:: HAHNBAN1 semantic presentation

REAL is non empty V35() V192() V193() V194() V198() set
NAT is V192() V193() V194() V195() V196() V197() V198() Element of bool REAL
bool REAL is set
F_Complex is non empty non degenerated non trivial left_complementable right_complementable complementable almost_left_invertible strict Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
the carrier of F_Complex is non empty non trivial set
COMPLEX is non empty V35() V192() V198() set
RAT is non empty V35() V192() V193() V194() V195() V198() set
INT is non empty V35() V192() V193() V194() V195() V196() V198() set
[:COMPLEX,COMPLEX:] is set
bool [:COMPLEX,COMPLEX:] is set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is set
[:REAL,REAL:] is set
bool [:REAL,REAL:] is set
[:[:REAL,REAL:],REAL:] is set
bool [:[:REAL,REAL:],REAL:] is set
[:RAT,RAT:] is set
bool [:RAT,RAT:] is set
[:[:RAT,RAT:],RAT:] is set
bool [:[:RAT,RAT:],RAT:] is set
[:INT,INT:] is set
bool [:INT,INT:] is set
[:[:INT,INT:],INT:] is set
bool [:[:INT,INT:],INT:] is set
[:NAT,NAT:] is set
[:[:NAT,NAT:],NAT:] is set
bool [:[:NAT,NAT:],NAT:] is set
omega is V192() V193() V194() V195() V196() V197() V198() set
bool omega is set
bool NAT is set
{} is empty Function-like functional V192() V193() V194() V195() V196() V197() V198() set
1 is non empty natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() Element of NAT
{{},1} is non empty set
K445() is L16()
the carrier of K445() is set
ExtREAL is non empty V193() set
0 is empty natural complex real ext-real Function-like functional V33() V34() V192() V193() V194() V195() V196() V197() V198() Element of NAT
<i> is complex Element of COMPLEX
K169() is V16() V19([:COMPLEX,COMPLEX:]) V20( COMPLEX ) Function-like V30([:COMPLEX,COMPLEX:], COMPLEX ) Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
K171() is V16() V19([:COMPLEX,COMPLEX:]) V20( COMPLEX ) Function-like V30([:COMPLEX,COMPLEX:], COMPLEX ) Element of bool [:[:COMPLEX,COMPLEX:],COMPLEX:]
1r is complex Element of COMPLEX
0c is empty complex Function-like functional V192() V193() V194() V195() V196() V197() V198() Element of COMPLEX
0. F_Complex is complex zero left_complementable right_complementable complementable Element of the carrier of F_Complex
the ZeroF of F_Complex is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
1_ F_Complex is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
1. F_Complex is complex non zero left_complementable right_complementable complementable Element of the carrier of F_Complex
the OneF of F_Complex is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive doubleLoopStr
the carrier of x is non empty set
n is left_complementable right_complementable complementable Element of the carrier of x
p1 is left_complementable right_complementable complementable Element of the carrier of x
- p1 is left_complementable right_complementable complementable Element of the carrier of x
n * (- p1) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (n,(- p1)) is left_complementable right_complementable complementable Element of the carrier of x
[n,(- p1)] is set
{n,(- p1)} is non empty set
{n} is non empty set
{{n,(- p1)},{n}} is non empty set
the multF of x . [n,(- p1)] is set
n * p1 is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . (n,p1) is left_complementable right_complementable complementable Element of the carrier of x
[n,p1] is set
{n,p1} is non empty set
{{n,p1},{n}} is non empty set
the multF of x . [n,p1] is set
- (n * p1) is left_complementable right_complementable complementable Element of the carrier of x
(n * p1) + (n * (- p1)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 addF of x . ((n * p1),(n * (- p1))) is left_complementable right_complementable complementable Element of the carrier of x
[(n * p1),(n * (- p1))] is set
{(n * p1),(n * (- p1))} is non empty set
{(n * p1)} is non empty set
{{(n * p1),(n * (- p1))},{(n * p1)}} is non empty set
the addF of x . [(n * p1),(n * (- p1))] is set
p1 + (- p1) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (p1,(- p1)) is left_complementable right_complementable complementable Element of the carrier of x
[p1,(- p1)] is set
{p1,(- p1)} is non empty set
{p1} is non empty set
{{p1,(- p1)},{p1}} is non empty set
the addF of x . [p1,(- p1)] is set
n * (p1 + (- p1)) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . (n,(p1 + (- p1))) is left_complementable right_complementable complementable Element of the carrier of x
[n,(p1 + (- p1))] is set
{n,(p1 + (- p1))} is non empty set
{{n,(p1 + (- p1))},{n}} is non empty set
the multF of x . [n,(p1 + (- p1))] is set
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
n * (0. x) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . (n,(0. x)) is left_complementable right_complementable complementable Element of the carrier of x
[n,(0. x)] is set
{n,(0. x)} is non empty set
{{n,(0. x)},{n}} is non empty set
the multF of x . [n,(0. x)] is set
x is complex real ext-real set
n is complex real ext-real set
n * <i> is complex set
x + (n * <i>) is complex set
p1 is complex real ext-real set
M is complex real ext-real set
M * <i> is complex set
p1 + (M * <i>) is complex set
(x + (n * <i>)) * (p1 + (M * <i>)) is complex set
x * p1 is complex real ext-real set
n * M is complex real ext-real set
(x * p1) - (n * M) is complex real ext-real set
x * M is complex real ext-real set
p1 * n is complex real ext-real set
(x * M) + (p1 * n) is complex real ext-real set
((x * M) + (p1 * n)) * <i> is complex set
((x * p1) - (n * M)) + (((x * M) + (p1 * n)) * <i>) is complex set
0 * <i> is complex set
x is complex Element of COMPLEX
|.x.| is complex real ext-real Element of REAL
Re x is complex real ext-real Element of REAL
(Re x) ^2 is complex real ext-real Element of REAL
(Re x) * (Re x) is complex real ext-real set
Im x is complex real ext-real Element of REAL
(Im x) ^2 is complex real ext-real Element of REAL
(Im x) * (Im x) is complex real ext-real set
((Re x) ^2) + ((Im x) ^2) is complex real ext-real Element of REAL
sqrt (((Re x) ^2) + ((Im x) ^2)) is complex real ext-real Element of REAL
|.x.| + (0 * <i>) is complex set
x *' is complex Element of COMPLEX
(Im x) * <i> is complex set
(Re x) - ((Im x) * <i>) is complex set
(x *') / (|.x.| + (0 * <i>)) is complex Element of COMPLEX
((x *') / (|.x.| + (0 * <i>))) * x is complex Element of COMPLEX
x * (x *') is complex Element of COMPLEX
Im (x * (x *')) is complex real ext-real Element of REAL
Re |.x.| is complex real ext-real Element of REAL
Im |.x.| is complex real ext-real Element of REAL
(x *') / |.x.| is complex Element of COMPLEX
((x *') / |.x.|) * x is complex Element of COMPLEX
(x * (x *')) / |.x.| is complex Element of COMPLEX
Re (x * (x *')) is complex real ext-real Element of REAL
Im (((x *') / |.x.|) * x) is complex real ext-real Element of REAL
|.x.| * 0 is complex real ext-real Element of REAL
(((Re x) ^2) + ((Im x) ^2)) * 0 is complex real ext-real Element of REAL
(|.x.| * 0) - ((((Re x) ^2) + ((Im x) ^2)) * 0) is complex real ext-real Element of REAL
|.x.| ^2 is complex real ext-real Element of REAL
|.x.| * |.x.| is complex real ext-real set
0 ^2 is complex real ext-real Element of REAL
0 * 0 is complex real ext-real set
(|.x.| ^2) + (0 ^2) is complex real ext-real Element of REAL
((|.x.| * 0) - ((((Re x) ^2) + ((Im x) ^2)) * 0)) / ((|.x.| ^2) + (0 ^2)) is complex real ext-real Element of REAL
Re (((x *') / |.x.|) * x) is complex real ext-real Element of REAL
(((Re x) ^2) + ((Im x) ^2)) * |.x.| is complex real ext-real Element of REAL
0 * 0 is natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() Element of NAT
((((Re x) ^2) + ((Im x) ^2)) * |.x.|) + (0 * 0) is complex real ext-real Element of REAL
(((((Re x) ^2) + ((Im x) ^2)) * |.x.|) + (0 * 0)) / ((|.x.| ^2) + (0 ^2)) is complex real ext-real Element of REAL
x * x is complex Element of COMPLEX
|.(x * x).| is complex real ext-real Element of REAL
Re (x * x) is complex real ext-real Element of REAL
(Re (x * x)) ^2 is complex real ext-real Element of REAL
(Re (x * x)) * (Re (x * x)) is complex real ext-real set
Im (x * x) is complex real ext-real Element of REAL
(Im (x * x)) ^2 is complex real ext-real Element of REAL
(Im (x * x)) * (Im (x * x)) is complex real ext-real set
((Re (x * x)) ^2) + ((Im (x * x)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (x * x)) ^2) + ((Im (x * x)) ^2)) is complex real ext-real Element of REAL
|.(x * x).| * |.x.| is complex real ext-real Element of REAL
|.x.| * |.x.| is complex real ext-real Element of REAL
(|.(x * x).| * |.x.|) / (|.x.| * |.x.|) is complex real ext-real Element of REAL
|.(x * x).| / |.x.| is complex real ext-real Element of REAL
(|.x.| * |.x.|) / |.x.| is complex real ext-real Element of REAL
x is complex real ext-real set
n is complex real ext-real set
n * <i> is complex set
x + (n * <i>) is complex set
1 * <i> is complex set
0 + (1 * <i>) is complex set
(0,1) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
() is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
() * () is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
[: the carrier of F_Complex, the carrier of F_Complex:] is set
[:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:] is set
the multF of F_Complex . ((),()) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(),()] is set
{(),()} is non empty V192() set
{()} is non empty V192() set
{{(),()},{()}} is non empty set
the multF of F_Complex . [(),()] is set
K147((),()) is complex Element of COMPLEX
- (1_ F_Complex) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
- 1r is complex Element of COMPLEX
(- (1_ F_Complex)) * (- (1_ F_Complex)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((- (1_ F_Complex)),(- (1_ F_Complex))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),(- (1_ F_Complex))] is set
{(- (1_ F_Complex)),(- (1_ F_Complex))} is non empty V192() set
{(- (1_ F_Complex))} is non empty V192() set
{{(- (1_ F_Complex)),(- (1_ F_Complex))},{(- (1_ F_Complex))}} is non empty set
the multF of F_Complex . [(- (1_ F_Complex)),(- (1_ F_Complex))] is set
K147((- (1_ F_Complex)),(- (1_ F_Complex))) is complex Element of COMPLEX
- 1r is complex Element of COMPLEX
x is complex real ext-real Element of REAL
n is complex real ext-real Element of REAL
(x,n) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n * <i> is complex set
x + (n * <i>) is complex set
p1 is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
(p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M * <i> is complex set
p1 + (M * <i>) is complex set
(x,n) + (p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((x,n),(p1,M)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(x,n),(p1,M)] is set
{(x,n),(p1,M)} is non empty V192() set
{(x,n)} is non empty V192() set
{{(x,n),(p1,M)},{(x,n)}} is non empty set
the addF of F_Complex . [(x,n),(p1,M)] is set
K145((x,n),(p1,M)) is complex Element of COMPLEX
x + p1 is complex real ext-real Element of REAL
n + M is complex real ext-real Element of REAL
((x + p1),(n + M)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(n + M) * <i> is complex set
(x + p1) + ((n + M) * <i>) is complex set
x is complex real ext-real set
n is complex real ext-real set
(x,n) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n * <i> is complex set
x + (n * <i>) is complex set
p1 is complex real ext-real set
M is complex real ext-real set
(p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M * <i> is complex set
p1 + (M * <i>) is complex set
(x,n) * (p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((x,n),(p1,M)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(x,n),(p1,M)] is set
{(x,n),(p1,M)} is non empty V192() set
{(x,n)} is non empty V192() set
{{(x,n),(p1,M)},{(x,n)}} is non empty set
the multF of F_Complex . [(x,n),(p1,M)] is set
K147((x,n),(p1,M)) is complex Element of COMPLEX
x * p1 is complex real ext-real set
n * M is complex real ext-real set
(x * p1) - (n * M) is complex real ext-real set
x * M is complex real ext-real set
p1 * n is complex real ext-real set
(x * M) + (p1 * n) is complex real ext-real set
(((x * p1) - (n * M)),((x * M) + (p1 * n))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
((x * M) + (p1 * n)) * <i> is complex set
((x * p1) - (n * M)) + (((x * M) + (p1 * n)) * <i>) is complex set
x is complex real ext-real Element of REAL
(x,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x + (0 * <i>) is complex set
|.(x,0).| is complex real ext-real Element of REAL
Re (x,0) is complex real ext-real Element of REAL
(Re (x,0)) ^2 is complex real ext-real Element of REAL
(Re (x,0)) * (Re (x,0)) is complex real ext-real set
Im (x,0) is complex real ext-real Element of REAL
(Im (x,0)) ^2 is complex real ext-real Element of REAL
(Im (x,0)) * (Im (x,0)) is complex real ext-real set
((Re (x,0)) ^2) + ((Im (x,0)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (x,0)) ^2) + ((Im (x,0)) ^2)) is complex real ext-real Element of REAL
abs x is complex real ext-real Element of REAL
Re x is complex real ext-real Element of REAL
(Re x) ^2 is complex real ext-real Element of REAL
(Re x) * (Re x) is complex real ext-real set
Im x is complex real ext-real Element of REAL
(Im x) ^2 is complex real ext-real Element of REAL
(Im x) * (Im x) is complex real ext-real set
((Re x) ^2) + ((Im x) ^2) is complex real ext-real Element of REAL
sqrt (((Re x) ^2) + ((Im x) ^2)) is complex real ext-real Element of REAL
x is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x + n is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (x,n) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[x,n] is set
{x,n} is non empty V192() set
{x} is non empty V192() set
{{x,n},{x}} is non empty set
the addF of F_Complex . [x,n] is set
K145(x,n) is complex Element of COMPLEX
Re (x + n) is complex real ext-real Element of REAL
Re x is complex real ext-real Element of REAL
Re n is complex real ext-real Element of REAL
(Re x) + (Re n) is complex real ext-real Element of REAL
p1 is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 + M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex . (p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[p1,M] is set
{p1,M} is non empty V192() set
{p1} is non empty V192() set
{{p1,M},{p1}} is non empty set
the addF of F_Complex . [p1,M] is set
K145(p1,M) is complex Element of COMPLEX
Im (p1 + M) is complex real ext-real Element of REAL
Im p1 is complex real ext-real Element of REAL
Im M is complex real ext-real Element of REAL
(Im p1) + (Im M) is complex real ext-real Element of REAL
x is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x * n is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (x,n) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[x,n] is set
{x,n} is non empty V192() set
{x} is non empty V192() set
{{x,n},{x}} is non empty set
the multF of F_Complex . [x,n] is set
K147(x,n) is complex Element of COMPLEX
Re (x * n) is complex real ext-real Element of REAL
Re x is complex real ext-real Element of REAL
Re n is complex real ext-real Element of REAL
(Re x) * (Re n) is complex real ext-real Element of REAL
Im x is complex real ext-real Element of REAL
Im n is complex real ext-real Element of REAL
(Im x) * (Im n) is complex real ext-real Element of REAL
((Re x) * (Re n)) - ((Im x) * (Im n)) is complex real ext-real Element of REAL
p1 is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 * M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (p1,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[p1,M] is set
{p1,M} is non empty V192() set
{p1} is non empty V192() set
{{p1,M},{p1}} is non empty set
the multF of F_Complex . [p1,M] is set
K147(p1,M) is complex Element of COMPLEX
Im (p1 * M) is complex real ext-real Element of REAL
Re p1 is complex real ext-real Element of REAL
Im M is complex real ext-real Element of REAL
(Re p1) * (Im M) is complex real ext-real Element of REAL
Re M is complex real ext-real Element of REAL
Im p1 is complex real ext-real Element of REAL
(Re M) * (Im p1) is complex real ext-real Element of REAL
((Re p1) * (Im M)) + ((Re M) * (Im p1)) is complex real ext-real Element of REAL
x is 1-sorted
x is non empty addLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
l is Element of the carrier of n
RVSM . l is Element of the carrier of x
p1 . l is Element of the carrier of x
M . l is Element of the carrier of x
(p1 . l) + (M . l) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((p1 . l),(M . l)) is Element of the carrier of x
[(p1 . l),(M . l)] is set
{(p1 . l),(M . l)} is non empty set
{(p1 . l)} is non empty set
{{(p1 . l),(M . l)},{(p1 . l)}} is non empty set
the addF of x . [(p1 . l),(M . l)] is set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
l is Element of the carrier of n
tcM . l is Element of the carrier of x
p1 . l is Element of the carrier of x
M . l is Element of the carrier of x
(p1 . l) + (M . l) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((p1 . l),(M . l)) is Element of the carrier of x
[(p1 . l),(M . l)] is set
{(p1 . l),(M . l)} is non empty set
{(p1 . l)} is non empty set
{{(p1 . l),(M . l)},{(p1 . l)}} is non empty set
the addF of x . [(p1 . l),(M . l)] is set
RVSM . l is Element of the carrier of x
x is non empty addLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
tcM . RVSM is Element of the carrier of x
p1 . RVSM is Element of the carrier of x
- (p1 . RVSM) is Element of the carrier of x
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
M . RVSM is Element of the carrier of x
p1 . RVSM is Element of the carrier of x
- (p1 . RVSM) is Element of the carrier of x
tcM . RVSM is Element of the carrier of x
x is non empty addLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,M)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
x is non empty multMagma
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is Element of the carrier of x
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
l is Element of the carrier of n
RVSM . l is Element of the carrier of x
M . l is Element of the carrier of x
p1 * (M . l) is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (p1,(M . l)) is Element of the carrier of x
[p1,(M . l)] is set
{p1,(M . l)} is non empty set
{p1} is non empty set
{{p1,(M . l)},{p1}} is non empty set
the multF of x . [p1,(M . l)] is set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
l is Element of the carrier of n
tcM . l is Element of the carrier of x
M . l is Element of the carrier of x
p1 * (M . l) is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (p1,(M . l)) is Element of the carrier of x
[p1,(M . l)] is set
{p1,(M . l)} is non empty set
{p1} is non empty set
{{p1,(M . l)},{p1}} is non empty set
the multF of x . [p1,(M . l)] is set
RVSM . l is Element of the carrier of x
x is non empty ZeroStr
the carrier of x is non empty set
n is VectSpStr over x
[#] n is non proper Element of bool the carrier of n
the carrier of n is set
bool the carrier of n is set
0. x is zero Element of the carrier of x
the ZeroF of x is Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
x is non empty multMagma
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
x is non empty ZeroStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
n is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
0. n is zero left_complementable right_complementable complementable Element of the carrier of n
the ZeroF of n is left_complementable right_complementable complementable Element of the carrier of n
p1 . (0. n) is left_complementable right_complementable complementable Element of the carrier of x
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
(0. x) * (0. n) is left_complementable right_complementable complementable Element of the carrier of n
the lmult of n is V16() V19([: the carrier of x, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of x, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of x, the carrier of n:], the carrier of n:]
[: the carrier of x, the carrier of n:] is set
[:[: the carrier of x, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of x, the carrier of n:], the carrier of n:] is set
the lmult of n . ((0. x),(0. n)) is left_complementable right_complementable complementable Element of the carrier of n
[(0. x),(0. n)] is set
{(0. x),(0. n)} is non empty set
{(0. x)} is non empty set
{{(0. x),(0. n)},{(0. x)}} is non empty set
the lmult of n . [(0. x),(0. n)] is set
p1 . ((0. x) * (0. n)) is left_complementable right_complementable complementable Element of the carrier of x
(0. x) * (p1 . (0. n)) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . ((0. x),(p1 . (0. n))) is left_complementable right_complementable complementable Element of the carrier of x
[(0. x),(p1 . (0. n))] is set
{(0. x),(p1 . (0. n))} is non empty set
{{(0. x),(p1 . (0. n))},{(0. x)}} is non empty set
the multF of x . [(0. x),(p1 . (0. n))] is set
x is non empty right_zeroed addLoopStr
n is non empty VectSpStr over x
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero Element of the carrier of x
the ZeroF of x is Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
p1 is Element of the carrier of n
M is Element of the carrier of n
p1 + M is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (p1,M) is Element of the carrier of n
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of n . [p1,M] is set
(x,n) . (p1 + M) is Element of the carrier of x
(x,n) . p1 is Element of the carrier of x
(x,n) . M is Element of the carrier of x
((x,n) . p1) + ((x,n) . M) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (((x,n) . p1),((x,n) . M)) is Element of the carrier of x
[((x,n) . p1),((x,n) . M)] is set
{((x,n) . p1),((x,n) . M)} is non empty set
{((x,n) . p1)} is non empty set
{{((x,n) . p1),((x,n) . M)},{((x,n) . p1)}} is non empty set
the addF of x . [((x,n) . p1),((x,n) . M)] is set
x is non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
n is non empty VectSpStr over x
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive Element of bool [: the carrier of n, the carrier of x:]
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero right_complementable Element of the carrier of x
the ZeroF of x is right_complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
p1 is Element of the carrier of n
(x,n) . p1 is right_complementable Element of the carrier of x
M is right_complementable Element of the carrier of x
M * p1 is Element of the carrier of n
the lmult of n is V16() V19([: the carrier of x, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of x, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of x, the carrier of n:], the carrier of n:]
[: the carrier of x, the carrier of n:] is set
[:[: the carrier of x, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of x, the carrier of n:], the carrier of n:] is set
the lmult of n . (M,p1) is Element of the carrier of n
[M,p1] is set
{M,p1} is non empty set
{M} is non empty set
{{M,p1},{M}} is non empty set
the lmult of n . [M,p1] is set
(x,n) . (M * p1) is right_complementable Element of the carrier of x
M * ((x,n) . p1) is right_complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (M,((x,n) . p1)) is right_complementable Element of the carrier of x
[M,((x,n) . p1)] is set
{M,((x,n) . p1)} is non empty set
{{M,((x,n) . p1)},{M}} is non empty set
the multF of x . [M,((x,n) . p1)] is set
x is non empty ZeroStr
n is non empty VectSpStr over x
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero Element of the carrier of x
the ZeroF of x is Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(x,n) . (0. n) is Element of the carrier of x
x is non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero right_complementable Element of the carrier of x
the ZeroF of x is right_complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
x is non empty Abelian addLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of n
