-infty})) is Element of bool X
((dom (R_EAL x)) /\ (dom (R_EAL y))) \ ((((R_EAL x) " {-infty}) /\ ((R_EAL y) " {+infty})) \/ (((R_EAL x) " {+infty}) /\ ((R_EAL y) " {-infty}))) is Element of bool X
|.(R_EAL y).| | f is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(|.(R_EAL y).| | f)) is ext-real Element of ExtREAL
max+ (|.(R_EAL y).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (|.(R_EAL y).| | f))) is ext-real Element of ExtREAL
max- (|.(R_EAL y).| | f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (|.(R_EAL y).| | f))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (|.(R_EAL y).| | f)))) - (integral+ (M,(max- (|.(R_EAL y).| | f)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (|.(R_EAL y).| | f))))) is ext-real set
K118((integral+ (M,(max+ (|.(R_EAL y).| | f)))),K119((integral+ (M,(max- (|.(R_EAL y).| | f)))))) is ext-real set
A is Element of S
Integral (M,|.(R_EAL y).|) is ext-real Element of ExtREAL
max+ |.(R_EAL y).| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ |.(R_EAL y).|)) is ext-real Element of ExtREAL
max- |.(R_EAL y).| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- |.(R_EAL y).|)) is ext-real Element of ExtREAL
(integral+ (M,(max+ |.(R_EAL y).|))) - (integral+ (M,(max- |.(R_EAL y).|))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- |.(R_EAL y).|)))) is ext-real set
K118((integral+ (M,(max+ |.(R_EAL y).|))),K119((integral+ (M,(max- |.(R_EAL y).|))))) is ext-real set
|.y.| is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,|.y.|) is ext-real Element of ExtREAL
R_EAL |.y.| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL |.y.|)) is ext-real Element of ExtREAL
max+ (R_EAL |.y.|) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL |.y.|))) is ext-real Element of ExtREAL
max- (R_EAL |.y.|) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL |.y.|))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL |.y.|)))) - (integral+ (M,(max- (R_EAL |.y.|)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL |.y.|))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL |.y.|)))),K119((integral+ (M,(max- (R_EAL |.y.|)))))) is ext-real set
|.(R_EAL x).| | g is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(|.(R_EAL x).| | g)) is ext-real Element of ExtREAL
max+ (|.(R_EAL x).| | g) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (|.(R_EAL x).| | g))) is ext-real Element of ExtREAL
max- (|.(R_EAL x).| | g) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (|.(R_EAL x).| | g))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (|.(R_EAL x).| | g)))) - (integral+ (M,(max- (|.(R_EAL x).| | g)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (|.(R_EAL x).| | g))))) is ext-real set
K118((integral+ (M,(max+ (|.(R_EAL x).| | g)))),K119((integral+ (M,(max- (|.(R_EAL x).| | g)))))) is ext-real set
A is Element of S
Integral (M,|.(R_EAL x).|) is ext-real Element of ExtREAL
max+ |.(R_EAL x).| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ |.(R_EAL x).|)) is ext-real Element of ExtREAL
max- |.(R_EAL x).| is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- |.(R_EAL x).|)))) is ext-real set
K118((integral+ (M,(max+ |.(R_EAL x).|))),K119((integral+ (M,(max- |.(R_EAL x).|))))) is ext-real set
|.x.| is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,|.x.|) is ext-real Element of ExtREAL
R_EAL |.x.| is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL |.x.|)) is ext-real Element of ExtREAL
max+ (R_EAL |.x.|) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL |.x.|))) is ext-real Element of ExtREAL
max- (R_EAL |.x.|) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL |.x.|))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL |.x.|)))),K119((integral+ (M,(max- (R_EAL |.x.|)))))) is ext-real set
(Integral (M,|.x.|)) + (Integral (M,|.y.|)) 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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty NORMSTR
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
(X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M), the ZeroF of (X,S,M), the addF of (X,S,M), the Mult of (X,S,M),(X,S,M) #) is strict NORMSTR
the carrier of (X,S,M) is non empty set
[:X,REAL:] is non empty V173() V174() V175() set
bool [:X,REAL:] is non empty set
x is Element of the carrier of (X,S,M)
||.x.|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
the U9 of (X,S,M) . x is V11() real ext-real Element of REAL
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs f)) is ext-real Element of ExtREAL
R_EAL (abs f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
[:X,ExtREAL:] is non empty V174() set
bool [:X,ExtREAL:] is non empty set
Integral (M,(R_EAL (abs f))) is ext-real Element of ExtREAL
max+ (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs f)))) is ext-real Element of ExtREAL
max- (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs f))))) - (integral+ (M,(max- (R_EAL (abs f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs f))))),K119((integral+ (M,(max- (R_EAL (abs f))))))) is ext-real set
a is V11() real ext-real Element of REAL