(x,n,p1,M) . tcM is Element of the carrier of x
p1 . tcM is Element of the carrier of x
M . tcM is Element of the carrier of x
(p1 . tcM) + (M . tcM) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((p1 . tcM),(M . tcM)) is Element of the carrier of x
[(p1 . tcM),(M . tcM)] is set
{(p1 . tcM),(M . tcM)} is non empty set
{(p1 . tcM)} is non empty set
{{(p1 . tcM),(M . tcM)},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),(M . tcM)] is set
(x,n,M,p1) . tcM is Element of the carrier of x
x is non empty add-associative addLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,p1,M),tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,M,tcM)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
(x,n,(x,n,p1,M),tcM) . RVSM is Element of the carrier of x
(x,n,p1,M) . RVSM is Element of the carrier of x
tcM . RVSM is Element of the carrier of x
((x,n,p1,M) . RVSM) + (tcM . RVSM) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (((x,n,p1,M) . RVSM),(tcM . RVSM)) is Element of the carrier of x
[((x,n,p1,M) . RVSM),(tcM . RVSM)] is set
{((x,n,p1,M) . RVSM),(tcM . RVSM)} is non empty set
{((x,n,p1,M) . RVSM)} is non empty set
{{((x,n,p1,M) . RVSM),(tcM . RVSM)},{((x,n,p1,M) . RVSM)}} is non empty set
the addF of x . [((x,n,p1,M) . RVSM),(tcM . RVSM)] is set
p1 . RVSM is Element of the carrier of x
M . RVSM is Element of the carrier of x
(p1 . RVSM) + (M . RVSM) is Element of the carrier of x
the addF of x . ((p1 . RVSM),(M . RVSM)) is Element of the carrier of x
[(p1 . RVSM),(M . RVSM)] is set
{(p1 . RVSM),(M . RVSM)} is non empty set
{(p1 . RVSM)} is non empty set
{{(p1 . RVSM),(M . RVSM)},{(p1 . RVSM)}} is non empty set
the addF of x . [(p1 . RVSM),(M . RVSM)] is set
((p1 . RVSM) + (M . RVSM)) + (tcM . RVSM) is Element of the carrier of x
the addF of x . (((p1 . RVSM) + (M . RVSM)),(tcM . RVSM)) is Element of the carrier of x
[((p1 . RVSM) + (M . RVSM)),(tcM . RVSM)] is set
{((p1 . RVSM) + (M . RVSM)),(tcM . RVSM)} is non empty set
{((p1 . RVSM) + (M . RVSM))} is non empty set
{{((p1 . RVSM) + (M . RVSM)),(tcM . RVSM)},{((p1 . RVSM) + (M . RVSM))}} is non empty set
the addF of x . [((p1 . RVSM) + (M . RVSM)),(tcM . RVSM)] is set
(M . RVSM) + (tcM . RVSM) is Element of the carrier of x
the addF of x . ((M . RVSM),(tcM . RVSM)) is Element of the carrier of x
[(M . RVSM),(tcM . RVSM)] is set
{(M . RVSM),(tcM . RVSM)} is non empty set
{(M . RVSM)} is non empty set
{{(M . RVSM),(tcM . RVSM)},{(M . RVSM)}} is non empty set
the addF of x . [(M . RVSM),(tcM . RVSM)] is set
(p1 . RVSM) + ((M . RVSM) + (tcM . RVSM)) is Element of the carrier of x
the addF of x . ((p1 . RVSM),((M . RVSM) + (tcM . RVSM))) is Element of the carrier of x
[(p1 . RVSM),((M . RVSM) + (tcM . RVSM))] is set
{(p1 . RVSM),((M . RVSM) + (tcM . RVSM))} is non empty set
{{(p1 . RVSM),((M . RVSM) + (tcM . RVSM))},{(p1 . RVSM)}} is non empty set
the addF of x . [(p1 . RVSM),((M . RVSM) + (tcM . RVSM))] is set
(x,n,M,tcM) . RVSM is Element of the carrier of x
(p1 . RVSM) + ((x,n,M,tcM) . RVSM) is Element of the carrier of x
the addF of x . ((p1 . RVSM),((x,n,M,tcM) . RVSM)) is Element of the carrier of x
[(p1 . RVSM),((x,n,M,tcM) . RVSM)] is set
{(p1 . RVSM),((x,n,M,tcM) . RVSM)} is non empty set
{{(p1 . RVSM),((x,n,M,tcM) . RVSM)},{(p1 . RVSM)}} is non empty set
the addF of x . [(p1 . RVSM),((x,n,M,tcM) . RVSM)] is set
(x,n,p1,(x,n,M,tcM)) . RVSM is Element of the carrier of x
x is non empty ZeroStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) (x,n) Element of bool [: the carrier of n, the carrier of x:]
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero Element of the carrier of x
the ZeroF of x is Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
p1 is Element of the carrier of n
(x,n) . p1 is Element of the carrier of x
x is non empty right_zeroed addLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero Element of the carrier of x
the ZeroF of x is Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of the carrier of n
(x,n,p1,(x,n)) . M is Element of the carrier of x
p1 . M is Element of the carrier of x
(x,n) . M is Element of the carrier of x
(p1 . M) + ((x,n) . M) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((p1 . M),((x,n) . M)) is Element of the carrier of x
[(p1 . M),((x,n) . M)] is set
{(p1 . M),((x,n) . M)} is non empty set
{(p1 . M)} is non empty set
{{(p1 . M),((x,n) . M)},{(p1 . M)}} is non empty set
the addF of x . [(p1 . M),((x,n) . M)] is set
(p1 . M) + (0. x) is Element of the carrier of x
the addF of x . ((p1 . M),(0. x)) is Element of the carrier of x
[(p1 . M),(0. x)] is set
{(p1 . M),(0. x)} is non empty set
{{(p1 . M),(0. x)},{(p1 . M)}} is non empty set
the addF of x . [(p1 . M),(0. x)] is set
x is non empty right_complementable add-associative right_zeroed addLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero right_complementable Element of the carrier of x
the ZeroF of x is right_complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,p1)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of the carrier of n
(x,n,p1,p1) . M is right_complementable Element of the carrier of x
p1 . M is right_complementable Element of the carrier of x
(x,n,p1) . M is right_complementable Element of the carrier of x
(p1 . M) + ((x,n,p1) . M) is right_complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((p1 . M),((x,n,p1) . M)) is right_complementable Element of the carrier of x
[(p1 . M),((x,n,p1) . M)] is set
{(p1 . M),((x,n,p1) . M)} is non empty set
{(p1 . M)} is non empty set
{{(p1 . M),((x,n,p1) . M)},{(p1 . M)}} is non empty set
the addF of x . [(p1 . M),((x,n,p1) . M)] is set
- (p1 . M) is right_complementable Element of the carrier of x
(p1 . M) + (- (p1 . M)) is right_complementable Element of the carrier of x
the addF of x . ((p1 . M),(- (p1 . M))) is right_complementable Element of the carrier of x
[(p1 . M),(- (p1 . M))] is set
{(p1 . M),(- (p1 . M))} is non empty set
{{(p1 . M),(- (p1 . M))},{(p1 . M)}} is non empty set
the addF of x . [(p1 . M),(- (p1 . M))] is set
(x,n) . M is right_complementable Element of the carrier of x
x is non empty right-distributive doubleLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is Element of the carrier of x
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,M,tcM)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,p1,M),(x,n,p1,tcM)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
(x,n,p1,(x,n,M,tcM)) . RVSM is Element of the carrier of x
(x,n,M,tcM) . RVSM is Element of the carrier of x
p1 * ((x,n,M,tcM) . RVSM) is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (p1,((x,n,M,tcM) . RVSM)) is Element of the carrier of x
[p1,((x,n,M,tcM) . RVSM)] is set
{p1,((x,n,M,tcM) . RVSM)} is non empty set
{p1} is non empty set
{{p1,((x,n,M,tcM) . RVSM)},{p1}} is non empty set
the multF of x . [p1,((x,n,M,tcM) . RVSM)] is set
M . RVSM is Element of the carrier of x
tcM . RVSM is Element of the carrier of x
(M . RVSM) + (tcM . RVSM) is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 addF of x . ((M . RVSM),(tcM . RVSM)) is Element of the carrier of x
[(M . RVSM),(tcM . RVSM)] is set
{(M . RVSM),(tcM . RVSM)} is non empty set
{(M . RVSM)} is non empty set
{{(M . RVSM),(tcM . RVSM)},{(M . RVSM)}} is non empty set
the addF of x . [(M . RVSM),(tcM . RVSM)] is set
p1 * ((M . RVSM) + (tcM . RVSM)) is Element of the carrier of x
the multF of x . (p1,((M . RVSM) + (tcM . RVSM))) is Element of the carrier of x
[p1,((M . RVSM) + (tcM . RVSM))] is set
{p1,((M . RVSM) + (tcM . RVSM))} is non empty set
{{p1,((M . RVSM) + (tcM . RVSM))},{p1}} is non empty set
the multF of x . [p1,((M . RVSM) + (tcM . RVSM))] is set
p1 * (M . RVSM) is Element of the carrier of x
the multF of x . (p1,(M . RVSM)) is Element of the carrier of x
[p1,(M . RVSM)] is set
{p1,(M . RVSM)} is non empty set
{{p1,(M . RVSM)},{p1}} is non empty set
the multF of x . [p1,(M . RVSM)] is set
p1 * (tcM . RVSM) is Element of the carrier of x
the multF of x . (p1,(tcM . RVSM)) is Element of the carrier of x
[p1,(tcM . RVSM)] is set
{p1,(tcM . RVSM)} is non empty set
{{p1,(tcM . RVSM)},{p1}} is non empty set
the multF of x . [p1,(tcM . RVSM)] is set
(p1 * (M . RVSM)) + (p1 * (tcM . RVSM)) is Element of the carrier of x
the addF of x . ((p1 * (M . RVSM)),(p1 * (tcM . RVSM))) is Element of the carrier of x
[(p1 * (M . RVSM)),(p1 * (tcM . RVSM))] is set
{(p1 * (M . RVSM)),(p1 * (tcM . RVSM))} is non empty set
{(p1 * (M . RVSM))} is non empty set
{{(p1 * (M . RVSM)),(p1 * (tcM . RVSM))},{(p1 * (M . RVSM))}} is non empty set
the addF of x . [(p1 * (M . RVSM)),(p1 * (tcM . RVSM))] is set
(x,n,p1,M) . RVSM is Element of the carrier of x
((x,n,p1,M) . RVSM) + (p1 * (tcM . RVSM)) is Element of the carrier of x
the addF of x . (((x,n,p1,M) . RVSM),(p1 * (tcM . RVSM))) is Element of the carrier of x
[((x,n,p1,M) . RVSM),(p1 * (tcM . RVSM))] is set
{((x,n,p1,M) . RVSM),(p1 * (tcM . RVSM))} is non empty set
{((x,n,p1,M) . RVSM)} is non empty set
{{((x,n,p1,M) . RVSM),(p1 * (tcM . RVSM))},{((x,n,p1,M) . RVSM)}} is non empty set
the addF of x . [((x,n,p1,M) . RVSM),(p1 * (tcM . RVSM))] is set
(x,n,p1,tcM) . RVSM is Element of the carrier of x
((x,n,p1,M) . RVSM) + ((x,n,p1,tcM) . RVSM) is Element of the carrier of x
the addF of x . (((x,n,p1,M) . RVSM),((x,n,p1,tcM) . RVSM)) is Element of the carrier of x
[((x,n,p1,M) . RVSM),((x,n,p1,tcM) . RVSM)] is set
{((x,n,p1,M) . RVSM),((x,n,p1,tcM) . RVSM)} is non empty set
{{((x,n,p1,M) . RVSM),((x,n,p1,tcM) . RVSM)},{((x,n,p1,M) . RVSM)}} is non empty set
the addF of x . [((x,n,p1,M) . RVSM),((x,n,p1,tcM) . RVSM)] is set
(x,n,(x,n,p1,M),(x,n,p1,tcM)) . RVSM is Element of the carrier of x
x is non empty left-distributive doubleLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is Element of the carrier of x
M is Element of the carrier of x
p1 + M is Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (p1,M) is Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of x . [p1,M] is set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(p1 + M),tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,p1,tcM),(x,n,M,tcM)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
(x,n,(p1 + M),tcM) . RVSM is Element of the carrier of x
tcM . RVSM is Element of the carrier of x
(p1 + M) * (tcM . RVSM) is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 multF of x . ((p1 + M),(tcM . RVSM)) is Element of the carrier of x
[(p1 + M),(tcM . RVSM)] is set
{(p1 + M),(tcM . RVSM)} is non empty set
{(p1 + M)} is non empty set
{{(p1 + M),(tcM . RVSM)},{(p1 + M)}} is non empty set
the multF of x . [(p1 + M),(tcM . RVSM)] is set
p1 * (tcM . RVSM) is Element of the carrier of x
the multF of x . (p1,(tcM . RVSM)) is Element of the carrier of x
[p1,(tcM . RVSM)] is set
{p1,(tcM . RVSM)} is non empty set
{{p1,(tcM . RVSM)},{p1}} is non empty set
the multF of x . [p1,(tcM . RVSM)] is set
M * (tcM . RVSM) is Element of the carrier of x
the multF of x . (M,(tcM . RVSM)) is Element of the carrier of x
[M,(tcM . RVSM)] is set
{M,(tcM . RVSM)} is non empty set
{M} is non empty set
{{M,(tcM . RVSM)},{M}} is non empty set
the multF of x . [M,(tcM . RVSM)] is set
(p1 * (tcM . RVSM)) + (M * (tcM . RVSM)) is Element of the carrier of x
the addF of x . ((p1 * (tcM . RVSM)),(M * (tcM . RVSM))) is Element of the carrier of x
[(p1 * (tcM . RVSM)),(M * (tcM . RVSM))] is set
{(p1 * (tcM . RVSM)),(M * (tcM . RVSM))} is non empty set
{(p1 * (tcM . RVSM))} is non empty set
{{(p1 * (tcM . RVSM)),(M * (tcM . RVSM))},{(p1 * (tcM . RVSM))}} is non empty set
the addF of x . [(p1 * (tcM . RVSM)),(M * (tcM . RVSM))] is set
(x,n,p1,tcM) . RVSM is Element of the carrier of x
((x,n,p1,tcM) . RVSM) + (M * (tcM . RVSM)) is Element of the carrier of x
the addF of x . (((x,n,p1,tcM) . RVSM),(M * (tcM . RVSM))) is Element of the carrier of x
[((x,n,p1,tcM) . RVSM),(M * (tcM . RVSM))] is set
{((x,n,p1,tcM) . RVSM),(M * (tcM . RVSM))} is non empty set
{((x,n,p1,tcM) . RVSM)} is non empty set
{{((x,n,p1,tcM) . RVSM),(M * (tcM . RVSM))},{((x,n,p1,tcM) . RVSM)}} is non empty set
the addF of x . [((x,n,p1,tcM) . RVSM),(M * (tcM . RVSM))] is set
(x,n,M,tcM) . RVSM is Element of the carrier of x
((x,n,p1,tcM) . RVSM) + ((x,n,M,tcM) . RVSM) is Element of the carrier of x
the addF of x . (((x,n,p1,tcM) . RVSM),((x,n,M,tcM) . RVSM)) is Element of the carrier of x
[((x,n,p1,tcM) . RVSM),((x,n,M,tcM) . RVSM)] is set
{((x,n,p1,tcM) . RVSM),((x,n,M,tcM) . RVSM)} is non empty set
{{((x,n,p1,tcM) . RVSM),((x,n,M,tcM) . RVSM)},{((x,n,p1,tcM) . RVSM)}} is non empty set
the addF of x . [((x,n,p1,tcM) . RVSM),((x,n,M,tcM) . RVSM)] is set
(x,n,(x,n,p1,tcM),(x,n,M,tcM)) . RVSM is Element of the carrier of x
x is non empty associative multMagma
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is Element of the carrier of x
M is Element of the carrier of x
p1 * M is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (p1,M) is Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the multF of x . [p1,M] is set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(p1 * M),tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,M,tcM)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
RVSM is Element of the carrier of n
(x,n,(p1 * M),tcM) . RVSM is Element of the carrier of x
tcM . RVSM is Element of the carrier of x
(p1 * M) * (tcM . RVSM) is Element of the carrier of x
the multF of x . ((p1 * M),(tcM . RVSM)) is Element of the carrier of x
[(p1 * M),(tcM . RVSM)] is set
{(p1 * M),(tcM . RVSM)} is non empty set
{(p1 * M)} is non empty set
{{(p1 * M),(tcM . RVSM)},{(p1 * M)}} is non empty set
the multF of x . [(p1 * M),(tcM . RVSM)] is set
M * (tcM . RVSM) is Element of the carrier of x
the multF of x . (M,(tcM . RVSM)) is Element of the carrier of x
[M,(tcM . RVSM)] is set
{M,(tcM . RVSM)} is non empty set
{M} is non empty set
{{M,(tcM . RVSM)},{M}} is non empty set
the multF of x . [M,(tcM . RVSM)] is set
p1 * (M * (tcM . RVSM)) is Element of the carrier of x
the multF of x . (p1,(M * (tcM . RVSM))) is Element of the carrier of x
[p1,(M * (tcM . RVSM))] is set
{p1,(M * (tcM . RVSM))} is non empty set
{{p1,(M * (tcM . RVSM))},{p1}} is non empty set
the multF of x . [p1,(M * (tcM . RVSM))] is set
(x,n,M,tcM) . RVSM is Element of the carrier of x
p1 * ((x,n,M,tcM) . RVSM) is Element of the carrier of x
the multF of x . (p1,((x,n,M,tcM) . RVSM)) is Element of the carrier of x
[p1,((x,n,M,tcM) . RVSM)] is set
{p1,((x,n,M,tcM) . RVSM)} is non empty set
{{p1,((x,n,M,tcM) . RVSM)},{p1}} is non empty set
the multF of x . [p1,((x,n,M,tcM) . RVSM)] is set
(x,n,p1,(x,n,M,tcM)) . RVSM is Element of the carrier of x
x is non empty left_unital doubleLoopStr
the carrier of x is non empty set
1. x is Element of the carrier of x
the OneF of x is Element of the carrier of x
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(1. x),p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of the carrier of n
(x,n,(1. x),p1) . M is Element of the carrier of x
p1 . M is Element of the carrier of x
(1. x) * (p1 . M) is Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . ((1. x),(p1 . M)) is Element of the carrier of x
[(1. x),(p1 . M)] is set
{(1. x),(p1 . M)} is non empty set
{(1. x)} is non empty set
{{(1. x),(p1 . M)},{(1. x)}} is non empty set
the multF of x . [(1. x),(p1 . M)] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of n
RVSM is Element of the carrier of n
tcM + RVSM is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (tcM,RVSM) is Element of the carrier of n
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of n . [tcM,RVSM] is set
(x,n,p1,M) . (tcM + RVSM) is left_complementable right_complementable complementable Element of the carrier of x
(x,n,p1,M) . tcM is left_complementable right_complementable complementable Element of the carrier of x
(x,n,p1,M) . RVSM is left_complementable right_complementable complementable Element of the carrier of x
((x,n,p1,M) . tcM) + ((x,n,p1,M) . RVSM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
[((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)] is set
{((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)} is non empty set
{((x,n,p1,M) . tcM)} is non empty set
{{((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)},{((x,n,p1,M) . tcM)}} is non empty set
the addF of x . [((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)] is set
p1 . (tcM + RVSM) is left_complementable right_complementable complementable Element of the carrier of x
M . (tcM + RVSM) is left_complementable right_complementable complementable Element of the carrier of x
(p1 . (tcM + RVSM)) + (M . (tcM + RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . (tcM + RVSM)),(M . (tcM + RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . (tcM + RVSM)),(M . (tcM + RVSM))] is set
{(p1 . (tcM + RVSM)),(M . (tcM + RVSM))} is non empty set
{(p1 . (tcM + RVSM))} is non empty set
{{(p1 . (tcM + RVSM)),(M . (tcM + RVSM))},{(p1 . (tcM + RVSM))}} is non empty set
the addF of x . [(p1 . (tcM + RVSM)),(M . (tcM + RVSM))] is set
p1 . tcM is left_complementable right_complementable complementable Element of the carrier of x
p1 . RVSM is left_complementable right_complementable complementable Element of the carrier of x
(p1 . tcM) + (p1 . RVSM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . tcM),(p1 . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . tcM),(p1 . RVSM)] is set
{(p1 . tcM),(p1 . RVSM)} is non empty set
{(p1 . tcM)} is non empty set
{{(p1 . tcM),(p1 . RVSM)},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),(p1 . RVSM)] is set
((p1 . tcM) + (p1 . RVSM)) + (M . (tcM + RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (((p1 . tcM) + (p1 . RVSM)),(M . (tcM + RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[((p1 . tcM) + (p1 . RVSM)),(M . (tcM + RVSM))] is set
{((p1 . tcM) + (p1 . RVSM)),(M . (tcM + RVSM))} is non empty set
{((p1 . tcM) + (p1 . RVSM))} is non empty set
{{((p1 . tcM) + (p1 . RVSM)),(M . (tcM + RVSM))},{((p1 . tcM) + (p1 . RVSM))}} is non empty set
the addF of x . [((p1 . tcM) + (p1 . RVSM)),(M . (tcM + RVSM))] is set
M . tcM is left_complementable right_complementable complementable Element of the carrier of x
M . RVSM is left_complementable right_complementable complementable Element of the carrier of x
(M . tcM) + (M . RVSM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((M . tcM),(M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
[(M . tcM),(M . RVSM)] is set
{(M . tcM),(M . RVSM)} is non empty set
{(M . tcM)} is non empty set
{{(M . tcM),(M . RVSM)},{(M . tcM)}} is non empty set
the addF of x . [(M . tcM),(M . RVSM)] is set
((p1 . tcM) + (p1 . RVSM)) + ((M . tcM) + (M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (((p1 . tcM) + (p1 . RVSM)),((M . tcM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[((p1 . tcM) + (p1 . RVSM)),((M . tcM) + (M . RVSM))] is set
{((p1 . tcM) + (p1 . RVSM)),((M . tcM) + (M . RVSM))} is non empty set
{{((p1 . tcM) + (p1 . RVSM)),((M . tcM) + (M . RVSM))},{((p1 . tcM) + (p1 . RVSM))}} is non empty set
the addF of x . [((p1 . tcM) + (p1 . RVSM)),((M . tcM) + (M . RVSM))] is set
(p1 . RVSM) + ((M . tcM) + (M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . RVSM),((M . tcM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . RVSM),((M . tcM) + (M . RVSM))] is set
{(p1 . RVSM),((M . tcM) + (M . RVSM))} is non empty set
{(p1 . RVSM)} is non empty set
{{(p1 . RVSM),((M . tcM) + (M . RVSM))},{(p1 . RVSM)}} is non empty set
the addF of x . [(p1 . RVSM),((M . tcM) + (M . RVSM))] is set
(p1 . tcM) + ((p1 . RVSM) + ((M . tcM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . tcM),((p1 . RVSM) + ((M . tcM) + (M . RVSM)))) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . tcM),((p1 . RVSM) + ((M . tcM) + (M . RVSM)))] is set
{(p1 . tcM),((p1 . RVSM) + ((M . tcM) + (M . RVSM)))} is non empty set
{{(p1 . tcM),((p1 . RVSM) + ((M . tcM) + (M . RVSM)))},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),((p1 . RVSM) + ((M . tcM) + (M . RVSM)))] is set
(p1 . RVSM) + (M . RVSM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . RVSM),(M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . RVSM),(M . RVSM)] is set
{(p1 . RVSM),(M . RVSM)} is non empty set
{{(p1 . RVSM),(M . RVSM)},{(p1 . RVSM)}} is non empty set
the addF of x . [(p1 . RVSM),(M . RVSM)] is set
(M . tcM) + ((p1 . RVSM) + (M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((M . tcM),((p1 . RVSM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[(M . tcM),((p1 . RVSM) + (M . RVSM))] is set
{(M . tcM),((p1 . RVSM) + (M . RVSM))} is non empty set
{{(M . tcM),((p1 . RVSM) + (M . RVSM))},{(M . tcM)}} is non empty set
the addF of x . [(M . tcM),((p1 . RVSM) + (M . RVSM))] is set
(p1 . tcM) + ((M . tcM) + ((p1 . RVSM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . tcM),((M . tcM) + ((p1 . RVSM) + (M . RVSM)))) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . tcM),((M . tcM) + ((p1 . RVSM) + (M . RVSM)))] is set
{(p1 . tcM),((M . tcM) + ((p1 . RVSM) + (M . RVSM)))} is non empty set
{{(p1 . tcM),((M . tcM) + ((p1 . RVSM) + (M . RVSM)))},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),((M . tcM) + ((p1 . RVSM) + (M . RVSM)))] is set
(p1 . tcM) + (M . tcM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . tcM),(M . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . tcM),(M . tcM)] is set
{(p1 . tcM),(M . tcM)} is non empty set
{{(p1 . tcM),(M . tcM)},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),(M . tcM)] is set
((p1 . tcM) + (M . tcM)) + ((p1 . RVSM) + (M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (((p1 . tcM) + (M . tcM)),((p1 . RVSM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[((p1 . tcM) + (M . tcM)),((p1 . RVSM) + (M . RVSM))] is set
{((p1 . tcM) + (M . tcM)),((p1 . RVSM) + (M . RVSM))} is non empty set
{((p1 . tcM) + (M . tcM))} is non empty set
{{((p1 . tcM) + (M . tcM)),((p1 . RVSM) + (M . RVSM))},{((p1 . tcM) + (M . tcM))}} is non empty set
the addF of x . [((p1 . tcM) + (M . tcM)),((p1 . RVSM) + (M . RVSM))] is set
((x,n,p1,M) . tcM) + ((p1 . RVSM) + (M . RVSM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (((x,n,p1,M) . tcM),((p1 . RVSM) + (M . RVSM))) is left_complementable right_complementable complementable Element of the carrier of x
[((x,n,p1,M) . tcM),((p1 . RVSM) + (M . RVSM))] is set
{((x,n,p1,M) . tcM),((p1 . RVSM) + (M . RVSM))} is non empty set
{{((x,n,p1,M) . tcM),((p1 . RVSM) + (M . RVSM))},{((x,n,p1,M) . tcM)}} is non empty set
the addF of x . [((x,n,p1,M) . tcM),((p1 . RVSM) + (M . RVSM))] is set
((x,n,p1,M) . tcM) + ((x,n,p1,M) . RVSM) is left_complementable right_complementable complementable Element of the carrier of x
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of the carrier of n
tcM is Element of the carrier of n
M + tcM is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (M,tcM) is Element of the carrier of n
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of n . [M,tcM] is set
(x,n,p1) . (M + tcM) is left_complementable right_complementable complementable Element of the carrier of x
(x,n,p1) . M is left_complementable right_complementable complementable Element of the carrier of x
(x,n,p1) . tcM is left_complementable right_complementable complementable Element of the carrier of x
((x,n,p1) . M) + ((x,n,p1) . tcM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (((x,n,p1) . M),((x,n,p1) . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
[((x,n,p1) . M),((x,n,p1) . tcM)] is set
{((x,n,p1) . M),((x,n,p1) . tcM)} is non empty set
{((x,n,p1) . M)} is non empty set
{{((x,n,p1) . M),((x,n,p1) . tcM)},{((x,n,p1) . M)}} is non empty set
the addF of x . [((x,n,p1) . M),((x,n,p1) . tcM)] is set
p1 . (M + tcM) is left_complementable right_complementable complementable Element of the carrier of x
- (p1 . (M + tcM)) is left_complementable right_complementable complementable Element of the carrier of x
p1 . M is left_complementable right_complementable complementable Element of the carrier of x
p1 . tcM is left_complementable right_complementable complementable Element of the carrier of x
(p1 . M) + (p1 . tcM) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((p1 . M),(p1 . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
[(p1 . M),(p1 . tcM)] is set
{(p1 . M),(p1 . tcM)} is non empty set
{(p1 . M)} is non empty set
{{(p1 . M),(p1 . tcM)},{(p1 . M)}} is non empty set
the addF of x . [(p1 . M),(p1 . tcM)] is set
- ((p1 . M) + (p1 . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
- (p1 . M) is left_complementable right_complementable complementable Element of the carrier of x
- (p1 . tcM) is left_complementable right_complementable complementable Element of the carrier of x
(- (p1 . M)) + (- (p1 . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((- (p1 . M)),(- (p1 . tcM))) is left_complementable right_complementable complementable Element of the carrier of x
[(- (p1 . M)),(- (p1 . tcM))] is set
{(- (p1 . M)),(- (p1 . tcM))} is non empty set
{(- (p1 . M))} is non empty set
{{(- (p1 . M)),(- (p1 . tcM))},{(- (p1 . M))}} is non empty set
the addF of x . [(- (p1 . M)),(- (p1 . tcM))] is set
((x,n,p1) . M) + (- (p1 . tcM)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (((x,n,p1) . M),(- (p1 . tcM))) is left_complementable right_complementable complementable Element of the carrier of x
[((x,n,p1) . M),(- (p1 . tcM))] is set
{((x,n,p1) . M),(- (p1 . tcM))} is non empty set
{{((x,n,p1) . M),(- (p1 . tcM))},{((x,n,p1) . M)}} is non empty set
the addF of x . [((x,n,p1) . M),(- (p1 . tcM))] is set
((x,n,p1) . M) + ((x,n,p1) . tcM) is left_complementable right_complementable complementable Element of the carrier of x
x is non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is right_complementable Element of the carrier of x
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of n
RVSM is Element of the carrier of n
tcM + RVSM is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (tcM,RVSM) is Element of the carrier of n
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of n . [tcM,RVSM] is set
(x,n,p1,M) . (tcM + RVSM) is right_complementable Element of the carrier of x
(x,n,p1,M) . tcM is right_complementable Element of the carrier of x
(x,n,p1,M) . RVSM is right_complementable Element of the carrier of x
((x,n,p1,M) . tcM) + ((x,n,p1,M) . RVSM) is right_complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)) is right_complementable Element of the carrier of x
[((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)] is set
{((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)} is non empty set
{((x,n,p1,M) . tcM)} is non empty set
{{((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)},{((x,n,p1,M) . tcM)}} is non empty set
the addF of x . [((x,n,p1,M) . tcM),((x,n,p1,M) . RVSM)] is set
M . (tcM + RVSM) is right_complementable Element of the carrier of x
p1 * (M . (tcM + RVSM)) is right_complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 multF of x . (p1,(M . (tcM + RVSM))) is right_complementable Element of the carrier of x
[p1,(M . (tcM + RVSM))] is set
{p1,(M . (tcM + RVSM))} is non empty set
{p1} is non empty set
{{p1,(M . (tcM + RVSM))},{p1}} is non empty set
the multF of x . [p1,(M . (tcM + RVSM))] is set
M . tcM is right_complementable Element of the carrier of x
M . RVSM is right_complementable Element of the carrier of x
(M . tcM) + (M . RVSM) is right_complementable Element of the carrier of x
the addF of x . ((M . tcM),(M . RVSM)) is right_complementable Element of the carrier of x
[(M . tcM),(M . RVSM)] is set
{(M . tcM),(M . RVSM)} is non empty set
{(M . tcM)} is non empty set
{{(M . tcM),(M . RVSM)},{(M . tcM)}} is non empty set
the addF of x . [(M . tcM),(M . RVSM)] is set
p1 * ((M . tcM) + (M . RVSM)) is right_complementable Element of the carrier of x
the multF of x . (p1,((M . tcM) + (M . RVSM))) is right_complementable Element of the carrier of x
[p1,((M . tcM) + (M . RVSM))] is set
{p1,((M . tcM) + (M . RVSM))} is non empty set
{{p1,((M . tcM) + (M . RVSM))},{p1}} is non empty set
the multF of x . [p1,((M . tcM) + (M . RVSM))] is set
p1 * (M . tcM) is right_complementable Element of the carrier of x
the multF of x . (p1,(M . tcM)) is right_complementable Element of the carrier of x
[p1,(M . tcM)] is set
{p1,(M . tcM)} is non empty set
{{p1,(M . tcM)},{p1}} is non empty set
the multF of x . [p1,(M . tcM)] is set
p1 * (M . RVSM) is right_complementable Element of the carrier of x
the multF of x . (p1,(M . RVSM)) is right_complementable Element of the carrier of x
[p1,(M . RVSM)] is set
{p1,(M . RVSM)} is non empty set
{{p1,(M . RVSM)},{p1}} is non empty set
the multF of x . [p1,(M . RVSM)] is set
(p1 * (M . tcM)) + (p1 * (M . RVSM)) is right_complementable Element of the carrier of x
the addF of x . ((p1 * (M . tcM)),(p1 * (M . RVSM))) is right_complementable Element of the carrier of x
[(p1 * (M . tcM)),(p1 * (M . RVSM))] is set
{(p1 * (M . tcM)),(p1 * (M . RVSM))} is non empty set
{(p1 * (M . tcM))} is non empty set
{{(p1 * (M . tcM)),(p1 * (M . RVSM))},{(p1 * (M . tcM))}} is non empty set
the addF of x . [(p1 * (M . tcM)),(p1 * (M . RVSM))] is set
((x,n,p1,M) . tcM) + (p1 * (M . RVSM)) is right_complementable Element of the carrier of x
the addF of x . (((x,n,p1,M) . tcM),(p1 * (M . RVSM))) is right_complementable Element of the carrier of x
[((x,n,p1,M) . tcM),(p1 * (M . RVSM))] is set
{((x,n,p1,M) . tcM),(p1 * (M . RVSM))} is non empty set
{{((x,n,p1,M) . tcM),(p1 * (M . RVSM))},{((x,n,p1,M) . tcM)}} is non empty set
the addF of x . [((x,n,p1,M) . tcM),(p1 * (M . RVSM))] is set
x is non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) (x,n) Element of bool [: the carrier of n, the carrier of x:]
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of n
(x,n,p1,M) . tcM is right_complementable Element of the carrier of x
RVSM is right_complementable Element of the carrier of x
RVSM * tcM is Element of the carrier of n
the lmult of n is V16() V19([: the carrier of x, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of x, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of x, the carrier of n:], the carrier of n:]
[: the carrier of x, the carrier of n:] is set
[:[: the carrier of x, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of x, the carrier of n:], the carrier of n:] is set
the lmult of n . (RVSM,tcM) is Element of the carrier of n
[RVSM,tcM] is set
{RVSM,tcM} is non empty set
{RVSM} is non empty set
{{RVSM,tcM},{RVSM}} is non empty set
the lmult of n . [RVSM,tcM] is set
(x,n,p1,M) . (RVSM * tcM) is right_complementable Element of the carrier of x
RVSM * ((x,n,p1,M) . tcM) is right_complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (RVSM,((x,n,p1,M) . tcM)) is right_complementable Element of the carrier of x
[RVSM,((x,n,p1,M) . tcM)] is set
{RVSM,((x,n,p1,M) . tcM)} is non empty set
{{RVSM,((x,n,p1,M) . tcM)},{RVSM}} is non empty set
the multF of x . [RVSM,((x,n,p1,M) . tcM)] is set
p1 . (RVSM * tcM) is right_complementable Element of the carrier of x
M . (RVSM * tcM) is right_complementable Element of the carrier of x
(p1 . (RVSM * tcM)) + (M . (RVSM * tcM)) is right_complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 addF of x . ((p1 . (RVSM * tcM)),(M . (RVSM * tcM))) is right_complementable Element of the carrier of x
[(p1 . (RVSM * tcM)),(M . (RVSM * tcM))] is set
{(p1 . (RVSM * tcM)),(M . (RVSM * tcM))} is non empty set
{(p1 . (RVSM * tcM))} is non empty set
{{(p1 . (RVSM * tcM)),(M . (RVSM * tcM))},{(p1 . (RVSM * tcM))}} is non empty set
the addF of x . [(p1 . (RVSM * tcM)),(M . (RVSM * tcM))] is set
p1 . tcM is right_complementable Element of the carrier of x
RVSM * (p1 . tcM) is right_complementable Element of the carrier of x
the multF of x . (RVSM,(p1 . tcM)) is right_complementable Element of the carrier of x
[RVSM,(p1 . tcM)] is set
{RVSM,(p1 . tcM)} is non empty set
{{RVSM,(p1 . tcM)},{RVSM}} is non empty set
the multF of x . [RVSM,(p1 . tcM)] is set
(RVSM * (p1 . tcM)) + (M . (RVSM * tcM)) is right_complementable Element of the carrier of x
the addF of x . ((RVSM * (p1 . tcM)),(M . (RVSM * tcM))) is right_complementable Element of the carrier of x
[(RVSM * (p1 . tcM)),(M . (RVSM * tcM))] is set
{(RVSM * (p1 . tcM)),(M . (RVSM * tcM))} is non empty set
{(RVSM * (p1 . tcM))} is non empty set
{{(RVSM * (p1 . tcM)),(M . (RVSM * tcM))},{(RVSM * (p1 . tcM))}} is non empty set
the addF of x . [(RVSM * (p1 . tcM)),(M . (RVSM * tcM))] is set
M . tcM is right_complementable Element of the carrier of x
RVSM * (M . tcM) is right_complementable Element of the carrier of x
the multF of x . (RVSM,(M . tcM)) is right_complementable Element of the carrier of x
[RVSM,(M . tcM)] is set
{RVSM,(M . tcM)} is non empty set
{{RVSM,(M . tcM)},{RVSM}} is non empty set
the multF of x . [RVSM,(M . tcM)] is set
(RVSM * (p1 . tcM)) + (RVSM * (M . tcM)) is right_complementable Element of the carrier of x
the addF of x . ((RVSM * (p1 . tcM)),(RVSM * (M . tcM))) is right_complementable Element of the carrier of x
[(RVSM * (p1 . tcM)),(RVSM * (M . tcM))] is set
{(RVSM * (p1 . tcM)),(RVSM * (M . tcM))} is non empty set
{{(RVSM * (p1 . tcM)),(RVSM * (M . tcM))},{(RVSM * (p1 . tcM))}} is non empty set
the addF of x . [(RVSM * (p1 . tcM)),(RVSM * (M . tcM))] is set
(p1 . tcM) + (M . tcM) is right_complementable Element of the carrier of x
the addF of x . ((p1 . tcM),(M . tcM)) is right_complementable Element of the carrier of x
[(p1 . tcM),(M . tcM)] is set
{(p1 . tcM),(M . tcM)} is non empty set
{(p1 . tcM)} is non empty set
{{(p1 . tcM),(M . tcM)},{(p1 . tcM)}} is non empty set
the addF of x . [(p1 . tcM),(M . tcM)] is set
RVSM * ((p1 . tcM) + (M . tcM)) is right_complementable Element of the carrier of x
the multF of x . (RVSM,((p1 . tcM) + (M . tcM))) is right_complementable Element of the carrier of x
[RVSM,((p1 . tcM) + (M . tcM))] is set
{RVSM,((p1 . tcM) + (M . tcM))} is non empty set
{{RVSM,((p1 . tcM) + (M . tcM))},{RVSM}} is non empty set
the multF of x . [RVSM,((p1 . tcM) + (M . tcM))] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of the carrier of n
(x,n,p1) . M is left_complementable right_complementable complementable Element of the carrier of x
tcM is left_complementable right_complementable complementable Element of the carrier of x
tcM * M is Element of the carrier of n
the lmult of n is V16() V19([: the carrier of x, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of x, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of x, the carrier of n:], the carrier of n:]
[: the carrier of x, the carrier of n:] is set
[:[: the carrier of x, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of x, the carrier of n:], the carrier of n:] is set
the lmult of n . (tcM,M) is Element of the carrier of n
[tcM,M] is set
{tcM,M} is non empty set
{tcM} is non empty set
{{tcM,M},{tcM}} is non empty set
the lmult of n . [tcM,M] is set
(x,n,p1) . (tcM * M) is left_complementable right_complementable complementable Element of the carrier of x
tcM * ((x,n,p1) . M) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (tcM,((x,n,p1) . M)) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,((x,n,p1) . M)] is set
{tcM,((x,n,p1) . M)} is non empty set
{{tcM,((x,n,p1) . M)},{tcM}} is non empty set
the multF of x . [tcM,((x,n,p1) . M)] is set
p1 . (tcM * M) is left_complementable right_complementable complementable Element of the carrier of x
- (p1 . (tcM * M)) is left_complementable right_complementable complementable Element of the carrier of x
p1 . M is left_complementable right_complementable complementable Element of the carrier of x
tcM * (p1 . M) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . (tcM,(p1 . M)) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,(p1 . M)] is set
{tcM,(p1 . M)} is non empty set
{{tcM,(p1 . M)},{tcM}} is non empty set
the multF of x . [tcM,(p1 . M)] is set
- (tcM * (p1 . M)) is left_complementable right_complementable complementable Element of the carrier of x
- (p1 . M) is left_complementable right_complementable complementable Element of the carrier of x
tcM * (- (p1 . M)) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . (tcM,(- (p1 . M))) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,(- (p1 . M))] is set
{tcM,(- (p1 . M))} is non empty set
{{tcM,(- (p1 . M))},{tcM}} is non empty set
the multF of x . [tcM,(- (p1 . M))] is set
x is non empty right_complementable add-associative right_zeroed associative commutative right-distributive doubleLoopStr
the carrier of x is non empty set
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 is right_complementable Element of the carrier of x
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of n
(x,n,p1,M) . tcM is right_complementable Element of the carrier of x
RVSM is right_complementable Element of the carrier of x
RVSM * tcM is Element of the carrier of n
the lmult of n is V16() V19([: the carrier of x, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of x, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of x, the carrier of n:], the carrier of n:]
[: the carrier of x, the carrier of n:] is set
[:[: the carrier of x, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of x, the carrier of n:], the carrier of n:] is set
the lmult of n . (RVSM,tcM) is Element of the carrier of n
[RVSM,tcM] is set
{RVSM,tcM} is non empty set
{RVSM} is non empty set
{{RVSM,tcM},{RVSM}} is non empty set
the lmult of n . [RVSM,tcM] is set
(x,n,p1,M) . (RVSM * tcM) is right_complementable Element of the carrier of x
RVSM * ((x,n,p1,M) . tcM) is right_complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the multF of x . (RVSM,((x,n,p1,M) . tcM)) is right_complementable Element of the carrier of x
[RVSM,((x,n,p1,M) . tcM)] is set
{RVSM,((x,n,p1,M) . tcM)} is non empty set
{{RVSM,((x,n,p1,M) . tcM)},{RVSM}} is non empty set
the multF of x . [RVSM,((x,n,p1,M) . tcM)] is set
M . (RVSM * tcM) is right_complementable Element of the carrier of x
p1 * (M . (RVSM * tcM)) is right_complementable Element of the carrier of x
the multF of x . (p1,(M . (RVSM * tcM))) is right_complementable Element of the carrier of x
[p1,(M . (RVSM * tcM))] is set
{p1,(M . (RVSM * tcM))} is non empty set
{p1} is non empty set
{{p1,(M . (RVSM * tcM))},{p1}} is non empty set
the multF of x . [p1,(M . (RVSM * tcM))] is set
M . tcM is right_complementable Element of the carrier of x
RVSM * (M . tcM) is right_complementable Element of the carrier of x
the multF of x . (RVSM,(M . tcM)) is right_complementable Element of the carrier of x
[RVSM,(M . tcM)] is set
{RVSM,(M . tcM)} is non empty set
{{RVSM,(M . tcM)},{RVSM}} is non empty set
the multF of x . [RVSM,(M . tcM)] is set
p1 * (RVSM * (M . tcM)) is right_complementable Element of the carrier of x
the multF of x . (p1,(RVSM * (M . tcM))) is right_complementable Element of the carrier of x
[p1,(RVSM * (M . tcM))] is set
{p1,(RVSM * (M . tcM))} is non empty set
{{p1,(RVSM * (M . tcM))},{p1}} is non empty set
the multF of x . [p1,(RVSM * (M . tcM))] is set
p1 * (M . tcM) is right_complementable Element of the carrier of x
the multF of x . (p1,(M . tcM)) is right_complementable Element of the carrier of x
[p1,(M . tcM)] is set
{p1,(M . tcM)} is non empty set
{{p1,(M . tcM)},{p1}} is non empty set
the multF of x . [p1,(M . tcM)] is set
RVSM * (p1 * (M . tcM)) is right_complementable Element of the carrier of x
the multF of x . (RVSM,(p1 * (M . tcM))) is right_complementable Element of the carrier of x
[RVSM,(p1 * (M . tcM))] is set
{RVSM,(p1 * (M . tcM))} is non empty set
{{RVSM,(p1 * (M . tcM))},{RVSM}} is non empty set
the multF of x . [RVSM,(p1 * (M . tcM))] is set
RVSM * ((x,n,p1,M) . tcM) is right_complementable Element of the carrier of x
x is non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive doubleLoopStr
n is non empty VectSpStr over x
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
{ b₁ where b₁ is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:] : verum } is set
M is set
p1 is non empty set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of p1
RVSM is Element of p1
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,l,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
L1 is Element of p1
[:p1,p1:] is set
[:[:p1,p1:],p1:] is set
bool [:[:p1,p1:],p1:] is set
tcM is V16() V19([:p1,p1:]) V20(p1) Function-like V30([:p1,p1:],p1) Element of bool [:[:p1,p1:],p1:]
RVSM is left_complementable right_complementable complementable Element of the carrier of x
l is Element of p1
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,RVSM,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
L1 is Element of p1
[: the carrier of x,p1:] is set
[:[: the carrier of x,p1:],p1:] is set
bool [:[: the carrier of x,p1:],p1:] is set
RVSM is V16() V19([: the carrier of x,p1:]) V20(p1) Function-like V30([: the carrier of x,p1:],p1) Element of bool [:[: the carrier of x,p1:],p1:]
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
prRl is Element of p1
L1 is left_complementable right_complementable complementable Element of the carrier of x
RVSM . (L1,prRl) is Element of p1
[L1,prRl] is set
{L1,prRl} is non empty set
{L1} is non empty set
{{L1,prRl},{L1}} is non empty set
RVSM . [L1,prRl] is set
RVSM . (L1,l) is set
[L1,l] is set
{L1,l} is non empty set
{{L1,l},{L1}} is non empty set
RVSM . [L1,l] is set
(x,n,L1,l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
L is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,L1,L) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
M is Element of p1
VectSpStr(# p1,tcM,M,RVSM #) is non empty strict VectSpStr over x
l is non empty strict VectSpStr over x
the carrier of l is non empty set
the addF of l is V16() V19([: the carrier of l, the carrier of l:]) V20( the carrier of l) Function-like V30([: the carrier of l, the carrier of l:], the carrier of l) Element of bool [:[: the carrier of l, the carrier of l:], the carrier of l:]
[: the carrier of l, the carrier of l:] is set
[:[: the carrier of l, the carrier of l:], the carrier of l:] is set
bool [:[: the carrier of l, the carrier of l:], the carrier of l:] is set
0. l is zero Element of the carrier of l
the ZeroF of l is Element of the carrier of l
the lmult of l is V16() V19([: the carrier of x, the carrier of l:]) V20( the carrier of l) Function-like V30([: the carrier of x, the carrier of l:], the carrier of l) Element of bool [:[: the carrier of x, the carrier of l:], the carrier of l:]
[: the carrier of x, the carrier of l:] is set
[:[: the carrier of x, the carrier of l:], the carrier of l:] is set
bool [:[: the carrier of x, the carrier of l:], the carrier of l:] is set
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
L1 is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
L is Element of p1
e is Element of p1
tcM . (L,e) is Element of p1
[L,e] is set
{L,e} is non empty set
{L} is non empty set
{{L,e},{L}} is non empty set
tcM . [L,e] is set
tcM . (prRl,L1) is set
[prRl,L1] is set
{prRl,L1} is non empty functional set
{prRl} is non empty functional set
{{prRl,L1},{prRl}} is non empty set
tcM . [prRl,L1] is set
(x,n,prRl,L1) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
Le is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
Ledz is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,Le,Ledz) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
prRl is set
L1 is set
L is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
e is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of l . (L,e) is set
[L,e] is set
{L,e} is non empty functional set
{L} is non empty functional set
{{L,e},{L}} is non empty set
the addF of l . [L,e] is set
(x,n,L,e) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
Ledz is left_complementable right_complementable complementable Element of the carrier of x
Le is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the lmult of l . (Ledz,Le) is set
[Ledz,Le] is set
{Ledz,Le} is non empty set
{Ledz} is non empty set
{{Ledz,Le},{Ledz}} is non empty set
the lmult of l . [Ledz,Le] is set
(x,n,Ledz,Le) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
p1 is non empty strict VectSpStr over x
the carrier of p1 is non empty set
the addF of p1 is V16() V19([: the carrier of p1, the carrier of p1:]) V20( the carrier of p1) Function-like V30([: the carrier of p1, the carrier of p1:], the carrier of p1) Element of bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:]
[: the carrier of p1, the carrier of p1:] is set
[:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
0. p1 is zero Element of the carrier of p1
the ZeroF of p1 is Element of the carrier of p1
the lmult of p1 is V16() V19([: the carrier of x, the carrier of p1:]) V20( the carrier of p1) Function-like V30([: the carrier of x, the carrier of p1:], the carrier of p1) Element of bool [:[: the carrier of x, the carrier of p1:], the carrier of p1:]
[: the carrier of x, the carrier of p1:] is set
[:[: the carrier of x, the carrier of p1:], the carrier of p1:] is set
bool [:[: the carrier of x, the carrier of p1:], the carrier of p1:] is set
M is non empty strict VectSpStr over x
the carrier of M is non empty set
the addF of M is V16() V19([: the carrier of M, the carrier of M:]) V20( the carrier of M) Function-like V30([: the carrier of M, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
0. M is zero Element of the carrier of M
the ZeroF of M is Element of the carrier of M
the lmult of M is V16() V19([: the carrier of x, the carrier of M:]) V20( the carrier of M) Function-like V30([: the carrier of x, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of x, the carrier of M:], the carrier of M:]
[: the carrier of x, the carrier of M:] is set
[:[: the carrier of x, the carrier of M:], the carrier of M:] is set
bool [:[: the carrier of x, the carrier of M:], the carrier of M:] is set
RVSM is Element of the carrier of p1
tcM is left_complementable right_complementable complementable Element of the carrier of x
the lmult of p1 . (tcM,RVSM) is Element of the carrier of p1
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the lmult of p1 . [tcM,RVSM] is set
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,tcM,l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the lmult of M . (tcM,RVSM) is set
the lmult of M . [tcM,RVSM] is set
tcM is set
tcM is Element of the carrier of p1
RVSM is Element of the carrier of p1
the addF of p1 . (tcM,RVSM) is Element of the carrier of p1
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of p1 . [tcM,RVSM] is set
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,l,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of M . (tcM,RVSM) is set
the addF of M . [tcM,RVSM] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive doubleLoopStr
n is non empty VectSpStr over x
(x,n) is non empty strict VectSpStr over x
the carrier of (x,n) is non empty set
p1 is Element of the carrier of (x,n)
M is Element of the carrier of (x,n)
p1 + M is Element of the carrier of (x,n)
the addF of (x,n) is V16() V19([: the carrier of (x,n), the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of (x,n), the carrier of (x,n):] is set
[:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
the addF of (x,n) . (p1,M) is Element of the carrier of (x,n)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x,n) . [p1,M] is set
M + p1 is Element of the carrier of (x,n)
the addF of (x,n) . (M,p1) is Element of the carrier of (x,n)
[M,p1] is set
{M,p1} is non empty set
{M} is non empty set
{{M,p1},{M}} is non empty set
the addF of (x,n) . [M,p1] is set
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