abs a is V11() real ext-real Element of REAL
(abs a) (#) (abs f) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,((abs a) (#) (abs f))) is ext-real Element of ExtREAL
R_EAL ((abs a) (#) (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL ((abs a) (#) (abs f)))) is ext-real Element of ExtREAL
max+ (R_EAL ((abs a) (#) (abs f))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL ((abs a) (#) (abs f))))) is ext-real Element of ExtREAL
max- (R_EAL ((abs a) (#) (abs f))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL ((abs a) (#) (abs f))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL ((abs a) (#) (abs f)))))) - (integral+ (M,(max- (R_EAL ((abs a) (#) (abs f)))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL ((abs a) (#) (abs f))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL ((abs a) (#) (abs f)))))),K119((integral+ (M,(max- (R_EAL ((abs a) (#) (abs f)))))))) is ext-real set
R_EAL (abs a) is ext-real Element of ExtREAL
(R_EAL (abs a)) * (Integral (M,(abs f))) is ext-real Element of ExtREAL
a (#) f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs (a (#) f) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs (a (#) f))) is ext-real Element of ExtREAL
R_EAL (abs (a (#) f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (a (#) f)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (a (#) f))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (a (#) f))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (a (#) f))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (a (#) f))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (a (#) f)))))) - (integral+ (M,(max- (R_EAL (abs (a (#) f)))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs (a (#) f))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs (a (#) f)))))),K119((integral+ (M,(max- (R_EAL (abs (a (#) f)))))))) is ext-real set
(abs a) * ||.x.|| is V11() real ext-real Element of REAL
(X,0) is V13() V16(X) V17( REAL ) V17( RAT ) V17( INT ) Function-like V173() V174() V175() V176() Element of bool [:X,REAL:]
f is Element of the carrier of (X,S,M)
(X,S,M) . f is V11() real ext-real Element of REAL
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs f)) is ext-real Element of ExtREAL
R_EAL (abs f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs f))) is ext-real Element of ExtREAL
max+ (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs f)))) is ext-real Element of ExtREAL
max- (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs f))))) - (integral+ (M,(max- (R_EAL (abs f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs f))))),K119((integral+ (M,(max- (R_EAL (abs f))))))) is ext-real set
(X,S,M) is non empty add-closed ((X)) Element of bool the carrier of (X)
(X) is non empty strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
PFuncs (X,REAL) is non empty functional PartFunc-set of X, REAL
(X) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of PFuncs (X,REAL)
X --> 0 is V13() V16(X) V17( NAT ) V17( RAT ) V17( INT ) Function-like V41(X, NAT ) V173() V174() V175() V176() Element of bool [:X,NAT:]
[:X,NAT:] is non empty V17( RAT ) V17( INT ) V173() V174() V175() V176() set
bool [:X,NAT:] is non empty set
addpfunc X is V13() V16([:(PFuncs (X,REAL)),(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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
(X) is V13() V16([:REAL,(PFuncs (X,REAL)):]) V17( PFuncs (X,REAL)) Function-like 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)),(X),(addpfunc X),(X) #) is non empty strict RLSStruct
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() 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,S,M,f) is Element of bool (X,S,M)
bool (X,S,M) is non empty set
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
(X,S,M) is non empty Element of bool (bool (X,S,M))
bool (bool (X,S,M)) is non empty set
{ (X,S,M,b₁) where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : b₁ in (X,S,M) } is set
(X,S,M,(X,0)) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & (X,0) in (X,S,M) & (X,S,M,(X,0),b₁) ) } is set
(X,S,M) is Element of (X,S,M)
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
0. (X,S,M) is V55((X,S,M)) Element of the carrier of (X,S,M)
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
(X,S,M,f) is Element of bool (X,S,M)
{ b₁ where b₁ is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:] : ( b₁ in (X,S,M) & f in (X,S,M) & (X,S,M,f,b₁) ) } is set
g is Element of the carrier of (X,S,M)
(X,S,M) . g is V11() real ext-real Element of REAL
abs f is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs f)) is ext-real Element of ExtREAL
R_EAL (abs f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs f))) is ext-real Element of ExtREAL
max+ (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs f)))) is ext-real Element of ExtREAL
max- (R_EAL (abs f)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs f)))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs f))))) - (integral+ (M,(max- (R_EAL (abs f))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs f)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs f))))),K119((integral+ (M,(max- (R_EAL (abs f))))))) is ext-real set