tcM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,tcM,RVSM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,RVSM,tcM) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() associative commutative right-distributive doubleLoopStr
n is non empty VectSpStr over x
(x,n) is non empty Abelian strict VectSpStr over x
the carrier of (x,n) is non empty set
p1 is Element of the carrier of (x,n)
M is Element of the carrier of (x,n)
p1 + M is Element of the carrier of (x,n)
the addF of (x,n) is V16() V19([: the carrier of (x,n), the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of (x,n), the carrier of (x,n):] is set
[:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
the addF of (x,n) . (p1,M) is Element of the carrier of (x,n)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x,n) . [p1,M] is set
tcM is Element of the carrier of (x,n)
(p1 + M) + tcM is Element of the carrier of (x,n)
the addF of (x,n) . ((p1 + M),tcM) is Element of the carrier of (x,n)
[(p1 + M),tcM] is set
{(p1 + M),tcM} is non empty set
{(p1 + M)} is non empty set
{{(p1 + M),tcM},{(p1 + M)}} is non empty set
the addF of (x,n) . [(p1 + M),tcM] is set
M + tcM is Element of the carrier of (x,n)
the addF of (x,n) . (M,tcM) is Element of the carrier of (x,n)
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of (x,n) . [M,tcM] is set
p1 + (M + tcM) is Element of the carrier of (x,n)
the addF of (x,n) . (p1,(M + tcM)) is Element of the carrier of (x,n)
[p1,(M + tcM)] is set
{p1,(M + tcM)} is non empty set
{{p1,(M + tcM)},{p1}} is non empty set
the addF of (x,n) . [p1,(M + tcM)] is set
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 + M is Element of the carrier of (x,n)
(p1 + M) + tcM is Element of the carrier of (x,n)
the addF of (x,n) . ((p1 + M),tcM) is Element of the carrier of (x,n)
[(p1 + M),tcM] is set
{(p1 + M),tcM} is non empty set
{(p1 + M)} is non empty set
{{(p1 + M),tcM},{(p1 + M)}} is non empty set
the addF of (x,n) . [(p1 + M),tcM] is set
RVSM is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,RVSM,l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of (x,n) . ((x,n,RVSM,l),tcM) is set
[(x,n,RVSM,l),tcM] is set
{(x,n,RVSM,l),tcM} is non empty set
{(x,n,RVSM,l)} is non empty functional set
{{(x,n,RVSM,l),tcM},{(x,n,RVSM,l)}} is non empty set
the addF of (x,n) . [(x,n,RVSM,l),tcM] is set
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,RVSM,l),prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,l,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,RVSM,(x,n,l,prRl)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of (x,n) . (p1,(x,n,l,prRl)) is set
[p1,(x,n,l,prRl)] is set
{p1,(x,n,l,prRl)} is non empty set
{{p1,(x,n,l,prRl)},{p1}} is non empty set
the addF of (x,n) . [p1,(x,n,l,prRl)] is set
M + tcM is Element of the carrier of (x,n)
p1 + (M + tcM) is Element of the carrier of (x,n)
the addF of (x,n) . (p1,(M + tcM)) is Element of the carrier of (x,n)
[p1,(M + tcM)] is set
{p1,(M + tcM)} is non empty set
{{p1,(M + tcM)},{p1}} is non empty set
the addF of (x,n) . [p1,(M + tcM)] is set
the carrier of (x,n) is non empty set
p1 is Element of the carrier of (x,n)
0. (x,n) is zero Element of the carrier of (x,n)
the ZeroF of (x,n) is Element of the carrier of (x,n)
p1 + (0. (x,n)) is Element of the carrier of (x,n)
the addF of (x,n) is V16() V19([: the carrier of (x,n), the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of (x,n), the carrier of (x,n):] is set
[:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
the addF of (x,n) . (p1,(0. (x,n))) is Element of the carrier of (x,n)
[p1,(0. (x,n))] is set
{p1,(0. (x,n))} is non empty set
{p1} is non empty set
{{p1,(0. (x,n))},{p1}} is non empty set
the addF of (x,n) . [p1,(0. (x,n))] is set
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
p1 + (0. (x,n)) is Element of the carrier of (x,n)
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
the addF of (x,n) . (p1,(x,n)) is set
[p1,(x,n)] is set
{p1,(x,n)} is non empty set
{{p1,(x,n)},{p1}} is non empty set
the addF of (x,n) . [p1,(x,n)] is set
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,(x,n)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the carrier of (x,n) is non empty set
p1 is Element of the carrier of (x,n)
the carrier of n is non empty set
the carrier of x is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
M is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
tcM is Element of the carrier of (x,n)
p1 + tcM is Element of the carrier of (x,n)
the addF of (x,n) is V16() V19([: the carrier of (x,n), the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of (x,n), the carrier of (x,n):] is set
[:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
the addF of (x,n) . (p1,tcM) is Element of the carrier of (x,n)
[p1,tcM] is set
{p1,tcM} is non empty set
{p1} is non empty set
{{p1,tcM},{p1}} is non empty set
the addF of (x,n) . [p1,tcM] is set
0. (x,n) is zero Element of the carrier of (x,n)
the ZeroF of (x,n) is Element of the carrier of (x,n)
p1 + tcM is Element of the carrier of (x,n)
(x,n,M,M) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,(x,n,M)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) (x,n) Element of bool [: the carrier of n, the carrier of x:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
([#] n) --> (0. x) is V16() V19( [#] n) V20( the carrier of x) Function-like V30( [#] n, the carrier of x) Element of bool [:([#] n), the carrier of x:]
[:([#] n), the carrier of x:] is set
bool [:([#] n), the carrier of x:] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() unital associative commutative right-distributive left-distributive right_unital distributive left_unital doubleLoopStr
n is non empty VectSpStr over x
(x,n) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() strict VectSpStr over x
the carrier of x is non empty set
the carrier of (x,n) is non empty set
the carrier of n is non empty set
[: the carrier of n, the carrier of x:] is set
bool [: the carrier of n, the carrier of x:] is set
tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
RVSM is left_complementable right_complementable complementable Element of the carrier of (x,n)
p1 is left_complementable right_complementable complementable Element of the carrier of x
tcM + RVSM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the addF of (x,n) is V16() V19([: the carrier of (x,n), the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of (x,n), the carrier of (x,n):] is set
[:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of (x,n), the carrier of (x,n):], the carrier of (x,n):] is set
the addF of (x,n) . (tcM,RVSM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of (x,n) . [tcM,RVSM] is set
p1 * (tcM + RVSM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) is V16() V19([: the carrier of x, the carrier of (x,n):]) V20( the carrier of (x,n)) Function-like V30([: the carrier of x, the carrier of (x,n):], the carrier of (x,n)) Element of bool [:[: the carrier of x, the carrier of (x,n):], the carrier of (x,n):]
[: the carrier of x, the carrier of (x,n):] is set
[:[: the carrier of x, the carrier of (x,n):], the carrier of (x,n):] is set
bool [:[: the carrier of x, the carrier of (x,n):], the carrier of (x,n):] is set
the lmult of (x,n) . (p1,(tcM + RVSM)) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[p1,(tcM + RVSM)] is set
{p1,(tcM + RVSM)} is non empty set
{p1} is non empty set
{{p1,(tcM + RVSM)},{p1}} is non empty set
the lmult of (x,n) . [p1,(tcM + RVSM)] is set
l is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
prRl is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,l,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the lmult of (x,n) . (p1,(x,n,l,prRl)) is set
[p1,(x,n,l,prRl)] is set
{p1,(x,n,l,prRl)} is non empty set
{{p1,(x,n,l,prRl)},{p1}} is non empty set
the lmult of (x,n) . [p1,(x,n,l,prRl)] is set
(x,n,p1,(x,n,l,prRl)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,prRl) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,p1,l),(x,n,p1,prRl)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of (x,n) . ((x,n,p1,l),(x,n,p1,prRl)) is set
[(x,n,p1,l),(x,n,p1,prRl)] is set
{(x,n,p1,l),(x,n,p1,prRl)} is non empty functional set
{(x,n,p1,l)} is non empty functional set
{{(x,n,p1,l),(x,n,p1,prRl)},{(x,n,p1,l)}} is non empty set
the addF of (x,n) . [(x,n,p1,l),(x,n,p1,prRl)] is set
the lmult of (x,n) . (p1,l) is set
[p1,l] is set
{p1,l} is non empty set
{{p1,l},{p1}} is non empty set
the lmult of (x,n) . [p1,l] is set
the addF of (x,n) . (( the lmult of (x,n) . (p1,l)),(x,n,p1,prRl)) is set
[( the lmult of (x,n) . (p1,l)),(x,n,p1,prRl)] is set
{( the lmult of (x,n) . (p1,l)),(x,n,p1,prRl)} is non empty set
{( the lmult of (x,n) . (p1,l))} is non empty set
{{( the lmult of (x,n) . (p1,l)),(x,n,p1,prRl)},{( the lmult of (x,n) . (p1,l))}} is non empty set
the addF of (x,n) . [( the lmult of (x,n) . (p1,l)),(x,n,p1,prRl)] is set
p1 * tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . (p1,tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[p1,tcM] is set
{p1,tcM} is non empty set
{{p1,tcM},{p1}} is non empty set
the lmult of (x,n) . [p1,tcM] is set
p1 * RVSM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . (p1,RVSM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[p1,RVSM] is set
{p1,RVSM} is non empty set
{{p1,RVSM},{p1}} is non empty set
the lmult of (x,n) . [p1,RVSM] is set
(p1 * tcM) + (p1 * RVSM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
the addF of (x,n) . ((p1 * tcM),(p1 * RVSM)) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[(p1 * tcM),(p1 * RVSM)] is set
{(p1 * tcM),(p1 * RVSM)} is non empty set
{(p1 * tcM)} is non empty set
{{(p1 * tcM),(p1 * RVSM)},{(p1 * tcM)}} is non empty set
the addF of (x,n) . [(p1 * tcM),(p1 * RVSM)] is set
M is left_complementable right_complementable complementable Element of the carrier of x
p1 + M is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (p1,M) is left_complementable right_complementable complementable Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{{p1,M},{p1}} is non empty set
the addF of x . [p1,M] is set
(p1 + M) * tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . ((p1 + M),tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[(p1 + M),tcM] is set
{(p1 + M),tcM} is non empty set
{(p1 + M)} is non empty set
{{(p1 + M),tcM},{(p1 + M)}} is non empty set
the lmult of (x,n) . [(p1 + M),tcM] is set
(x,n,(p1 + M),l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,M,l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,(x,n,p1,l),(x,n,M,l)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the addF of (x,n) . ((x,n,p1,l),(x,n,M,l)) is set
[(x,n,p1,l),(x,n,M,l)] is set
{(x,n,p1,l),(x,n,M,l)} is non empty functional set
{{(x,n,p1,l),(x,n,M,l)},{(x,n,p1,l)}} is non empty set
the addF of (x,n) . [(x,n,p1,l),(x,n,M,l)] is set
the addF of (x,n) . (( the lmult of (x,n) . (p1,l)),(x,n,M,l)) is set
[( the lmult of (x,n) . (p1,l)),(x,n,M,l)] is set
{( the lmult of (x,n) . (p1,l)),(x,n,M,l)} is non empty set
{{( the lmult of (x,n) . (p1,l)),(x,n,M,l)},{( the lmult of (x,n) . (p1,l))}} is non empty set
the addF of (x,n) . [( the lmult of (x,n) . (p1,l)),(x,n,M,l)] is set
M * tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . (M,tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the lmult of (x,n) . [M,tcM] is set
(p1 * tcM) + (M * tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
the addF of (x,n) . ((p1 * tcM),(M * tcM)) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[(p1 * tcM),(M * tcM)] is set
{(p1 * tcM),(M * tcM)} is non empty set
{{(p1 * tcM),(M * tcM)},{(p1 * tcM)}} is non empty set
the addF of (x,n) . [(p1 * tcM),(M * tcM)] is set
p1 * M is left_complementable right_complementable complementable Element of the carrier of x
the multF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 multF of x . (p1,M) is left_complementable right_complementable complementable Element of the carrier of x
the multF of x . [p1,M] is set
(p1 * M) * tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . ((p1 * M),tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[(p1 * M),tcM] is set
{(p1 * M),tcM} is non empty set
{(p1 * M)} is non empty set
{{(p1 * M),tcM},{(p1 * M)}} is non empty set
the lmult of (x,n) . [(p1 * M),tcM] is set
(x,n,(p1 * M),l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
(x,n,p1,(x,n,M,l)) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
the lmult of (x,n) . (p1,(x,n,M,l)) is set
[p1,(x,n,M,l)] is set
{p1,(x,n,M,l)} is non empty set
{{p1,(x,n,M,l)},{p1}} is non empty set
the lmult of (x,n) . [p1,(x,n,M,l)] is set
p1 * (M * tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . (p1,(M * tcM)) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[p1,(M * tcM)] is set
{p1,(M * tcM)} is non empty set
{{p1,(M * tcM)},{p1}} is non empty set
the lmult of (x,n) . [p1,(M * tcM)] is set
1. x is left_complementable right_complementable complementable Element of the carrier of x
the OneF of x is left_complementable right_complementable complementable Element of the carrier of x
(1. x) * tcM is left_complementable right_complementable complementable Element of the carrier of (x,n)
the lmult of (x,n) . ((1. x),tcM) is left_complementable right_complementable complementable Element of the carrier of (x,n)
[(1. x),tcM] is set
{(1. x),tcM} is non empty set
{(1. x)} is non empty set
{{(1. x),tcM},{(1. x)}} is non empty set
the lmult of (x,n) . [(1. x),tcM] is set
(x,n,(1. x),l) is V16() V19( the carrier of n) V20( the carrier of x) Function-like V30( the carrier of n, the carrier of x) additive (x,n) Element of bool [: the carrier of n, the carrier of x:]
x is 1-sorted
x is 1-sorted
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
x is 1-sorted
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
x is non empty vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
p1 is Element of the carrier of x
() * p1 is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((),p1) is Element of the carrier of x
[(),p1] is set
{(),p1} is non empty set
{{(),p1},{()}} is non empty set
the lmult of x . [(),p1] is set
n . (() * p1) is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
(0,M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M * <i> is complex set
0 + (M * <i>) is complex set
(0,M) * p1 is Element of the carrier of x
the lmult of x . ((0,M),p1) is Element of the carrier of x
[(0,M),p1] is set
{(0,M),p1} is non empty set
{(0,M)} is non empty V192() set
{{(0,M),p1},{(0,M)}} is non empty set
the lmult of x . [(0,M),p1] is set
n . ((0,M) * p1) is complex real ext-real Element of REAL
M * (n . (() * p1)) is complex real ext-real Element of REAL
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
(M,0) * () is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((M,0),()) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(M,0),()] is set
{(M,0),()} is non empty V192() set
{(M,0)} is non empty V192() set
{{(M,0),()},{(M,0)}} is non empty set
the multF of F_Complex . [(M,0),()] is set
K147((M,0),()) is complex Element of COMPLEX
((M,0) * ()) * p1 is Element of the carrier of x
the lmult of x . (((M,0) * ()),p1) is Element of the carrier of x
[((M,0) * ()),p1] is set
{((M,0) * ()),p1} is non empty set
{((M,0) * ())} is non empty V192() set
{{((M,0) * ()),p1},{((M,0) * ())}} is non empty set
the lmult of x . [((M,0) * ()),p1] is set
n . (((M,0) * ()) * p1) is complex real ext-real Element of REAL
(M,0) * (() * p1) is Element of the carrier of x
the lmult of x . ((M,0),(() * p1)) is Element of the carrier of x
[(M,0),(() * p1)] is set
{(M,0),(() * p1)} is non empty set
{{(M,0),(() * p1)},{(M,0)}} is non empty set
the lmult of x . [(M,0),(() * p1)] is set
n . ((M,0) * (() * p1)) is complex real ext-real Element of REAL
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
x is 1-sorted
n is VectSpStr over x
the carrier of n is set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
x is 1-sorted
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
p1 is V16() V19( the carrier of n) V20( REAL ) Function-like V30( the carrier of n, REAL ) Element of bool [: the carrier of n,REAL:]
M is Element of the carrier of n
tcM is Element of the carrier of n
M + tcM is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (M,tcM) is Element of the carrier of n
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of n . [M,tcM] is set
p1 . (M + tcM) is complex real ext-real Element of REAL
p1 . M is complex real ext-real Element of REAL
p1 . tcM is complex real ext-real Element of REAL
(p1 . M) + (p1 . tcM) is complex real ext-real Element of REAL
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
(0,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
0 + (0 * <i>) is complex set
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
n . (0. x) is complex real ext-real Element of REAL
(0. F_Complex) * (0. x) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((0. F_Complex),(0. x)) is left_complementable right_complementable complementable Element of the carrier of x
[(0. F_Complex),(0. x)] is set
{(0. F_Complex),(0. x)} is non empty set
{(0. F_Complex)} is non empty V192() set
{{(0. F_Complex),(0. x)},{(0. F_Complex)}} is non empty set
the lmult of x . [(0. F_Complex),(0. x)] is set
n . ((0. F_Complex) * (0. x)) is complex real ext-real Element of REAL
0 * (n . (0. x)) is complex real ext-real Element of REAL
x is 1-sorted
n is VectSpStr over x
[#] n is non proper Element of bool the carrier of n
the carrier of n is set
bool the carrier of n is set
([#] n) --> 0 is V16() V19( [#] n) V20( REAL ) Function-like V30( [#] n, REAL ) Element of bool [:([#] n),REAL:]
[:([#] n),REAL:] is set
bool [:([#] n),REAL:] is set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
x is 1-sorted
n is non empty VectSpStr over x
(x,n) is V16() V19( the carrier of n) V20( REAL ) Function-like V30( the carrier of n, REAL ) Element of bool [: the carrier of n,REAL:]
the carrier of n is non empty set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
([#] n) --> 0 is V16() V19( [#] n) V20( REAL ) Function-like V30( [#] n, REAL ) Element of bool [:([#] n),REAL:]
[:([#] n),REAL:] is set
bool [:([#] n),REAL:] is set
p1 is Element of the carrier of n
M is Element of the carrier of n
p1 + M is Element of the carrier of n
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . (p1,M) is Element of the carrier of n
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of n . [p1,M] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(x,n) . (0. n) is complex real ext-real Element of REAL
x is non empty VectSpStr over F_Complex
(F_Complex,x) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
[#] x is non empty non proper Element of bool the carrier of x
bool the carrier of x is set
([#] x) --> 0 is V16() V19( [#] x) V20( REAL ) Function-like V30( [#] x, REAL ) Element of bool [:([#] x),REAL:]
[:([#] x),REAL:] is set
bool [:([#] x),REAL:] is set
n is Element of the carrier of x
(F_Complex,x) . n is complex real ext-real Element of REAL
p1 is complex real ext-real Element of REAL
(p1,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 + (0 * <i>) is complex set
(p1,0) * n is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((p1,0),n) is Element of the carrier of x
[(p1,0),n] is set
{(p1,0),n} is non empty set
{(p1,0)} is non empty V192() set
{{(p1,0),n},{(p1,0)}} is non empty set
the lmult of x . [(p1,0),n] is set
(F_Complex,x) . ((p1,0) * n) is complex real ext-real Element of REAL
p1 * ((F_Complex,x) . n) is complex real ext-real Element of REAL
n is Element of the carrier of x
(F_Complex,x) . n is complex real ext-real Element of REAL
p1 is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 * n is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . (p1,n) is Element of the carrier of x
[p1,n] is set
{p1,n} is non empty set
{p1} is non empty V192() set
{{p1,n},{p1}} is non empty set
the lmult of x . [p1,n] is set
(F_Complex,x) . (p1 * n) is complex real ext-real Element of REAL
|.p1.| is complex real ext-real Element of REAL
Re p1 is complex real ext-real Element of REAL
(Re p1) ^2 is complex real ext-real Element of REAL
(Re p1) * (Re p1) is complex real ext-real set
Im p1 is complex real ext-real Element of REAL
(Im p1) ^2 is complex real ext-real Element of REAL
(Im p1) * (Im p1) is complex real ext-real set
((Re p1) ^2) + ((Im p1) ^2) is complex real ext-real Element of REAL
sqrt (((Re p1) ^2) + ((Im p1) ^2)) is complex real ext-real Element of REAL
|.p1.| * ((F_Complex,x) . n) is complex real ext-real Element of REAL
x is 1-sorted
n is non empty VectSpStr over x
the carrier of n is non empty set
[: the carrier of n,REAL:] is set
bool [: the carrier of n,REAL:] is set
(x,n) is V16() V19( the carrier of n) V20( REAL ) Function-like V30( the carrier of n, REAL ) (x,n) (x,n) (x,n) Element of bool [: the carrier of n,REAL:]
[#] n is non empty non proper Element of bool the carrier of n
bool the carrier of n is set
([#] n) --> 0 is V16() V19( [#] n) V20( REAL ) Function-like V30( [#] n, REAL ) Element of bool [:([#] n),REAL:]
[:([#] n),REAL:] is set
bool [:([#] n),REAL:] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(F_Complex,x) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) (x) (x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
[#] x is non empty non proper Element of bool the carrier of x
bool the carrier of x is set
([#] x) --> 0 is V16() V19( [#] x) V20( REAL ) Function-like V30( [#] x, REAL ) Element of bool [:([#] x),REAL:]
[:([#] x),REAL:] is set
bool [:([#] x),REAL:] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
[:REAL, the carrier of x:] is set
[:[:REAL, the carrier of x:], the carrier of x:] is set
bool [:[:REAL, the carrier of x:], the carrier of x:] is set
n is V16() V19([:REAL, the carrier of x:]) V20( the carrier of x) Function-like V30([:REAL, the carrier of x:], the carrier of x) Element of bool [:[:REAL, the carrier of x:], the carrier of x:]
0. x is zero Element of the carrier of x
RLSStruct(# the carrier of x,(0. x), the addF of x,n #) is non empty strict RLSStruct
p1 is non empty strict RLSStruct
the carrier of p1 is non empty set
the addF of p1 is V16() V19([: the carrier of p1, the carrier of p1:]) V20( the carrier of p1) Function-like V30([: the carrier of p1, the carrier of p1:], the carrier of p1) Element of bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:]
[: the carrier of p1, the carrier of p1:] is set
[:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
the ZeroF of p1 is Element of the carrier of p1
addLoopStr(# the carrier of p1, the addF of p1, the ZeroF of p1 #) is non empty strict addLoopStr
the Mult of p1 is V16() V19([:REAL, the carrier of p1:]) V20( the carrier of p1) Function-like V30([:REAL, the carrier of p1:], the carrier of p1) Element of bool [:[:REAL, the carrier of p1:], the carrier of p1:]
[:REAL, the carrier of p1:] is set
[:[:REAL, the carrier of p1:], the carrier of p1:] is set
bool [:[:REAL, the carrier of p1:], the carrier of p1:] is set
M is complex real ext-real Element of REAL
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
tcM is Element of the carrier of x
the Mult of p1 . (M,tcM) is set
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty V192() V193() V194() set
{{M,tcM},{M}} is non empty set
the Mult of p1 . [M,tcM] is set
(M,0) * tcM is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),tcM) is Element of the carrier of x
[(M,0),tcM] is set
{(M,0),tcM} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),tcM},{(M,0)}} is non empty set
the lmult of x . [(M,0),tcM] is set
n is strict RLSStruct
the carrier of n is set
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the ZeroF of n is Element of the carrier of n
addLoopStr(# the carrier of n, the addF of n, the ZeroF of n #) is strict addLoopStr
the Mult of n is V16() V19([:REAL, the carrier of n:]) V20( the carrier of n) Function-like V30([:REAL, the carrier of n:], the carrier of n) Element of bool [:[:REAL, the carrier of n:], the carrier of n:]
[:REAL, the carrier of n:] is set
[:[:REAL, the carrier of n:], the carrier of n:] is set
bool [:[:REAL, the carrier of n:], the carrier of n:] is set
p1 is strict RLSStruct
the carrier of p1 is set
the addF of p1 is V16() V19([: the carrier of p1, the carrier of p1:]) V20( the carrier of p1) Function-like V30([: the carrier of p1, the carrier of p1:], the carrier of p1) Element of bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:]
[: the carrier of p1, the carrier of p1:] is set
[:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
bool [:[: the carrier of p1, the carrier of p1:], the carrier of p1:] is set
the ZeroF of p1 is Element of the carrier of p1
addLoopStr(# the carrier of p1, the addF of p1, the ZeroF of p1 #) is strict addLoopStr
the Mult of p1 is V16() V19([:REAL, the carrier of p1:]) V20( the carrier of p1) Function-like V30([:REAL, the carrier of p1:], the carrier of p1) Element of bool [:[:REAL, the carrier of p1:], the carrier of p1:]
[:REAL, the carrier of p1:] is set
[:[:REAL, the carrier of p1:], the carrier of p1:] is set
bool [:[:REAL, the carrier of p1:], the carrier of p1:] is set
M is complex real ext-real Element of REAL
tcM is Element of the carrier of x
the Mult of n . (M,tcM) is set
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty V192() V193() V194() set
{{M,tcM},{M}} is non empty set
the Mult of n . [M,tcM] is set
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
(M,0) * tcM is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),tcM) is Element of the carrier of x
[(M,0),tcM] is set
{(M,0),tcM} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),tcM},{(M,0)}} is non empty set
the lmult of x . [(M,0),tcM] is set
the Mult of p1 . (M,tcM) is set
the Mult of p1 . [M,tcM] is set
x is non empty VectSpStr over F_Complex
(x) is strict RLSStruct
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the carrier of (x) is set
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is strict addLoopStr
x is non empty Abelian VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
n is Element of the carrier of (x)
p1 is Element of the carrier of (x)
n + p1 is Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (n,p1) is Element of the carrier of (x)
[n,p1] is set
{n,p1} is non empty set
{n} is non empty set
{{n,p1},{n}} is non empty set
the addF of (x) . [n,p1] is set
p1 + n is Element of the carrier of (x)
the addF of (x) . (p1,n) is Element of the carrier of (x)
[p1,n] is set
{p1,n} is non empty set
{p1} is non empty set
{{p1,n},{p1}} is non empty set
the addF of (x) . [p1,n] is set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
M is Element of the carrier of x
tcM is Element of the carrier of x
M + tcM is Element of the carrier of x
the addF of x . (M,tcM) is Element of the carrier of x
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of x . [M,tcM] is set
tcM + M is Element of the carrier of x
the addF of x . (tcM,M) is Element of the carrier of x
[tcM,M] is set
{tcM,M} is non empty set
{tcM} is non empty set
{{tcM,M},{tcM}} is non empty set
the addF of x . [tcM,M] is set
x is non empty add-associative VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
n is Element of the carrier of (x)
p1 is Element of the carrier of (x)
n + p1 is Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (n,p1) is Element of the carrier of (x)
[n,p1] is set
{n,p1} is non empty set
{n} is non empty set
{{n,p1},{n}} is non empty set
the addF of (x) . [n,p1] is set
M is Element of the carrier of (x)
(n + p1) + M is Element of the carrier of (x)
the addF of (x) . ((n + p1),M) is Element of the carrier of (x)
[(n + p1),M] is set
{(n + p1),M} is non empty set
{(n + p1)} is non empty set
{{(n + p1),M},{(n + p1)}} is non empty set
the addF of (x) . [(n + p1),M] is set
p1 + M is Element of the carrier of (x)
the addF of (x) . (p1,M) is Element of the carrier of (x)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x) . [p1,M] is set
n + (p1 + M) is Element of the carrier of (x)
the addF of (x) . (n,(p1 + M)) is Element of the carrier of (x)
[n,(p1 + M)] is set
{n,(p1 + M)} is non empty set
{{n,(p1 + M)},{n}} is non empty set
the addF of (x) . [n,(p1 + M)] is set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is Element of the carrier of x
RVSM is Element of the carrier of x
tcM + RVSM is Element of the carrier of x
the addF of x . (tcM,RVSM) is Element of the carrier of x
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of x . [tcM,RVSM] is set
l is Element of the carrier of x
(tcM + RVSM) + l is Element of the carrier of x
the addF of x . ((tcM + RVSM),l) is Element of the carrier of x
[(tcM + RVSM),l] is set
{(tcM + RVSM),l} is non empty set
{(tcM + RVSM)} is non empty set
{{(tcM + RVSM),l},{(tcM + RVSM)}} is non empty set
the addF of x . [(tcM + RVSM),l] is set
RVSM + l is Element of the carrier of x
the addF of x . (RVSM,l) is Element of the carrier of x
[RVSM,l] is set
{RVSM,l} is non empty set
{RVSM} is non empty set
{{RVSM,l},{RVSM}} is non empty set
the addF of x . [RVSM,l] is set
tcM + (RVSM + l) is Element of the carrier of x
the addF of x . (tcM,(RVSM + l)) is Element of the carrier of x
[tcM,(RVSM + l)] is set
{tcM,(RVSM + l)} is non empty set
{{tcM,(RVSM + l)},{tcM}} is non empty set
the addF of x . [tcM,(RVSM + l)] is set
x is non empty right_zeroed VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
n is Element of the carrier of (x)
0. (x) is zero Element of the carrier of (x)
the ZeroF of (x) is Element of the carrier of (x)
n + (0. (x)) is Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (n,(0. (x))) is Element of the carrier of (x)
[n,(0. (x))] is set
{n,(0. (x))} is non empty set
{n} is non empty set
{{n,(0. (x))},{n}} is non empty set
the addF of (x) . [n,(0. (x))] is set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
p1 is Element of the carrier of x
0. x is zero Element of the carrier of x
p1 + (0. x) is Element of the carrier of x
the addF of x . (p1,(0. x)) is Element of the carrier of x
[p1,(0. x)] is set
{p1,(0. x)} is non empty set
{p1} is non empty set
{{p1,(0. x)},{p1}} is non empty set
the addF of x . [p1,(0. x)] is set
x is non empty right_complementable VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
n is Element of the carrier of (x)
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is right_complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
p1 is right_complementable Element of the carrier of x
0. x is zero right_complementable Element of the carrier of x
M is right_complementable Element of the carrier of x
p1 + M is right_complementable Element of the carrier of x
the addF of x . (p1,M) is right_complementable Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of x . [p1,M] is set
tcM is Element of the carrier of (x)
n + tcM is Element of the carrier of (x)
the addF of (x) . (n,tcM) is Element of the carrier of (x)
[n,tcM] is set
{n,tcM} is non empty set
{n} is non empty set
{{n,tcM},{n}} is non empty set
the addF of (x) . [n,tcM] is set
0. (x) is zero Element of the carrier of (x)
x is non empty vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
n is complex real ext-real set
M is Element of the carrier of (x)
tcM is Element of the carrier of (x)
M + tcM is Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (M,tcM) is Element of the carrier of (x)
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of (x) . [M,tcM] is set
n * (M + tcM) is Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (n,(M + tcM)) is set
[n,(M + tcM)] is set
{n,(M + tcM)} is non empty set
{n} is non empty V192() V193() V194() set
{{n,(M + tcM)},{n}} is non empty set
the Mult of (x) . [n,(M + tcM)] is set
n * M is Element of the carrier of (x)
the Mult of (x) . (n,M) is set
[n,M] is set
{n,M} is non empty set
{{n,M},{n}} is non empty set
the Mult of (x) . [n,M] is set
n * tcM is Element of the carrier of (x)
the Mult of (x) . (n,tcM) is set
[n,tcM] is set
{n,tcM} is non empty set
{{n,tcM},{n}} is non empty set
the Mult of (x) . [n,tcM] is set
(n * M) + (n * tcM) is Element of the carrier of (x)
the addF of (x) . ((n * M),(n * tcM)) is Element of the carrier of (x)
[(n * M),(n * tcM)] is set
{(n * M),(n * tcM)} is non empty set
{(n * M)} is non empty set
{{(n * M),(n * tcM)},{(n * M)}} is non empty set
the addF of (x) . [(n * M),(n * tcM)] is set
p1 is complex real ext-real Element of REAL
(p1,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 + (0 * <i>) is complex set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
p1 * (M + tcM) is Element of the carrier of (x)
the Mult of (x) . (p1,(M + tcM)) is set
[p1,(M + tcM)] is set
{p1,(M + tcM)} is non empty set
{p1} is non empty V192() V193() V194() set
{{p1,(M + tcM)},{p1}} is non empty set
the Mult of (x) . [p1,(M + tcM)] is set
l is Element of the carrier of x
prRl is Element of the carrier of x
l + prRl is Element of the carrier of x
the addF of x . (l,prRl) is Element of the carrier of x
[l,prRl] is set
{l,prRl} is non empty set
{l} is non empty set
{{l,prRl},{l}} is non empty set
the addF of x . [l,prRl] is set
(p1,0) * (l + prRl) is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((p1,0),(l + prRl)) is Element of the carrier of x
[(p1,0),(l + prRl)] is set
{(p1,0),(l + prRl)} is non empty set
{(p1,0)} is non empty V192() set
{{(p1,0),(l + prRl)},{(p1,0)}} is non empty set
the lmult of x . [(p1,0),(l + prRl)] is set
(p1,0) * l is Element of the carrier of x
the lmult of x . ((p1,0),l) is Element of the carrier of x
[(p1,0),l] is set
{(p1,0),l} is non empty set
{{(p1,0),l},{(p1,0)}} is non empty set
the lmult of x . [(p1,0),l] is set
(p1,0) * prRl is Element of the carrier of x
the lmult of x . ((p1,0),prRl) is Element of the carrier of x
[(p1,0),prRl] is set
{(p1,0),prRl} is non empty set
{{(p1,0),prRl},{(p1,0)}} is non empty set
the lmult of x . [(p1,0),prRl] is set
((p1,0) * l) + ((p1,0) * prRl) is Element of the carrier of x
the addF of x . (((p1,0) * l),((p1,0) * prRl)) is Element of the carrier of x
[((p1,0) * l),((p1,0) * prRl)] is set
{((p1,0) * l),((p1,0) * prRl)} is non empty set
{((p1,0) * l)} is non empty set
{{((p1,0) * l),((p1,0) * prRl)},{((p1,0) * l)}} is non empty set
the addF of x . [((p1,0) * l),((p1,0) * prRl)] is set
the Mult of (x) . (p1,l) is set
[p1,l] is set
{p1,l} is non empty set
{{p1,l},{p1}} is non empty set
the Mult of (x) . [p1,l] is set
[( the Mult of (x) . (p1,l)),((p1,0) * prRl)] is set
{( the Mult of (x) . (p1,l)),((p1,0) * prRl)} is non empty set
{( the Mult of (x) . (p1,l))} is non empty set
{{( the Mult of (x) . (p1,l)),((p1,0) * prRl)},{( the Mult of (x) . (p1,l))}} is non empty set
the addF of x . [( the Mult of (x) . (p1,l)),((p1,0) * prRl)] is set
p1 * M is Element of the carrier of (x)
the Mult of (x) . (p1,M) is set
[p1,M] is set
{p1,M} is non empty set
{{p1,M},{p1}} is non empty set
the Mult of (x) . [p1,M] is set
p1 * tcM is Element of the carrier of (x)
the Mult of (x) . (p1,tcM) is set
[p1,tcM] is set
{p1,tcM} is non empty set
{{p1,tcM},{p1}} is non empty set
the Mult of (x) . [p1,tcM] is set
(p1 * M) + (p1 * tcM) is Element of the carrier of (x)
the addF of (x) . ((p1 * M),(p1 * tcM)) is Element of the carrier of (x)
[(p1 * M),(p1 * tcM)] is set
{(p1 * M),(p1 * tcM)} is non empty set
{(p1 * M)} is non empty set
{{(p1 * M),(p1 * tcM)},{(p1 * M)}} is non empty set
the addF of (x) . [(p1 * M),(p1 * tcM)] is set
n is complex real ext-real set
p1 is complex real ext-real set
n + p1 is complex real ext-real set
RVSM is Element of the carrier of (x)
(n + p1) * RVSM is Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . ((n + p1),RVSM) is set
[(n + p1),RVSM] is set
{(n + p1),RVSM} is non empty set
{(n + p1)} is non empty V192() V193() V194() set
{{(n + p1),RVSM},{(n + p1)}} is non empty set
the Mult of (x) . [(n + p1),RVSM] is set
n * RVSM is Element of the carrier of (x)
the Mult of (x) . (n,RVSM) is set
[n,RVSM] is set
{n,RVSM} is non empty set
{n} is non empty V192() V193() V194() set
{{n,RVSM},{n}} is non empty set
the Mult of (x) . [n,RVSM] is set
p1 * RVSM is Element of the carrier of (x)
the Mult of (x) . (p1,RVSM) is set
[p1,RVSM] is set
{p1,RVSM} is non empty set
{p1} is non empty V192() V193() V194() set
{{p1,RVSM},{p1}} is non empty set
the Mult of (x) . [p1,RVSM] is set
(n * RVSM) + (p1 * RVSM) is Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . ((n * RVSM),(p1 * RVSM)) is Element of the carrier of (x)
[(n * RVSM),(p1 * RVSM)] is set
{(n * RVSM),(p1 * RVSM)} is non empty set
{(n * RVSM)} is non empty set
{{(n * RVSM),(p1 * RVSM)},{(n * RVSM)}} is non empty set
the addF of (x) . [(n * RVSM),(p1 * RVSM)] is set
M is complex real ext-real Element of REAL
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
tcM is complex real ext-real Element of REAL
(tcM,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
tcM + (0 * <i>) is complex set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(M,0) + (tcM,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((M,0),(tcM,0)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(M,0),(tcM,0)] is set
{(M,0),(tcM,0)} is non empty V192() set
{(M,0)} is non empty V192() set
{{(M,0),(tcM,0)},{(M,0)}} is non empty set
the addF of F_Complex . [(M,0),(tcM,0)] is set
K145((M,0),(tcM,0)) is complex Element of COMPLEX
M + tcM is complex real ext-real Element of REAL
((M + tcM),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(M + tcM) + (0 * <i>) is complex set
(M + tcM) * RVSM is Element of the carrier of (x)
the Mult of (x) . ((M + tcM),RVSM) is set
[(M + tcM),RVSM] is set
{(M + tcM),RVSM} is non empty set
{(M + tcM)} is non empty V192() V193() V194() set
{{(M + tcM),RVSM},{(M + tcM)}} is non empty set
the Mult of (x) . [(M + tcM),RVSM] is set
L1 is Element of the carrier of x
((M,0) + (tcM,0)) * L1 is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . (((M,0) + (tcM,0)),L1) is Element of the carrier of x
[((M,0) + (tcM,0)),L1] is set
{((M,0) + (tcM,0)),L1} is non empty set
{((M,0) + (tcM,0))} is non empty V192() set
{{((M,0) + (tcM,0)),L1},{((M,0) + (tcM,0))}} is non empty set
the lmult of x . [((M,0) + (tcM,0)),L1] is set
(M,0) * L1 is Element of the carrier of x
the lmult of x . ((M,0),L1) is Element of the carrier of x
[(M,0),L1] is set
{(M,0),L1} is non empty set
{{(M,0),L1},{(M,0)}} is non empty set
the lmult of x . [(M,0),L1] is set
(tcM,0) * L1 is Element of the carrier of x
the lmult of x . ((tcM,0),L1) is Element of the carrier of x
[(tcM,0),L1] is set
{(tcM,0),L1} is non empty set
{(tcM,0)} is non empty V192() set
{{(tcM,0),L1},{(tcM,0)}} is non empty set
the lmult of x . [(tcM,0),L1] is set
((M,0) * L1) + ((tcM,0) * L1) is Element of the carrier of x
the addF of x . (((M,0) * L1),((tcM,0) * L1)) is Element of the carrier of x
[((M,0) * L1),((tcM,0) * L1)] is set
{((M,0) * L1),((tcM,0) * L1)} is non empty set
{((M,0) * L1)} is non empty set
{{((M,0) * L1),((tcM,0) * L1)},{((M,0) * L1)}} is non empty set
the addF of x . [((M,0) * L1),((tcM,0) * L1)] is set
the Mult of (x) . (M,RVSM) is Element of the carrier of (x)
[M,RVSM] is set
{M,RVSM} is non empty set
{M} is non empty V192() V193() V194() set
{{M,RVSM},{M}} is non empty set
the Mult of (x) . [M,RVSM] is set
[( the Mult of (x) . (M,RVSM)),((tcM,0) * L1)] is set
{( the Mult of (x) . (M,RVSM)),((tcM,0) * L1)} is non empty set
{( the Mult of (x) . (M,RVSM))} is non empty set
{{( the Mult of (x) . (M,RVSM)),((tcM,0) * L1)},{( the Mult of (x) . (M,RVSM))}} is non empty set
the addF of (x) . [( the Mult of (x) . (M,RVSM)),((tcM,0) * L1)] is set
M * RVSM is Element of the carrier of (x)
the Mult of (x) . (M,RVSM) is set
tcM * RVSM is Element of the carrier of (x)
the Mult of (x) . (tcM,RVSM) is set
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty V192() V193() V194() set
{{tcM,RVSM},{tcM}} is non empty set
the Mult of (x) . [tcM,RVSM] is set
(M * RVSM) + (tcM * RVSM) is Element of the carrier of (x)
the addF of (x) . ((M * RVSM),(tcM * RVSM)) is Element of the carrier of (x)
[(M * RVSM),(tcM * RVSM)] is set
{(M * RVSM),(tcM * RVSM)} is non empty set
{(M * RVSM)} is non empty set
{{(M * RVSM),(tcM * RVSM)},{(M * RVSM)}} is non empty set
the addF of (x) . [(M * RVSM),(tcM * RVSM)] is set
n is complex real ext-real set
p1 is complex real ext-real set
n * p1 is complex real ext-real set
RVSM is Element of the carrier of (x)
(n * p1) * RVSM is Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . ((n * p1),RVSM) is set
[(n * p1),RVSM] is set
{(n * p1),RVSM} is non empty set
{(n * p1)} is non empty V192() V193() V194() set
{{(n * p1),RVSM},{(n * p1)}} is non empty set
the Mult of (x) . [(n * p1),RVSM] is set
p1 * RVSM is Element of the carrier of (x)
the Mult of (x) . (p1,RVSM) is set
[p1,RVSM] is set
{p1,RVSM} is non empty set
{p1} is non empty V192() V193() V194() set
{{p1,RVSM},{p1}} is non empty set
the Mult of (x) . [p1,RVSM] is set
n * (p1 * RVSM) is Element of the carrier of (x)
the Mult of (x) . (n,(p1 * RVSM)) is set
[n,(p1 * RVSM)] is set
{n,(p1 * RVSM)} is non empty set
{n} is non empty V192() V193() V194() set
{{n,(p1 * RVSM)},{n}} is non empty set
the Mult of (x) . [n,(p1 * RVSM)] is set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
M is complex real ext-real Element of REAL
tcM is complex real ext-real Element of REAL
M * tcM is complex real ext-real Element of REAL
((M * tcM),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(M * tcM) + (0 * <i>) is complex set
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
(tcM,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
tcM + (0 * <i>) is complex set
(M,0) * (tcM,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((M,0),(tcM,0)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(M,0),(tcM,0)] is set
{(M,0),(tcM,0)} is non empty V192() set
{(M,0)} is non empty V192() set
{{(M,0),(tcM,0)},{(M,0)}} is non empty set
the multF of F_Complex . [(M,0),(tcM,0)] is set
K147((M,0),(tcM,0)) is complex Element of COMPLEX
(M * tcM) * RVSM is Element of the carrier of (x)
the Mult of (x) . ((M * tcM),RVSM) is set
[(M * tcM),RVSM] is set
{(M * tcM),RVSM} is non empty set
{(M * tcM)} is non empty V192() V193() V194() set
{{(M * tcM),RVSM},{(M * tcM)}} is non empty set
the Mult of (x) . [(M * tcM),RVSM] is set
l is Element of the carrier of x
((M,0) * (tcM,0)) * l is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . (((M,0) * (tcM,0)),l) is Element of the carrier of x
[((M,0) * (tcM,0)),l] is set
{((M,0) * (tcM,0)),l} is non empty set
{((M,0) * (tcM,0))} is non empty V192() set
{{((M,0) * (tcM,0)),l},{((M,0) * (tcM,0))}} is non empty set
the lmult of x . [((M,0) * (tcM,0)),l] is set
(tcM,0) * l is Element of the carrier of x
the lmult of x . ((tcM,0),l) is Element of the carrier of x
[(tcM,0),l] is set
{(tcM,0),l} is non empty set
{(tcM,0)} is non empty V192() set
{{(tcM,0),l},{(tcM,0)}} is non empty set
the lmult of x . [(tcM,0),l] is set
(M,0) * ((tcM,0) * l) is Element of the carrier of x
the lmult of x . ((M,0),((tcM,0) * l)) is Element of the carrier of x
[(M,0),((tcM,0) * l)] is set
{(M,0),((tcM,0) * l)} is non empty set
{{(M,0),((tcM,0) * l)},{(M,0)}} is non empty set
the lmult of x . [(M,0),((tcM,0) * l)] is set
the Mult of (x) . (M,((tcM,0) * l)) is set
[M,((tcM,0) * l)] is set
{M,((tcM,0) * l)} is non empty set
{M} is non empty V192() V193() V194() set
{{M,((tcM,0) * l)},{M}} is non empty set
the Mult of (x) . [M,((tcM,0) * l)] is set
tcM * RVSM is Element of the carrier of (x)
the Mult of (x) . (tcM,RVSM) is set
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty V192() V193() V194() set
{{tcM,RVSM},{tcM}} is non empty set
the Mult of (x) . [tcM,RVSM] is set
M * (tcM * RVSM) is Element of the carrier of (x)
the Mult of (x) . (M,(tcM * RVSM)) is set
[M,(tcM * RVSM)] is set
{M,(tcM * RVSM)} is non empty set
{{M,(tcM * RVSM)},{M}} is non empty set
the Mult of (x) . [M,(tcM * RVSM)] is set
n is Element of the carrier of (x)
1 * n is Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (1,n) is set
[1,n] is set
{1,n} is non empty set
{1} is non empty V192() V193() V194() V195() V196() V197() set
{{1,n},{1}} is non empty set
the Mult of (x) . [1,n] is set
the carrier of x is non empty set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(1,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
1 + (0 * <i>) is complex set
p1 is Element of the carrier of x
(1,0) * p1 is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((1,0),p1) is Element of the carrier of x
[(1,0),p1] is set
{(1,0),p1} is non empty set
{(1,0)} is non empty V192() set
{{(1,0),p1},{(1,0)}} is non empty set
the lmult of x . [(1,0),p1] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
n is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of x
(n) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of n is non empty set
the carrier of x is non empty set
the lmult of n is V16() V19([: the carrier of F_Complex, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of F_Complex, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of F_Complex, the carrier of n:], the carrier of n:]
[: the carrier of F_Complex, the carrier of n:] is set
[:[: the carrier of F_Complex, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of F_Complex, the carrier of n:], the carrier of n:] is set
[: the carrier of F_Complex, the carrier of x:] is set
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x | [: the carrier of F_Complex, the carrier of n:] is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
the addF of n is V16() V19([: the carrier of n, the carrier of n:]) V20( the carrier of n) Function-like V30([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the ZeroF of n is left_complementable right_complementable complementable Element of the carrier of n
addLoopStr(# the carrier of n, the addF of n, the ZeroF of n #) is non empty strict addLoopStr
the carrier of (n) is non empty set
the addF of (n) is V16() V19([: the carrier of (n), the carrier of (n):]) V20( the carrier of (n)) Function-like V30([: the carrier of (n), the carrier of (n):], the carrier of (n)) Element of bool [:[: the carrier of (n), the carrier of (n):], the carrier of (n):]
[: the carrier of (n), the carrier of (n):] is set
[:[: the carrier of (n), the carrier of (n):], the carrier of (n):] is set
bool [:[: the carrier of (n), the carrier of (n):], the carrier of (n):] is set
the ZeroF of (n) is left_complementable right_complementable complementable Element of the carrier of (n)
addLoopStr(# the carrier of (n), the addF of (n), the ZeroF of (n) #) is non empty strict addLoopStr
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the carrier of (x) is non empty set
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
0. (n) is zero left_complementable right_complementable complementable Element of the carrier of (n)
0. (x) is zero left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) || the carrier of (n) is set
the Mult of (n) is V16() V19([:REAL, the carrier of (n):]) V20( the carrier of (n)) Function-like V30([:REAL, the carrier of (n):], the carrier of (n)) Element of bool [:[:REAL, the carrier of (n):], the carrier of (n):]
[:REAL, the carrier of (n):] is set
[:[:REAL, the carrier of (n):], the carrier of (n):] is set
bool [:[:REAL, the carrier of (n):], the carrier of (n):] is set
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
K69( the Mult of (x),[:REAL, the carrier of (n):]) is V16() Function-like set
dom the Mult of (x) is V16() V19( REAL ) V20( the carrier of (x)) Element of bool [:REAL, the carrier of (x):]
bool [:REAL, the carrier of (x):] is set
the Mult of (x) | [:REAL, the carrier of (n):] is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like Element of bool [:[:REAL, the carrier of (x):], the carrier of (x):]