Integral (M,f) is ext-real Element of ExtREAL
R_EAL f is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL f)) is ext-real Element of ExtREAL
max+ (R_EAL f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL f))) is ext-real Element of ExtREAL
max- (R_EAL f) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL f))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL f)))) - (integral+ (M,(max- (R_EAL f)))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL f))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL f)))),K119((integral+ (M,(max- (R_EAL f)))))) is ext-real set
|.(Integral (M,f)).| is ext-real Element of ExtREAL
y is Element of the carrier of (X,S,M)
||.y.|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) . y is V11() real ext-real Element of REAL
g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
abs g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs g)) is ext-real Element of ExtREAL
R_EAL (abs g) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs g))) is ext-real Element of ExtREAL
max+ (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs g)))) is ext-real Element of ExtREAL
max- (R_EAL (abs g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() 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
K119((integral+ (M,(max- (R_EAL (abs g)))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs g))))),K119((integral+ (M,(max- (R_EAL (abs g))))))) is ext-real set
f + g is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
x + y is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the addF of (X,S,M) . (x,y) is Element of the carrier of (X,S,M)
[x,y] is set
{x,y} is non empty set
{x} is non empty set
{{x,y},{x}} is non empty set
the addF of (X,S,M) . [x,y] is set
||.(x + y).|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) . (x + y) is V11() real ext-real Element of REAL
abs (f + g) is V13() V16(X) V17( REAL ) Function-like V173() V174() V175() Element of bool [:X,REAL:]
Integral (M,(abs (f + g))) is ext-real Element of ExtREAL
R_EAL (abs (f + g)) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
Integral (M,(R_EAL (abs (f + g)))) is ext-real Element of ExtREAL
max+ (R_EAL (abs (f + g))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max+ (R_EAL (abs (f + g))))) is ext-real Element of ExtREAL
max- (R_EAL (abs (f + g))) is V13() V16(X) V17( ExtREAL ) Function-like V174() Element of bool [:X,ExtREAL:]
integral+ (M,(max- (R_EAL (abs (f + g))))) is ext-real Element of ExtREAL
(integral+ (M,(max+ (R_EAL (abs (f + g)))))) - (integral+ (M,(max- (R_EAL (abs (f + g)))))) is ext-real Element of ExtREAL
K119((integral+ (M,(max- (R_EAL (abs (f + g))))))) is ext-real set
K118((integral+ (M,(max+ (R_EAL (abs (f + g)))))),K119((integral+ (M,(max- (R_EAL (abs (f + g)))))))) is ext-real set
(Integral (M,(abs f))) + (Integral (M,(abs g))) is ext-real Element of ExtREAL
||.x.|| + ||.y.|| is V11() real ext-real Element of REAL
a * x is Element of the carrier of (X,S,M)
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) . (a,x) is set
[a,x] is set
{a,x} is non empty set
{a} is non empty V166() V167() V168() set
{{a,x},{a}} is non empty set
the Mult of (X,S,M) . [a,x] is set
||.(a * x).|| is V11() real ext-real Element of REAL
the U9 of (X,S,M) . (a * x) is V11() real ext-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 V162() V163() V164() sigma-additive Element of bool (bool X)
[:S,ExtREAL:] is non empty V174() set
bool [:S,ExtREAL:] is non empty set
M is V13() V16(S) V17( ExtREAL ) Function-like V41(S, ExtREAL ) V174() V182() nonnegative sigma-additive Element of bool [:S,ExtREAL:]
(X,S,M) is non empty NORMSTR
(X,S,M) is non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() RLSStruct
the carrier of (X,S,M) is non empty set
the ZeroF of (X,S,M) is Element of the carrier of (X,S,M)
the addF of (X,S,M) is V13() V16([: the carrier of (X,S,M), the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):]
[: the carrier of (X,S,M), the carrier of (X,S,M):] is non empty set
[:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[: the carrier of (X,S,M), the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
the Mult of (X,S,M) is V13() V16([:REAL, the carrier of (X,S,M):]) V17( the carrier of (X,S,M)) Function-like V41([:REAL, the carrier of (X,S,M):], the carrier of (X,S,M)) Element of bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):]
[:REAL, the carrier of (X,S,M):] is non empty set
[:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
bool [:[:REAL, the carrier of (X,S,M):], the carrier of (X,S,M):] is non empty set
(X,S,M) is V13() V16( the carrier of (X,S,M)) V17( REAL ) Function-like V41( the carrier of (X,S,M), REAL ) V173() V174() V175() Element of bool [: the carrier of (X,S,M),REAL:]
[: the carrier of (X,S,M),REAL:] is non empty V173() V174() V175() set
bool [: the carrier of (X,S,M),REAL:] is non empty set
NORMSTR(# the carrier of (X,S,M), the ZeroF of (X,S,M), the addF of (X,S,M), the Mult of (X,S,M),(X,S,M) #) is strict NORMSTR