dom ( the Mult of (x) | [:REAL, the carrier of (n):]) is V16() V19( REAL ) V20( the carrier of (x)) Element of bool [:REAL, the carrier of (x):]
rng ( the Mult of (x) | [:REAL, the carrier of (n):]) is Element of bool the carrier of (x)
bool the carrier of (x) is set
p1 is set
M is set
( the Mult of (x) | [:REAL, the carrier of (n):]) . M is set
tcM is set
RVSM is set
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
prRl is left_complementable right_complementable complementable Element of the carrier of (n)
L1 is left_complementable right_complementable complementable Element of the carrier of n
l is complex real ext-real Element of REAL
(l,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
l + (0 * <i>) is complex set
L is left_complementable right_complementable complementable Element of the carrier of x
[(l,0),L] is set
{(l,0),L} is non empty set
{(l,0)} is non empty V192() set
{{(l,0),L},{(l,0)}} is non empty set
[(l,0),L1] is set
{(l,0),L1} is non empty set
{{(l,0),L1},{(l,0)}} is non empty set
dom the lmult of x is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
bool [: the carrier of F_Complex, the carrier of x:] is set
(dom the lmult of x) /\ [: the carrier of F_Complex, the carrier of n:] is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
dom ( the lmult of x | [: the carrier of F_Complex, the carrier of n:]) is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
the Mult of (x) . (l,prRl) is set
[l,prRl] is set
{l,prRl} is non empty set
{l} is non empty V192() V193() V194() set
{{l,prRl},{l}} is non empty set
the Mult of (x) . [l,prRl] is set
(l,0) * L is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((l,0),L) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . [(l,0),L] is set
(l,0) * L1 is left_complementable right_complementable complementable Element of the carrier of n
the lmult of n . ((l,0),L1) is left_complementable right_complementable complementable Element of the carrier of n
the lmult of n . [(l,0),L1] is set
the Mult of (n) . (l,prRl) is left_complementable right_complementable complementable Element of the carrier of (n)
the Mult of (n) . [l,prRl] is set
0. n is zero left_complementable right_complementable complementable Element of the carrier of n
0. x is zero left_complementable right_complementable complementable Element of the carrier of x
tcM is left_complementable right_complementable complementable Element of the carrier of (n)
RVSM is left_complementable right_complementable complementable Element of the carrier of n
M is complex real ext-real Element of REAL
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
l is left_complementable right_complementable complementable Element of the carrier of x
[(M,0),l] is set
{(M,0),l} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),l},{(M,0)}} is non empty set
[(M,0),RVSM] is set
{(M,0),RVSM} is non empty set
{{(M,0),RVSM},{(M,0)}} is non empty set
dom the lmult of x is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
bool [: the carrier of F_Complex, the carrier of x:] is set
(dom the lmult of x) /\ [: the carrier of F_Complex, the carrier of n:] is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
dom ( the lmult of x | [: the carrier of F_Complex, the carrier of n:]) is V16() V19( the carrier of F_Complex) V20( the carrier of x) Element of bool [: the carrier of F_Complex, the carrier of x:]
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty V192() V193() V194() set
{{M,tcM},{M}} is non empty set
(dom the Mult of (x)) /\ [:REAL, the carrier of (n):] is V16() V19( REAL ) V20( the carrier of (x)) Element of bool [:REAL, the carrier of (x):]
p1 is V16() V19([:REAL, the carrier of (n):]) V20( the carrier of (n)) Function-like V30([:REAL, the carrier of (n):], the carrier of (n)) Element of bool [:[:REAL, the carrier of (n):], the carrier of (n):]
dom p1 is V16() V19( REAL ) V20( the carrier of (n)) Element of bool [:REAL, the carrier of (n):]
bool [:REAL, the carrier of (n):] is set
the Mult of (n) . (M,tcM) is left_complementable right_complementable complementable Element of the carrier of (n)
the Mult of (n) . [M,tcM] is set
(M,0) * RVSM is left_complementable right_complementable complementable Element of the carrier of n
the lmult of n . ((M,0),RVSM) is left_complementable right_complementable complementable Element of the carrier of n
the lmult of n . [(M,0),RVSM] is set
(M,0) * l is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((M,0),l) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . [(M,0),l] is set
the Mult of (x) . (M,tcM) is set
the Mult of (x) . [M,tcM] is set
p1 . (M,tcM) is left_complementable right_complementable complementable Element of the carrier of (n)
p1 . [M,tcM] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) (x) Element of bool [: the carrier of x,REAL:]
p1 is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
M is left_complementable right_complementable complementable Element of the carrier of (x)
p1 . M is complex real ext-real Element of REAL
tcM is complex real ext-real Element of REAL
tcM * M is left_complementable right_complementable complementable Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (tcM,M) is set
[tcM,M] is set
{tcM,M} is non empty set
{tcM} is non empty V192() V193() V194() set
{{tcM,M},{tcM}} is non empty set
the Mult of (x) . [tcM,M] is set
p1 . (tcM * M) is complex real ext-real Element of REAL
tcM * (p1 . M) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(tcM,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
tcM + (0 * <i>) is complex set
RVSM is left_complementable right_complementable complementable Element of the carrier of x
(tcM,0) * RVSM is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((tcM,0),RVSM) is left_complementable right_complementable complementable Element of the carrier of x
[(tcM,0),RVSM] is set
{(tcM,0),RVSM} is non empty set
{(tcM,0)} is non empty V192() set
{{(tcM,0),RVSM},{(tcM,0)}} is non empty set
the lmult of x . [(tcM,0),RVSM] is set
abs tcM is complex real ext-real Element of REAL
Re tcM is complex real ext-real Element of REAL
(Re tcM) ^2 is complex real ext-real Element of REAL
(Re tcM) * (Re tcM) is complex real ext-real set
Im tcM is complex real ext-real Element of REAL
(Im tcM) ^2 is complex real ext-real Element of REAL
(Im tcM) * (Im tcM) is complex real ext-real set
((Re tcM) ^2) + ((Im tcM) ^2) is complex real ext-real Element of REAL
sqrt (((Re tcM) ^2) + ((Im tcM) ^2)) is complex real ext-real Element of REAL
(abs tcM) * (p1 . M) is complex real ext-real Element of REAL
M is left_complementable right_complementable complementable Element of the carrier of (x)
tcM is left_complementable right_complementable complementable Element of the carrier of (x)
M + tcM is left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (M,tcM) is left_complementable right_complementable complementable Element of the carrier of (x)
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty set
{{M,tcM},{M}} is non empty set
the addF of (x) . [M,tcM] is set
p1 . (M + tcM) is complex real ext-real Element of REAL
p1 . M is complex real ext-real Element of REAL
p1 . tcM is complex real ext-real Element of REAL
(p1 . M) + (p1 . tcM) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
RVSM is left_complementable right_complementable complementable Element of the carrier of x
l is left_complementable right_complementable complementable Element of the carrier of x
RVSM + l is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (RVSM,l) is left_complementable right_complementable complementable Element of the carrier of x
[RVSM,l] is set
{RVSM,l} is non empty set
{RVSM} is non empty set
{{RVSM,l},{RVSM}} is non empty set
the addF of x . [RVSM,l] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
p1 is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
M is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
tcM is Element of the carrier of x
M . tcM is set
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (n . tcM) is complex real ext-real Element of REAL
p1 is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
M is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is Element of the carrier of (x)
p1 . tcM is complex real ext-real Element of REAL
RVSM is Element of the carrier of x
n . RVSM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (n . RVSM) is complex real ext-real Element of REAL
M . tcM is complex real ext-real Element of REAL
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
p1 is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
M is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
tcM is Element of the carrier of x
M . tcM is set
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Im (n . tcM) is complex real ext-real Element of REAL
p1 is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
M is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is Element of the carrier of (x)
p1 . tcM is complex real ext-real Element of REAL
RVSM is Element of the carrier of x
n . RVSM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Im (n . RVSM) is complex real ext-real Element of REAL
M . tcM is complex real ext-real Element of REAL
x is non empty VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) subadditive additive Element of bool [: the carrier of (x),REAL:]
(x,n) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
p1 is left_complementable right_complementable complementable Element of the carrier of x
M is left_complementable right_complementable complementable Element of the carrier of x
p1 + M is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (p1,M) is left_complementable right_complementable complementable Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of x . [p1,M] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
p1 + M is left_complementable right_complementable complementable Element of the carrier of x
tcM is left_complementable right_complementable complementable Element of the carrier of (x)
RVSM is left_complementable right_complementable complementable Element of the carrier of (x)
tcM + RVSM is left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) . (tcM,RVSM) is left_complementable right_complementable complementable Element of the carrier of (x)
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of (x) . [tcM,RVSM] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
(x,n) is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
M is left_complementable right_complementable complementable Element of the carrier of (x)
p1 + M is left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (p1,M) is left_complementable right_complementable complementable Element of the carrier of (x)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x) . [p1,M] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is left_complementable right_complementable complementable Element of the carrier of x
RVSM is left_complementable right_complementable complementable Element of the carrier of x
tcM + RVSM is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (tcM,RVSM) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of x . [tcM,RVSM] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) homogeneous positively_homogeneous Element of bool [: the carrier of (x),REAL:]
(x,n) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
p1 is left_complementable right_complementable complementable Element of the carrier of x
(x,n) . p1 is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
(M,0) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(M,0),p1] is set
{(M,0),p1} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),p1},{(M,0)}} is non empty set
the lmult of x . [(M,0),p1] is set
(x,n) . ((M,0) * p1) is complex real ext-real Element of REAL
M * ((x,n) . p1) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is left_complementable right_complementable complementable Element of the carrier of (x)
M * tcM is left_complementable right_complementable complementable Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (M,tcM) is set
[M,tcM] is set
{M,tcM} is non empty set
{M} is non empty V192() V193() V194() set
{{M,tcM},{M}} is non empty set
the Mult of (x) . [M,tcM] is set
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) (x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
(x,n) is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
(x,n) . p1 is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
M * p1 is left_complementable right_complementable complementable Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (M,p1) is set
[M,p1] is set
{M,p1} is non empty set
{M} is non empty V192() V193() V194() set
{{M,p1},{M}} is non empty set
the Mult of (x) . [M,p1] is set
(x,n) . (M * p1) is complex real ext-real Element of REAL
M * ((x,n) . p1) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
tcM is left_complementable right_complementable complementable Element of the carrier of x
(M,0) * tcM is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),tcM) is left_complementable right_complementable complementable Element of the carrier of x
[(M,0),tcM] is set
{(M,0),tcM} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),tcM},{(M,0)}} is non empty set
the lmult of x . [(M,0),tcM] is set
x is non empty VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
p1 is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
M is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
tcM is Element of the carrier of x
M . tcM is complex real ext-real Element of REAL
() * tcM is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((),tcM) is Element of the carrier of x
[(),tcM] is set
{(),tcM} is non empty set
{{(),tcM},{()}} is non empty set
the lmult of x . [(),tcM] is set
n . (() * tcM) is complex real ext-real Element of REAL
p1 is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
M is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
tcM is Element of the carrier of x
p1 . tcM is complex real ext-real Element of REAL
() * tcM is Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((),tcM) is Element of the carrier of x
[(),tcM] is set
{(),tcM} is non empty set
{{(),tcM},{()}} is non empty set
the lmult of x . [(),tcM] is set
n . (() * tcM) is complex real ext-real Element of REAL
M . tcM is complex real ext-real Element of REAL
x is non empty VectSpStr over F_Complex
(x) is non empty strict RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
n is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
(x,n) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x,(x,n)) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
p1 is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
M is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
tcM is Element of the carrier of x
M . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,n) . tcM is complex real ext-real Element of REAL
(x,(x,n)) . tcM is complex real ext-real Element of REAL
- ((x,(x,n)) . tcM) is complex real ext-real Element of REAL
(((x,n) . tcM),(- ((x,(x,n)) . tcM))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . tcM)) * <i> is complex set
((x,n) . tcM) + ((- ((x,(x,n)) . tcM)) * <i>) is complex set
p1 is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
M is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
tcM is Element of the carrier of x
p1 . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,n) . tcM is complex real ext-real Element of REAL
(x,(x,n)) . tcM is complex real ext-real Element of REAL
- ((x,(x,n)) . tcM) is complex real ext-real Element of REAL
(((x,n) . tcM),(- ((x,(x,n)) . tcM))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . tcM)) * <i> is complex set
((x,n) . tcM) + ((- ((x,(x,n)) . tcM)) * <i>) is complex set
M . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) additive ( F_Complex ,x) ( F_Complex ,x) Element of bool [: the carrier of x, the carrier of F_Complex:]
(x,n) is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
(x,n) . p1 is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
M * p1 is left_complementable right_complementable complementable Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (M,p1) is set
[M,p1] is set
{M,p1} is non empty set
{M} is non empty V192() V193() V194() set
{{M,p1},{M}} is non empty set
the Mult of (x) . [M,p1] is set
(x,n) . (M * p1) is complex real ext-real Element of REAL
M * ((x,n) . p1) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
tcM is left_complementable right_complementable complementable Element of the carrier of x
(M,0) * tcM is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),tcM) is left_complementable right_complementable complementable Element of the carrier of x
[(M,0),tcM] is set
{(M,0),tcM} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),tcM},{(M,0)}} is non empty set
the lmult of x . [(M,0),tcM] is set
n . ((M,0) * tcM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (n . ((M,0) * tcM)) is complex real ext-real Element of REAL
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(M,0) * (n . tcM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((M,0),(n . tcM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(M,0),(n . tcM)] is set
{(M,0),(n . tcM)} is non empty V192() set
{{(M,0),(n . tcM)},{(M,0)}} is non empty set
the multF of F_Complex . [(M,0),(n . tcM)] is set
K147((M,0),(n . tcM)) is complex Element of COMPLEX
Re ((M,0) * (n . tcM)) is complex real ext-real Element of REAL
Re (M,0) is complex real ext-real Element of REAL
Re (n . tcM) is complex real ext-real Element of REAL
(Re (M,0)) * (Re (n . tcM)) is complex real ext-real Element of REAL
Im (M,0) is complex real ext-real Element of REAL
Im (n . tcM) is complex real ext-real Element of REAL
(Im (M,0)) * (Im (n . tcM)) is complex real ext-real Element of REAL
((Re (M,0)) * (Re (n . tcM))) - ((Im (M,0)) * (Im (n . tcM))) is complex real ext-real Element of REAL
0 * (Im (n . tcM)) is complex real ext-real Element of REAL
((Re (M,0)) * (Re (n . tcM))) - (0 * (Im (n . tcM))) is complex real ext-real Element of REAL
M * (Re (n . tcM)) is complex real ext-real Element of REAL
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
M is left_complementable right_complementable complementable Element of the carrier of (x)
p1 + M is left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (p1,M) is left_complementable right_complementable complementable Element of the carrier of (x)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x) . [p1,M] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is left_complementable right_complementable complementable Element of the carrier of x
RVSM is left_complementable right_complementable complementable Element of the carrier of x
tcM + RVSM is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (tcM,RVSM) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of x . [tcM,RVSM] is set
n . (tcM + RVSM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (n . (tcM + RVSM)) is complex real ext-real Element of REAL
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n . RVSM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(n . tcM) + (n . RVSM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((n . tcM),(n . RVSM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(n . tcM),(n . RVSM)] is set
{(n . tcM),(n . RVSM)} is non empty V192() set
{(n . tcM)} is non empty V192() set
{{(n . tcM),(n . RVSM)},{(n . tcM)}} is non empty set
the addF of F_Complex . [(n . tcM),(n . RVSM)] is set
K145((n . tcM),(n . RVSM)) is complex Element of COMPLEX
Re ((n . tcM) + (n . RVSM)) is complex real ext-real Element of REAL
Re (n . tcM) is complex real ext-real Element of REAL
Re (n . RVSM) is complex real ext-real Element of REAL
(Re (n . tcM)) + (Re (n . RVSM)) is complex real ext-real Element of REAL
(Re (n . tcM)) + ((x,n) . M) is complex real ext-real Element of REAL
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
n is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) additive ( F_Complex ,x) ( F_Complex ,x) Element of bool [: the carrier of x, the carrier of F_Complex:]
(x,n) is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) Element of bool [: the carrier of (x),REAL:]
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
(x,n) . p1 is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
M * p1 is left_complementable right_complementable complementable Element of the carrier of (x)
the Mult of (x) is V16() V19([:REAL, the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: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 set
[:[:REAL, the carrier of (x):], the carrier of (x):] is set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is set
the Mult of (x) . (M,p1) is set
[M,p1] is set
{M,p1} is non empty set
{M} is non empty V192() V193() V194() set
{{M,p1},{M}} is non empty set
the Mult of (x) . [M,p1] is set
(x,n) . (M * p1) is complex real ext-real Element of REAL
M * ((x,n) . p1) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
(M,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M + (0 * <i>) is complex set
tcM is left_complementable right_complementable complementable Element of the carrier of x
(M,0) * tcM is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((M,0),tcM) is left_complementable right_complementable complementable Element of the carrier of x
[(M,0),tcM] is set
{(M,0),tcM} is non empty set
{(M,0)} is non empty V192() set
{{(M,0),tcM},{(M,0)}} is non empty set
the lmult of x . [(M,0),tcM] is set
n . ((M,0) * tcM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Im (n . ((M,0) * tcM)) is complex real ext-real Element of REAL
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(M,0) * (n . tcM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((M,0),(n . tcM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(M,0),(n . tcM)] is set
{(M,0),(n . tcM)} is non empty V192() set
{{(M,0),(n . tcM)},{(M,0)}} is non empty set
the multF of F_Complex . [(M,0),(n . tcM)] is set
K147((M,0),(n . tcM)) is complex Element of COMPLEX
Im ((M,0) * (n . tcM)) is complex real ext-real Element of REAL
Re (M,0) is complex real ext-real Element of REAL
Im (n . tcM) is complex real ext-real Element of REAL
(Re (M,0)) * (Im (n . tcM)) is complex real ext-real Element of REAL
Re (n . tcM) is complex real ext-real Element of REAL
Im (M,0) is complex real ext-real Element of REAL
(Re (n . tcM)) * (Im (M,0)) is complex real ext-real Element of REAL
((Re (M,0)) * (Im (n . tcM))) + ((Re (n . tcM)) * (Im (M,0))) is complex real ext-real Element of REAL
(Re (n . tcM)) * 0 is complex real ext-real Element of REAL
((Re (M,0)) * (Im (n . tcM))) + ((Re (n . tcM)) * 0) is complex real ext-real Element of REAL
M * (Im (n . tcM)) is complex real ext-real Element of REAL
p1 is left_complementable right_complementable complementable Element of the carrier of (x)
M is left_complementable right_complementable complementable Element of the carrier of (x)
p1 + M is left_complementable right_complementable complementable Element of the carrier of (x)
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the addF of (x) . (p1,M) is left_complementable right_complementable complementable Element of the carrier of (x)
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of (x) . [p1,M] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
tcM is left_complementable right_complementable complementable Element of the carrier of x
RVSM is left_complementable right_complementable complementable Element of the carrier of x
tcM + RVSM is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . (tcM,RVSM) is left_complementable right_complementable complementable Element of the carrier of x
[tcM,RVSM] is set
{tcM,RVSM} is non empty set
{tcM} is non empty set
{{tcM,RVSM},{tcM}} is non empty set
the addF of x . [tcM,RVSM] is set
n . (tcM + RVSM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Im (n . (tcM + RVSM)) is complex real ext-real Element of REAL
n . tcM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
n . RVSM is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(n . tcM) + (n . RVSM) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . ((n . tcM),(n . RVSM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(n . tcM),(n . RVSM)] is set
{(n . tcM),(n . RVSM)} is non empty V192() set
{(n . tcM)} is non empty V192() set
{{(n . tcM),(n . RVSM)},{(n . tcM)}} is non empty set
the addF of F_Complex . [(n . tcM),(n . RVSM)] is set
K145((n . tcM),(n . RVSM)) is complex Element of COMPLEX
Im ((n . tcM) + (n . RVSM)) is complex real ext-real Element of REAL
Im (n . tcM) is complex real ext-real Element of REAL
Im (n . RVSM) is complex real ext-real Element of REAL
(Im (n . tcM)) + (Im (n . RVSM)) is complex real ext-real Element of REAL
(Im (n . tcM)) + ((x,n) . M) is complex real ext-real Element of REAL
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
the carrier of x is non empty set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
n is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) subadditive additive homogeneous positively_homogeneous Element of bool [: the carrier of (x),REAL:]
(x,n) is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
p1 is left_complementable right_complementable complementable Element of the carrier of x
(x,n) . p1 is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . (M,p1) is left_complementable right_complementable complementable Element of the carrier of x
[M,p1] is set
{M,p1} is non empty set
{M} is non empty V192() set
{{M,p1},{M}} is non empty set
the lmult of x . [M,p1] is set
(x,n) . (M * p1) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
M * ((x,n) . p1) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (M,((x,n) . p1)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[M,((x,n) . p1)] is set
{M,((x,n) . p1)} is non empty V192() set
{{M,((x,n) . p1)},{M}} is non empty set
the multF of F_Complex . [M,((x,n) . p1)] is set
K147(M,((x,n) . p1)) is complex Element of COMPLEX
tcM is complex Element of COMPLEX
Re tcM is complex real ext-real Element of REAL
Im tcM is complex real ext-real Element of REAL
(Im tcM) * <i> is complex set
(Re tcM) + ((Im tcM) * <i>) is complex set
- 1 is non empty complex real ext-real Element of REAL
((- 1),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- 1) + (0 * <i>) is complex set
() * (- (1_ F_Complex)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((),(- (1_ F_Complex))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(),(- (1_ F_Complex))] is set
{(),(- (1_ F_Complex))} is non empty V192() set
{{(),(- (1_ F_Complex))},{()}} is non empty set
the multF of F_Complex . [(),(- (1_ F_Complex))] is set
K147((),(- (1_ F_Complex))) is complex Element of COMPLEX
() * (() * (- (1_ F_Complex))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((),(() * (- (1_ F_Complex)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(),(() * (- (1_ F_Complex)))] is set
{(),(() * (- (1_ F_Complex)))} is non empty V192() set
{{(),(() * (- (1_ F_Complex)))},{()}} is non empty set
the multF of F_Complex . [(),(() * (- (1_ F_Complex)))] is set
K147((),(() * (- (1_ F_Complex)))) is complex Element of COMPLEX
(() * (() * (- (1_ F_Complex)))) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((() * (() * (- (1_ F_Complex)))),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(() * (() * (- (1_ F_Complex)))),p1] is set
{(() * (() * (- (1_ F_Complex)))),p1} is non empty set
{(() * (() * (- (1_ F_Complex))))} is non empty V192() set
{{(() * (() * (- (1_ F_Complex)))),p1},{(() * (() * (- (1_ F_Complex))))}} is non empty set
the lmult of x . [(() * (() * (- (1_ F_Complex)))),p1] is set
(- (1_ F_Complex)) * () is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((- (1_ F_Complex)),()) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(- (1_ F_Complex)),()] is set
{(- (1_ F_Complex)),()} is non empty V192() set
{{(- (1_ F_Complex)),()},{(- (1_ F_Complex))}} is non empty set
the multF of F_Complex . [(- (1_ F_Complex)),()] is set
K147((- (1_ F_Complex)),()) is complex Element of COMPLEX
((- (1_ F_Complex)) * ()) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . (((- (1_ F_Complex)) * ()),p1) is left_complementable right_complementable complementable Element of the carrier of x
[((- (1_ F_Complex)) * ()),p1] is set
{((- (1_ F_Complex)) * ()),p1} is non empty set
{((- (1_ F_Complex)) * ())} is non empty V192() set
{{((- (1_ F_Complex)) * ()),p1},{((- (1_ F_Complex)) * ())}} is non empty set
the lmult of x . [((- (1_ F_Complex)) * ()),p1] is set
() * (((- (1_ F_Complex)) * ()) * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((),(((- (1_ F_Complex)) * ()) * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(),(((- (1_ F_Complex)) * ()) * p1)] is set
{(),(((- (1_ F_Complex)) * ()) * p1)} is non empty set
{{(),(((- (1_ F_Complex)) * ()) * p1)},{()}} is non empty set
the lmult of x . [(),(((- (1_ F_Complex)) * ()) * p1)] is set
(x,n) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) (x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
() * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(),p1] is set
{(),p1} is non empty set
{{(),p1},{()}} is non empty set
the lmult of x . [(),p1] is set
(x,n) . (() * p1) is complex real ext-real Element of REAL
- ((x,n) . (() * p1)) is complex real ext-real Element of REAL
(Re tcM) * (- ((x,n) . (() * p1))) is complex real ext-real Element of REAL
(x,n) . p1 is complex real ext-real Element of REAL
((x,n) . p1) * (Im tcM) is complex real ext-real Element of REAL
((Re tcM) * (- ((x,n) . (() * p1)))) + (((x,n) . p1) * (Im tcM)) is complex real ext-real Element of REAL
(Re tcM) * ((x,n) . (() * p1)) is complex real ext-real Element of REAL
- ((Re tcM) * ((x,n) . (() * p1))) is complex real ext-real Element of REAL
(x,n) . (() * (((- (1_ F_Complex)) * ()) * p1)) is complex real ext-real Element of REAL
(Im tcM) * ((x,n) . (() * (((- (1_ F_Complex)) * ()) * p1))) is complex real ext-real Element of REAL
(- ((Re tcM) * ((x,n) . (() * p1)))) + ((Im tcM) * ((x,n) . (() * (((- (1_ F_Complex)) * ()) * p1)))) is complex real ext-real Element of REAL
((Re tcM),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(Re tcM) + (0 * <i>) is complex set
((Re tcM),0) * (() * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . (((Re tcM),0),(() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[((Re tcM),0),(() * p1)] is set
{((Re tcM),0),(() * p1)} is non empty set
{((Re tcM),0)} is non empty V192() set
{{((Re tcM),0),(() * p1)},{((Re tcM),0)}} is non empty set
the lmult of x . [((Re tcM),0),(() * p1)] is set
(x,n) . (((Re tcM),0) * (() * p1)) is complex real ext-real Element of REAL
- ((x,n) . (((Re tcM),0) * (() * p1))) is complex real ext-real Element of REAL
- (Im tcM) is complex real ext-real Element of REAL
(- (Im tcM)) * ((x,n) . (() * (((- (1_ F_Complex)) * ()) * p1))) is complex real ext-real Element of REAL
- ((- (Im tcM)) * ((x,n) . (() * (((- (1_ F_Complex)) * ()) * p1)))) is complex real ext-real Element of REAL
(- ((x,n) . (((Re tcM),0) * (() * p1)))) + (- ((- (Im tcM)) * ((x,n) . (() * (((- (1_ F_Complex)) * ()) * p1))))) is complex real ext-real Element of REAL
(0,(- (Im tcM))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- (Im tcM)) * <i> is complex set
0 + ((- (Im tcM)) * <i>) is complex set
(0,(- (Im tcM))) * (((- (1_ F_Complex)) * ()) * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((0,(- (Im tcM))),(((- (1_ F_Complex)) * ()) * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(0,(- (Im tcM))),(((- (1_ F_Complex)) * ()) * p1)] is set
{(0,(- (Im tcM))),(((- (1_ F_Complex)) * ()) * p1)} is non empty set
{(0,(- (Im tcM)))} is non empty V192() set
{{(0,(- (Im tcM))),(((- (1_ F_Complex)) * ()) * p1)},{(0,(- (Im tcM)))}} is non empty set
the lmult of x . [(0,(- (Im tcM))),(((- (1_ F_Complex)) * ()) * p1)] is set
(x,n) . ((0,(- (Im tcM))) * (((- (1_ F_Complex)) * ()) * p1)) is complex real ext-real Element of REAL
- ((x,n) . ((0,(- (Im tcM))) * (((- (1_ F_Complex)) * ()) * p1))) is complex real ext-real Element of REAL
(- ((x,n) . (((Re tcM),0) * (() * p1)))) + (- ((x,n) . ((0,(- (Im tcM))) * (((- (1_ F_Complex)) * ()) * p1)))) is complex real ext-real Element of REAL
(- (1_ F_Complex)) * (() * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((- (1_ F_Complex)),(() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(- (1_ F_Complex)),(() * p1)] is set
{(- (1_ F_Complex)),(() * p1)} is non empty set
{{(- (1_ F_Complex)),(() * p1)},{(- (1_ F_Complex))}} is non empty set
the lmult of x . [(- (1_ F_Complex)),(() * p1)] is set
(0,(- (Im tcM))) * ((- (1_ F_Complex)) * (() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((0,(- (Im tcM))),((- (1_ F_Complex)) * (() * p1))) is left_complementable right_complementable complementable Element of the carrier of x
[(0,(- (Im tcM))),((- (1_ F_Complex)) * (() * p1))] is set
{(0,(- (Im tcM))),((- (1_ F_Complex)) * (() * p1))} is non empty set
{{(0,(- (Im tcM))),((- (1_ F_Complex)) * (() * p1))},{(0,(- (Im tcM)))}} is non empty set
the lmult of x . [(0,(- (Im tcM))),((- (1_ F_Complex)) * (() * p1))] is set
(x,n) . ((0,(- (Im tcM))) * ((- (1_ F_Complex)) * (() * p1))) is complex real ext-real Element of REAL
- ((x,n) . ((0,(- (Im tcM))) * ((- (1_ F_Complex)) * (() * p1)))) is complex real ext-real Element of REAL
(- ((x,n) . (((Re tcM),0) * (() * p1)))) + (- ((x,n) . ((0,(- (Im tcM))) * ((- (1_ F_Complex)) * (() * p1))))) is complex real ext-real Element of REAL
(0,(- (Im tcM))) * (- (1_ F_Complex)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((0,(- (Im tcM))),(- (1_ F_Complex))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(0,(- (Im tcM))),(- (1_ F_Complex))] is set
{(0,(- (Im tcM))),(- (1_ F_Complex))} is non empty V192() set
{{(0,(- (Im tcM))),(- (1_ F_Complex))},{(0,(- (Im tcM)))}} is non empty set
the multF of F_Complex . [(0,(- (Im tcM))),(- (1_ F_Complex))] is set
K147((0,(- (Im tcM))),(- (1_ F_Complex))) is complex Element of COMPLEX
((0,(- (Im tcM))) * (- (1_ F_Complex))) * (() * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . (((0,(- (Im tcM))) * (- (1_ F_Complex))),(() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[((0,(- (Im tcM))) * (- (1_ F_Complex))),(() * p1)] is set
{((0,(- (Im tcM))) * (- (1_ F_Complex))),(() * p1)} is non empty set
{((0,(- (Im tcM))) * (- (1_ F_Complex)))} is non empty V192() set
{{((0,(- (Im tcM))) * (- (1_ F_Complex))),(() * p1)},{((0,(- (Im tcM))) * (- (1_ F_Complex)))}} is non empty set
the lmult of x . [((0,(- (Im tcM))) * (- (1_ F_Complex))),(() * p1)] is set
(x,n) . (((0,(- (Im tcM))) * (- (1_ F_Complex))) * (() * p1)) is complex real ext-real Element of REAL
- ((x,n) . (((0,(- (Im tcM))) * (- (1_ F_Complex))) * (() * p1))) is complex real ext-real Element of REAL
(- ((x,n) . (((Re tcM),0) * (() * p1)))) + (- ((x,n) . (((0,(- (Im tcM))) * (- (1_ F_Complex))) * (() * p1)))) is complex real ext-real Element of REAL
(0,(Im tcM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
0 + ((Im tcM) * <i>) is complex set
(0,(Im tcM)) * (() * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((0,(Im tcM)),(() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(0,(Im tcM)),(() * p1)] is set
{(0,(Im tcM)),(() * p1)} is non empty set
{(0,(Im tcM))} is non empty V192() set
{{(0,(Im tcM)),(() * p1)},{(0,(Im tcM))}} is non empty set
the lmult of x . [(0,(Im tcM)),(() * p1)] is set
(x,n) . ((0,(Im tcM)) * (() * p1)) is complex real ext-real Element of REAL
((x,n) . (((Re tcM),0) * (() * p1))) + ((x,n) . ((0,(Im tcM)) * (() * p1))) is complex real ext-real Element of REAL
- (((x,n) . (((Re tcM),0) * (() * p1))) + ((x,n) . ((0,(Im tcM)) * (() * p1)))) is complex real ext-real Element of REAL
(((Re tcM),0) * (() * p1)) + ((0,(Im tcM)) * (() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . ((((Re tcM),0) * (() * p1)),((0,(Im tcM)) * (() * p1))) is left_complementable right_complementable complementable Element of the carrier of x
[(((Re tcM),0) * (() * p1)),((0,(Im tcM)) * (() * p1))] is set
{(((Re tcM),0) * (() * p1)),((0,(Im tcM)) * (() * p1))} is non empty set
{(((Re tcM),0) * (() * p1))} is non empty set
{{(((Re tcM),0) * (() * p1)),((0,(Im tcM)) * (() * p1))},{(((Re tcM),0) * (() * p1))}} is non empty set
the addF of x . [(((Re tcM),0) * (() * p1)),((0,(Im tcM)) * (() * p1))] is set
(x,n) . ((((Re tcM),0) * (() * p1)) + ((0,(Im tcM)) * (() * p1))) is complex real ext-real Element of REAL
- ((x,n) . ((((Re tcM),0) * (() * p1)) + ((0,(Im tcM)) * (() * p1)))) is complex real ext-real Element of REAL
((Re tcM),0) + (0,(Im tcM)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (((Re tcM),0),(0,(Im tcM))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[((Re tcM),0),(0,(Im tcM))] is set
{((Re tcM),0),(0,(Im tcM))} is non empty V192() set
{{((Re tcM),0),(0,(Im tcM))},{((Re tcM),0)}} is non empty set
the addF of F_Complex . [((Re tcM),0),(0,(Im tcM))] is set
K145(((Re tcM),0),(0,(Im tcM))) is complex Element of COMPLEX
(((Re tcM),0) + (0,(Im tcM))) * (() * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((((Re tcM),0) + (0,(Im tcM))),(() * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(((Re tcM),0) + (0,(Im tcM))),(() * p1)] is set
{(((Re tcM),0) + (0,(Im tcM))),(() * p1)} is non empty set
{(((Re tcM),0) + (0,(Im tcM)))} is non empty V192() set
{{(((Re tcM),0) + (0,(Im tcM))),(() * p1)},{(((Re tcM),0) + (0,(Im tcM)))}} is non empty set
the lmult of x . [(((Re tcM),0) + (0,(Im tcM))),(() * p1)] is set
(x,n) . ((((Re tcM),0) + (0,(Im tcM))) * (() * p1)) is complex real ext-real Element of REAL
- ((x,n) . ((((Re tcM),0) + (0,(Im tcM))) * (() * p1))) is complex real ext-real Element of REAL
() * M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((),M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(),M] is set
{(),M} is non empty V192() set
{{(),M},{()}} is non empty set
the multF of F_Complex . [(),M] is set
K147((),M) is complex Element of COMPLEX
(() * M) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((() * M),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(() * M),p1] is set
{(() * M),p1} is non empty set
{(() * M)} is non empty V192() set
{{(() * M),p1},{(() * M)}} is non empty set
the lmult of x . [(() * M),p1] is set
(x,n) . ((() * M) * p1) is complex real ext-real Element of REAL
- ((x,n) . ((() * M) * p1)) is complex real ext-real Element of REAL
(Re tcM) * ((x,n) . p1) is complex real ext-real Element of REAL
(Im tcM) * (- ((x,n) . (() * p1))) is complex real ext-real Element of REAL
((Re tcM) * ((x,n) . p1)) - ((Im tcM) * (- ((x,n) . (() * p1)))) is complex real ext-real Element of REAL
(Im tcM) * ((x,n) . (() * p1)) is complex real ext-real Element of REAL
((Re tcM) * ((x,n) . p1)) + ((Im tcM) * ((x,n) . (() * p1))) is complex real ext-real Element of REAL
((Re tcM),0) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . (((Re tcM),0),p1) is left_complementable right_complementable complementable Element of the carrier of x
[((Re tcM),0),p1] is set
{((Re tcM),0),p1} is non empty set
{{((Re tcM),0),p1},{((Re tcM),0)}} is non empty set
the lmult of x . [((Re tcM),0),p1] is set
(x,n) . (((Re tcM),0) * p1) is complex real ext-real Element of REAL
((x,n) . (((Re tcM),0) * p1)) + ((Im tcM) * ((x,n) . (() * p1))) is complex real ext-real Element of REAL
(0,(Im tcM)) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((0,(Im tcM)),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(0,(Im tcM)),p1] is set
{(0,(Im tcM)),p1} is non empty set
{{(0,(Im tcM)),p1},{(0,(Im tcM))}} is non empty set
the lmult of x . [(0,(Im tcM)),p1] is set
(x,n) . ((0,(Im tcM)) * p1) is complex real ext-real Element of REAL
((x,n) . (((Re tcM),0) * p1)) + ((x,n) . ((0,(Im tcM)) * p1)) is complex real ext-real Element of REAL
(((Re tcM),0) * p1) + ((0,(Im tcM)) * p1) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((((Re tcM),0) * p1),((0,(Im tcM)) * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(((Re tcM),0) * p1),((0,(Im tcM)) * p1)] is set
{(((Re tcM),0) * p1),((0,(Im tcM)) * p1)} is non empty set
{(((Re tcM),0) * p1)} is non empty set
{{(((Re tcM),0) * p1),((0,(Im tcM)) * p1)},{(((Re tcM),0) * p1)}} is non empty set
the addF of x . [(((Re tcM),0) * p1),((0,(Im tcM)) * p1)] is set
(x,n) . ((((Re tcM),0) * p1) + ((0,(Im tcM)) * p1)) is complex real ext-real Element of REAL
(((Re tcM),0) + (0,(Im tcM))) * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((((Re tcM),0) + (0,(Im tcM))),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(((Re tcM),0) + (0,(Im tcM))),p1] is set
{(((Re tcM),0) + (0,(Im tcM))),p1} is non empty set
{{(((Re tcM),0) + (0,(Im tcM))),p1},{(((Re tcM),0) + (0,(Im tcM)))}} is non empty set
the lmult of x . [(((Re tcM),0) + (0,(Im tcM))),p1] is set
(x,n) . ((((Re tcM),0) + (0,(Im tcM))) * p1) is complex real ext-real Element of REAL
(x,n) . (M * p1) is complex real ext-real Element of REAL
(x,(x,n)) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
(x,(x,n)) . (M * p1) is complex real ext-real Element of REAL
- ((x,(x,n)) . (M * p1)) is complex real ext-real Element of REAL
(((x,n) . (M * p1)),(- ((x,(x,n)) . (M * p1)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . (M * p1))) * <i> is complex set
((x,n) . (M * p1)) + ((- ((x,(x,n)) . (M * p1))) * <i>) is complex set
() * (M * p1) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((),(M * p1)) is left_complementable right_complementable complementable Element of the carrier of x
[(),(M * p1)] is set
{(),(M * p1)} is non empty set
{{(),(M * p1)},{()}} is non empty set
the lmult of x . [(),(M * p1)] is set
(x,n) . (() * (M * p1)) is complex real ext-real Element of REAL
- ((x,n) . (() * (M * p1))) is complex real ext-real Element of REAL
(((x,n) . (M * p1)),(- ((x,n) . (() * (M * p1))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . (() * (M * p1)))) * <i> is complex set
((x,n) . (M * p1)) + ((- ((x,n) . (() * (M * p1)))) * <i>) is complex set
(((Re tcM) * (- ((x,n) . (() * p1)))) + (((x,n) . p1) * (Im tcM))) * <i> is complex set
((x,n) . (M * p1)) + ((((Re tcM) * (- ((x,n) . (() * p1)))) + (((x,n) . p1) * (Im tcM))) * <i>) is complex set
(((x,n) . p1),(- ((x,n) . (() * p1)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . (() * p1))) * <i> is complex set
((x,n) . p1) + ((- ((x,n) . (() * p1))) * <i>) is complex set
M * (((x,n) . p1),(- ((x,n) . (() * p1)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (M,(((x,n) . p1),(- ((x,n) . (() * p1))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[M,(((x,n) . p1),(- ((x,n) . (() * p1))))] is set
{M,(((x,n) . p1),(- ((x,n) . (() * p1))))} is non empty V192() set
{{M,(((x,n) . p1),(- ((x,n) . (() * p1))))},{M}} is non empty set
the multF of F_Complex . [M,(((x,n) . p1),(- ((x,n) . (() * p1))))] is set
K147(M,(((x,n) . p1),(- ((x,n) . (() * p1))))) is complex Element of COMPLEX
(x,(x,n)) . p1 is complex real ext-real Element of REAL
- ((x,(x,n)) . p1) is complex real ext-real Element of REAL
(((x,n) . p1),(- ((x,(x,n)) . p1))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . p1)) * <i> is complex set
((x,n) . p1) + ((- ((x,(x,n)) . p1)) * <i>) is complex set
M * (((x,n) . p1),(- ((x,(x,n)) . p1))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (M,(((x,n) . p1),(- ((x,(x,n)) . p1)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[M,(((x,n) . p1),(- ((x,(x,n)) . p1)))] is set
{M,(((x,n) . p1),(- ((x,(x,n)) . p1)))} is non empty V192() set
{{M,(((x,n) . p1),(- ((x,(x,n)) . p1)))},{M}} is non empty set
the multF of F_Complex . [M,(((x,n) . p1),(- ((x,(x,n)) . p1)))] is set
K147(M,(((x,n) . p1),(- ((x,(x,n)) . p1)))) is complex Element of COMPLEX
M * ((x,n) . p1) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
p1 is left_complementable right_complementable complementable Element of the carrier of x
M is left_complementable right_complementable complementable Element of the carrier of x
p1 + M is left_complementable right_complementable complementable Element of the carrier of x
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the addF of x . (p1,M) is left_complementable right_complementable complementable Element of the carrier of x
[p1,M] is set
{p1,M} is non empty set
{p1} is non empty set
{{p1,M},{p1}} is non empty set
the addF of x . [p1,M] is set
(x,n) . (p1 + M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,n) . p1 is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,n) . M is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
((x,n) . p1) + ((x,n) . M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex is V16() V19([: the carrier of F_Complex, the carrier of F_Complex:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex, the carrier of F_Complex:], the carrier of F_Complex:]
the addF of F_Complex . (((x,n) . p1),((x,n) . M)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[((x,n) . p1),((x,n) . M)] is set
{((x,n) . p1),((x,n) . M)} is non empty V192() set
{((x,n) . p1)} is non empty V192() set
{{((x,n) . p1),((x,n) . M)},{((x,n) . p1)}} is non empty set
the addF of F_Complex . [((x,n) . p1),((x,n) . M)] is set
K145(((x,n) . p1),((x,n) . M)) is complex Element of COMPLEX
p1 + M is left_complementable right_complementable complementable Element of the carrier of x
(x,n) . (p1 + M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,n) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) (x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
(x,n) . (p1 + M) is complex real ext-real Element of REAL
(x,(x,n)) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
(x,(x,n)) . (p1 + M) is complex real ext-real Element of REAL
- ((x,(x,n)) . (p1 + M)) is complex real ext-real Element of REAL
(((x,n) . (p1 + M)),(- ((x,(x,n)) . (p1 + M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . (p1 + M))) * <i> is complex set
((x,n) . (p1 + M)) + ((- ((x,(x,n)) . (p1 + M))) * <i>) is complex set
() * (p1 + M) is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((),(p1 + M)) is left_complementable right_complementable complementable Element of the carrier of x
[(),(p1 + M)] is set
{(),(p1 + M)} is non empty set
{{(),(p1 + M)},{()}} is non empty set
the lmult of x . [(),(p1 + M)] is set
(x,n) . (() * (p1 + M)) is complex real ext-real Element of REAL
- ((x,n) . (() * (p1 + M))) is complex real ext-real Element of REAL
(((x,n) . (p1 + M)),(- ((x,n) . (() * (p1 + M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . (() * (p1 + M)))) * <i> is complex set
((x,n) . (p1 + M)) + ((- ((x,n) . (() * (p1 + M)))) * <i>) is complex set
(x,n) . p1 is complex real ext-real Element of REAL
(x,n) . M is complex real ext-real Element of REAL
((x,n) . p1) + ((x,n) . M) is complex real ext-real Element of REAL
((((x,n) . p1) + ((x,n) . M)),(- ((x,n) . (() * (p1 + M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(((x,n) . p1) + ((x,n) . M)) + ((- ((x,n) . (() * (p1 + M)))) * <i>) is complex set
() * p1 is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((),p1) is left_complementable right_complementable complementable Element of the carrier of x
[(),p1] is set
{(),p1} is non empty set
{{(),p1},{()}} is non empty set
the lmult of x . [(),p1] is set
() * M is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . ((),M) is left_complementable right_complementable complementable Element of the carrier of x
[(),M] is set
{(),M} is non empty set
{{(),M},{()}} is non empty set
the lmult of x . [(),M] is set
(() * p1) + (() * M) is left_complementable right_complementable complementable Element of the carrier of x
the addF of x . ((() * p1),(() * M)) is left_complementable right_complementable complementable Element of the carrier of x
[(() * p1),(() * M)] is set
{(() * p1),(() * M)} is non empty set
{(() * p1)} is non empty set
{{(() * p1),(() * M)},{(() * p1)}} is non empty set
the addF of x . [(() * p1),(() * M)] is set
(x,n) . ((() * p1) + (() * M)) is complex real ext-real Element of REAL
- ((x,n) . ((() * p1) + (() * M))) is complex real ext-real Element of REAL
((((x,n) . p1) + ((x,n) . M)),(- ((x,n) . ((() * p1) + (() * M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . ((() * p1) + (() * M)))) * <i> is complex set
(((x,n) . p1) + ((x,n) . M)) + ((- ((x,n) . ((() * p1) + (() * M)))) * <i>) is complex set
(x,n) . (() * p1) is complex real ext-real Element of REAL
(x,n) . (() * M) is complex real ext-real Element of REAL
((x,n) . (() * p1)) + ((x,n) . (() * M)) is complex real ext-real Element of REAL
- (((x,n) . (() * p1)) + ((x,n) . (() * M))) is complex real ext-real Element of REAL
((((x,n) . p1) + ((x,n) . M)),(- (((x,n) . (() * p1)) + ((x,n) . (() * M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- (((x,n) . (() * p1)) + ((x,n) . (() * M)))) * <i> is complex set
(((x,n) . p1) + ((x,n) . M)) + ((- (((x,n) . (() * p1)) + ((x,n) . (() * M)))) * <i>) is complex set
- ((x,n) . (() * p1)) is complex real ext-real Element of REAL
(((x,n) . p1),(- ((x,n) . (() * p1)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . (() * p1))) * <i> is complex set
((x,n) . p1) + ((- ((x,n) . (() * p1))) * <i>) is complex set
- ((x,n) . (() * M)) is complex real ext-real Element of REAL
(((x,n) . M),(- ((x,n) . (() * M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,n) . (() * M))) * <i> is complex set
((x,n) . M) + ((- ((x,n) . (() * M))) * <i>) is complex set
(((x,n) . p1),(- ((x,n) . (() * p1)))) + (((x,n) . M),(- ((x,n) . (() * M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex . ((((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))] is set
{(((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))} is non empty V192() set
{(((x,n) . p1),(- ((x,n) . (() * p1))))} is non empty V192() set
{{(((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))},{(((x,n) . p1),(- ((x,n) . (() * p1))))}} is non empty set
the addF of F_Complex . [(((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))] is set
K145((((x,n) . p1),(- ((x,n) . (() * p1)))),(((x,n) . M),(- ((x,n) . (() * M))))) is complex Element of COMPLEX
(x,(x,n)) . p1 is complex real ext-real Element of REAL
- ((x,(x,n)) . p1) is complex real ext-real Element of REAL
(((x,n) . p1),(- ((x,(x,n)) . p1))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . p1)) * <i> is complex set
((x,n) . p1) + ((- ((x,(x,n)) . p1)) * <i>) is complex set
(((x,n) . p1),(- ((x,(x,n)) . p1))) + (((x,n) . M),(- ((x,n) . (() * M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex . ((((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))] is set
{(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))} is non empty V192() set
{(((x,n) . p1),(- ((x,(x,n)) . p1)))} is non empty V192() set
{{(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))},{(((x,n) . p1),(- ((x,(x,n)) . p1)))}} is non empty set
the addF of F_Complex . [(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))] is set
K145((((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,n) . (() * M))))) is complex Element of COMPLEX
(x,(x,n)) . M is complex real ext-real Element of REAL
- ((x,(x,n)) . M) is complex real ext-real Element of REAL
(((x,n) . M),(- ((x,(x,n)) . M))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,n)) . M)) * <i> is complex set
((x,n) . M) + ((- ((x,(x,n)) . M)) * <i>) is complex set
(((x,n) . p1),(- ((x,(x,n)) . p1))) + (((x,n) . M),(- ((x,(x,n)) . M))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex . ((((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))] is set
{(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))} is non empty V192() set
{{(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))},{(((x,n) . p1),(- ((x,(x,n)) . p1)))}} is non empty set
the addF of F_Complex . [(((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))] is set
K145((((x,n) . p1),(- ((x,(x,n)) . p1))),(((x,n) . M),(- ((x,(x,n)) . M)))) is complex Element of COMPLEX
((x,n) . p1) + (((x,n) . M),(- ((x,(x,n)) . M))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the addF of F_Complex . (((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))] is set
{((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))} is non empty V192() set
{{((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))},{((x,n) . p1)}} is non empty set
the addF of F_Complex . [((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))] is set
K145(((x,n) . p1),(((x,n) . M),(- ((x,(x,n)) . M)))) is complex Element of COMPLEX
((x,n) . p1) + ((x,n) . M) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over F_Complex
the carrier of x is non empty set
[: the carrier of x,REAL:] is set
bool [: the carrier of x,REAL:] is set
[: the carrier of x, the carrier of F_Complex:] is set
bool [: the carrier of x, the carrier of F_Complex:] is set
n is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) (x) Element of bool [: the carrier of x,REAL:]
(x) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
the carrier of (x) is non empty set
[: the carrier of (x),REAL:] is set
bool [: the carrier of (x),REAL:] is set
M is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V105() vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of x
the carrier of M is non empty set
[: the carrier of M, the carrier of F_Complex:] is set
bool [: the carrier of M, the carrier of F_Complex:] is set
bool the carrier of x is set
(M) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() RLSStruct
l is V16() V19( the carrier of M) V20( the carrier of F_Complex) Function-like V30( the carrier of M, the carrier of F_Complex) additive ( F_Complex ,M) ( F_Complex ,M) Element of bool [: the carrier of M, the carrier of F_Complex:]
RVSM is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V105() Subspace of (x)
the carrier of RVSM is non empty set
[: the carrier of RVSM,REAL:] is set
bool [: the carrier of RVSM,REAL:] is set
(M,l) is V16() V19( the carrier of (M)) V20( REAL ) Function-like V30( the carrier of (M), REAL ) Element of bool [: the carrier of (M),REAL:]
the carrier of (M) is non empty set
[: the carrier of (M),REAL:] is set
bool [: the carrier of (M),REAL:] is set
the addF of x is V16() V19([: the carrier of x, the carrier of x:]) V20( the carrier of x) Function-like V30([: 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 set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the ZeroF of x is left_complementable right_complementable complementable Element of the carrier of x
addLoopStr(# the carrier of x, the addF of x, the ZeroF of x #) is non empty strict addLoopStr
the addF of (x) is V16() V19([: the carrier of (x), the carrier of (x):]) V20( the carrier of (x)) Function-like V30([: 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 set
[:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
bool [:[: the carrier of (x), the carrier of (x):], the carrier of (x):] is set
the ZeroF of (x) is left_complementable right_complementable complementable Element of the carrier of (x)
addLoopStr(# the carrier of (x), the addF of (x), the ZeroF of (x) #) is non empty strict addLoopStr
the addF of M is V16() V19([: the carrier of M, the carrier of M:]) V20( the carrier of M) Function-like V30([: the carrier of M, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
the ZeroF of M is left_complementable right_complementable complementable Element of the carrier of M
addLoopStr(# the carrier of M, the addF of M, the ZeroF of M #) is non empty strict addLoopStr
the addF of (M) is V16() V19([: the carrier of (M), the carrier of (M):]) V20( the carrier of (M)) Function-like V30([: the carrier of (M), the carrier of (M):], the carrier of (M)) Element of bool [:[: the carrier of (M), the carrier of (M):], the carrier of (M):]
[: the carrier of (M), the carrier of (M):] is set
[:[: the carrier of (M), the carrier of (M):], the carrier of (M):] is set
bool [:[: the carrier of (M), the carrier of (M):], the carrier of (M):] is set
the ZeroF of (M) is left_complementable right_complementable complementable Element of the carrier of (M)
addLoopStr(# the carrier of (M), the addF of (M), the ZeroF of (M) #) is non empty strict addLoopStr
prRl is V16() V19( the carrier of RVSM) V20( REAL ) Function-like V30( the carrier of RVSM, REAL ) subadditive additive homogeneous positively_homogeneous Element of bool [: the carrier of RVSM,REAL:]
p1 is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) subadditive positively_homogeneous Element of bool [: the carrier of (x),REAL:]
L1 is left_complementable right_complementable complementable Element of the carrier of RVSM
prRl . L1 is complex real ext-real Element of REAL
e is left_complementable right_complementable complementable Element of the carrier of (x)
p1 . e is complex real ext-real Element of REAL
L is left_complementable right_complementable complementable Element of the carrier of M
l . L is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (l . L) is complex real ext-real Element of REAL
|.(l . L).| is complex real ext-real Element of REAL
(Re (l . L)) ^2 is complex real ext-real Element of REAL
(Re (l . L)) * (Re (l . L)) is complex real ext-real set
Im (l . L) is complex real ext-real Element of REAL
(Im (l . L)) ^2 is complex real ext-real Element of REAL
(Im (l . L)) * (Im (l . L)) is complex real ext-real set
((Re (l . L)) ^2) + ((Im (l . L)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (l . L)) ^2) + ((Im (l . L)) ^2)) is complex real ext-real Element of REAL
Le is left_complementable right_complementable complementable Element of the carrier of x
n . Le is complex real ext-real Element of REAL
L1 is V16() V19( the carrier of (x)) V20( REAL ) Function-like V30( the carrier of (x), REAL ) subadditive additive homogeneous positively_homogeneous Element of bool [: the carrier of (x),REAL:]
L1 | the carrier of RVSM is V16() V19( the carrier of (x)) V20( REAL ) Function-like Element of bool [: the carrier of (x),REAL:]
(x,L1) is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) Element of bool [: the carrier of x, the carrier of F_Complex:]
L is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like V30( the carrier of x, the carrier of F_Complex) additive ( F_Complex ,x) ( F_Complex ,x) Element of bool [: the carrier of x, the carrier of F_Complex:]
L | the carrier of M is V16() V19( the carrier of x) V20( the carrier of F_Complex) Function-like Element of bool [: the carrier of x, the carrier of F_Complex:]
e is left_complementable right_complementable complementable Element of the carrier of M
Le is left_complementable right_complementable complementable Element of the carrier of x
() * Le is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x is V16() V19([: the carrier of F_Complex, the carrier of x:]) V20( the carrier of x) Function-like V30([: the carrier of F_Complex, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:]
[: the carrier of F_Complex, the carrier of x:] is set
[:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of F_Complex, the carrier of x:], the carrier of x:] is set
the lmult of x . ((),Le) is left_complementable right_complementable complementable Element of the carrier of x
[(),Le] is set
{(),Le} is non empty set
{{(),Le},{()}} is non empty set
the lmult of x . [(),Le] is set
l . e is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Ledze is complex Element of COMPLEX
Re Ledze is complex real ext-real Element of REAL
Im Ledze is complex real ext-real Element of REAL
(Im Ledze) * <i> is complex set
(Re Ledze) + ((Im Ledze) * <i>) is complex set
() * (l . e) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . ((),(l . e)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[(),(l . e)] is set
{(),(l . e)} is non empty V192() set
{{(),(l . e)},{()}} is non empty set
the multF of F_Complex . [(),(l . e)] is set
K147((),(l . e)) is complex Element of COMPLEX
0 * (Re Ledze) is complex real ext-real Element of REAL
1 * (Im Ledze) is complex real ext-real Element of REAL
(0 * (Re Ledze)) - (1 * (Im Ledze)) is complex real ext-real Element of REAL
0 * (Im Ledze) is complex real ext-real Element of REAL
1 * (Re Ledze) is complex real ext-real Element of REAL
(0 * (Im Ledze)) + (1 * (Re Ledze)) is complex real ext-real Element of REAL
((0 * (Im Ledze)) + (1 * (Re Ledze))) * <i> is complex set
((0 * (Re Ledze)) - (1 * (Im Ledze))) + (((0 * (Im Ledze)) + (1 * (Re Ledze))) * <i>) is complex set
() * e is left_complementable right_complementable complementable Element of the carrier of M
the lmult of M is V16() V19([: the carrier of F_Complex, the carrier of M:]) V20( the carrier of M) Function-like V30([: the carrier of F_Complex, the carrier of M:], the carrier of M) Element of bool [:[: the carrier of F_Complex, the carrier of M:], the carrier of M:]
[: the carrier of F_Complex, the carrier of M:] is set
[:[: the carrier of F_Complex, the carrier of M:], the carrier of M:] is set
bool [:[: the carrier of F_Complex, the carrier of M:], the carrier of M:] is set
the lmult of M . ((),e) is left_complementable right_complementable complementable Element of the carrier of M
[(),e] is set
{(),e} is non empty set
{{(),e},{()}} is non empty set
the lmult of M . [(),e] is set
e1 is left_complementable right_complementable complementable Element of the carrier of (x)
L1 . e1 is complex real ext-real Element of REAL
(M,l) . e1 is set
l . (() * e) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Re (l . (() * e)) is complex real ext-real Element of REAL
- (Im Ledze) is complex real ext-real Element of REAL
(Re Ledze) * <i> is complex set
(- (Im Ledze)) + ((Re Ledze) * <i>) is complex set
Re ((- (Im Ledze)) + ((Re Ledze) * <i>)) is complex real ext-real Element of REAL
Im (l . e) is complex real ext-real Element of REAL
- (Im (l . e)) is complex real ext-real Element of REAL
Ledz is left_complementable right_complementable complementable Element of the carrier of (x)
L1 . Ledz is complex real ext-real Element of REAL
(M,l) . Ledz is set
Re (l . e) is complex real ext-real Element of REAL
tcM is Element of bool the carrier of x
L | tcM is V16() V19(tcM) V20( the carrier of F_Complex) Function-like V30(tcM, the carrier of F_Complex) Element of bool [:tcM, the carrier of F_Complex:]
[:tcM, the carrier of F_Complex:] is set
bool [:tcM, the carrier of F_Complex:] is set
(L | tcM) . e is set
L . e is set
(x,L1) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) ( F_Complex ,x) ( F_Complex ,x) (x) ( F_Complex ,x) Element of bool [: the carrier of x,REAL:]
(x,L1) . Le is complex real ext-real Element of REAL
(x,(x,L1)) is V16() V19( the carrier of x) V20( REAL ) Function-like V30( the carrier of x, REAL ) Element of bool [: the carrier of x,REAL:]
(x,(x,L1)) . Le is complex real ext-real Element of REAL
- ((x,(x,L1)) . Le) is complex real ext-real Element of REAL
(((x,L1) . Le),(- ((x,(x,L1)) . Le))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,L1)) . Le)) * <i> is complex set
((x,L1) . Le) + ((- ((x,(x,L1)) . Le)) * <i>) is complex set
(x,L1) . (() * Le) is complex real ext-real Element of REAL
- ((x,L1) . (() * Le)) is complex real ext-real Element of REAL
((Re (l . e)),(- ((x,L1) . (() * Le)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,L1) . (() * Le))) * <i> is complex set
(Re (l . e)) + ((- ((x,L1) . (() * Le))) * <i>) is complex set
e is left_complementable right_complementable complementable Element of the carrier of x
L . e is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
|.(L . e).| is complex real ext-real Element of REAL
Re (L . e) is complex real ext-real Element of REAL
(Re (L . e)) ^2 is complex real ext-real Element of REAL
(Re (L . e)) * (Re (L . e)) is complex real ext-real set
Im (L . e) is complex real ext-real Element of REAL
(Im (L . e)) ^2 is complex real ext-real Element of REAL
(Im (L . e)) * (Im (L . e)) is complex real ext-real set
((Re (L . e)) ^2) + ((Im (L . e)) ^2) is complex real ext-real Element of REAL
sqrt (((Re (L . e)) ^2) + ((Im (L . e)) ^2)) is complex real ext-real Element of REAL
n . e is complex real ext-real Element of REAL
Le is complex Element of COMPLEX
Le *' is complex Element of COMPLEX
Re Le is complex real ext-real Element of REAL
Im Le is complex real ext-real Element of REAL
(Im Le) * <i> is complex set
(Re Le) - ((Im Le) * <i>) is complex set
|.Le.| is complex real ext-real Element of REAL
(Re Le) ^2 is complex real ext-real Element of REAL
(Re Le) * (Re Le) is complex real ext-real set
(Im Le) ^2 is complex real ext-real Element of REAL
(Im Le) * (Im Le) is complex real ext-real set
((Re Le) ^2) + ((Im Le) ^2) is complex real ext-real Element of REAL
sqrt (((Re Le) ^2) + ((Im Le) ^2)) is complex real ext-real Element of REAL
(Le *') / |.Le.| is complex Element of COMPLEX
Ledz is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
Ledz * e is left_complementable right_complementable complementable Element of the carrier of x
the lmult of x . (Ledz,e) is left_complementable right_complementable complementable Element of the carrier of x
[Ledz,e] is set
{Ledz,e} is non empty set
{Ledz} is non empty V192() set
{{Ledz,e},{Ledz}} is non empty set
the lmult of x . [Ledz,e] is set
|.Ledz.| is complex real ext-real Element of REAL
Re Ledz is complex real ext-real Element of REAL
(Re Ledz) ^2 is complex real ext-real Element of REAL
(Re Ledz) * (Re Ledz) is complex real ext-real set
Im Ledz is complex real ext-real Element of REAL
(Im Ledz) ^2 is complex real ext-real Element of REAL
(Im Ledz) * (Im Ledz) is complex real ext-real set
((Re Ledz) ^2) + ((Im Ledz) ^2) is complex real ext-real Element of REAL
sqrt (((Re Ledz) ^2) + ((Im Ledz) ^2)) is complex real ext-real Element of REAL
|.(Le *').| is complex real ext-real Element of REAL
Re (Le *') is complex real ext-real Element of REAL
(Re (Le *')) ^2 is complex real ext-real Element of REAL
(Re (Le *')) * (Re (Le *')) is complex real ext-real set
Im (Le *') is complex real ext-real Element of REAL
(Im (Le *')) ^2 is complex real ext-real Element of REAL
(Im (Le *')) * (Im (Le *')) is complex real ext-real set
((Re (Le *')) ^2) + ((Im (Le *')) ^2) is complex real ext-real Element of REAL
sqrt (((Re (Le *')) ^2) + ((Im (Le *')) ^2)) is complex real ext-real Element of REAL
|.|.Le.|.| is complex real ext-real Element of REAL
Re |.Le.| is complex real ext-real Element of REAL
(Re |.Le.|) ^2 is complex real ext-real Element of REAL
(Re |.Le.|) * (Re |.Le.|) is complex real ext-real set
Im |.Le.| is complex real ext-real Element of REAL
(Im |.Le.|) ^2 is complex real ext-real Element of REAL
(Im |.Le.|) * (Im |.Le.|) is complex real ext-real set
((Re |.Le.|) ^2) + ((Im |.Le.|) ^2) is complex real ext-real Element of REAL
sqrt (((Re |.Le.|) ^2) + ((Im |.Le.|) ^2)) is complex real ext-real Element of REAL
|.(Le *').| / |.|.Le.|.| is complex real ext-real Element of REAL
|.Le.| / |.Le.| is complex real ext-real Element of REAL
|.Le.| + (0 * <i>) is complex set
Ledz * (L . e) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (Ledz,(L . e)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[Ledz,(L . e)] is set
{Ledz,(L . e)} is non empty V192() set
{{Ledz,(L . e)},{Ledz}} is non empty set
the multF of F_Complex . [Ledz,(L . e)] is set
K147(Ledz,(L . e)) is complex Element of COMPLEX
L . (Ledz * e) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x,L1) . (Ledz * e) is complex real ext-real Element of REAL
(x,(x,L1)) . (Ledz * e) is complex real ext-real Element of REAL
- ((x,(x,L1)) . (Ledz * e)) is complex real ext-real Element of REAL
(((x,L1) . (Ledz * e)),(- ((x,(x,L1)) . (Ledz * e)))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,L1)) . (Ledz * e))) * <i> is complex set
((x,L1) . (Ledz * e)) + ((- ((x,(x,L1)) . (Ledz * e))) * <i>) is complex set
Ledze is left_complementable right_complementable complementable Element of the carrier of (x)
L1 . Ledze is complex real ext-real Element of REAL
(L1 . Ledze) + ((- ((x,(x,L1)) . (Ledz * e))) * <i>) is complex set
p1 . Ledze is complex real ext-real Element of REAL
|.Ledz.| * (n . e) is complex real ext-real Element of REAL
(x,L1) . e is complex real ext-real Element of REAL
(x,(x,L1)) . e is complex real ext-real Element of REAL
- ((x,(x,L1)) . e) is complex real ext-real Element of REAL
(((x,L1) . e),(- ((x,(x,L1)) . e))) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(- ((x,(x,L1)) . e)) * <i> is complex set
((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>) is complex set
|.(((x,L1) . e),(- ((x,(x,L1)) . e))).| is complex real ext-real Element of REAL
Re (((x,L1) . e),(- ((x,(x,L1)) . e))) is complex real ext-real Element of REAL
(Re (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2 is complex real ext-real Element of REAL
(Re (((x,L1) . e),(- ((x,(x,L1)) . e)))) * (Re (((x,L1) . e),(- ((x,(x,L1)) . e)))) is complex real ext-real set
Im (((x,L1) . e),(- ((x,(x,L1)) . e))) is complex real ext-real Element of REAL
(Im (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2 is complex real ext-real Element of REAL
(Im (((x,L1) . e),(- ((x,(x,L1)) . e)))) * (Im (((x,L1) . e),(- ((x,(x,L1)) . e)))) is complex real ext-real set
((Re (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2) + ((Im (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2) + ((Im (((x,L1) . e),(- ((x,(x,L1)) . e)))) ^2)) is complex real ext-real Element of REAL
|.(((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>)).| is complex real ext-real Element of REAL
Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>)) is complex real ext-real Element of REAL
(Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2 is complex real ext-real Element of REAL
(Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) * (Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) is complex real ext-real set
Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>)) is complex real ext-real Element of REAL
(Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2 is complex real ext-real Element of REAL
(Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) * (Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) is complex real ext-real set
((Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2) + ((Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2) is complex real ext-real Element of REAL
sqrt (((Re (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2) + ((Im (((x,L1) . e) + ((- ((x,(x,L1)) . e)) * <i>))) ^2)) is complex real ext-real Element of REAL
0 + (0 * <i>) is complex set
e1 is left_complementable right_complementable complementable Element of the carrier of (x)
L1 . e1 is complex real ext-real Element of REAL
power F_Complex is V16() V19([: the carrier of F_Complex,NAT:]) V20( the carrier of F_Complex) Function-like V30([: the carrier of F_Complex,NAT:], the carrier of F_Complex) Element of bool [:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:]
[: the carrier of F_Complex,NAT:] is set
[:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:] is set
bool [:[: the carrier of F_Complex,NAT:], the carrier of F_Complex:] is set
x is complex real ext-real Element of REAL
(x,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
x + (0 * <i>) is complex set
n is natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() Element of NAT
(power F_Complex) . ((x,0),n) is set
[(x,0),n] is set
{(x,0),n} is non empty V192() set
{(x,0)} is non empty V192() set
{{(x,0),n},{(x,0)}} is non empty set
(power F_Complex) . [(x,0),n] is set
x to_power n is complex real ext-real set
((x to_power n),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x to_power n) + (0 * <i>) is complex set
n + 1 is natural complex real ext-real V33() V34() V192() V193() V194() V195() V196() V197() Element of NAT
(power F_Complex) . ((x,0),(n + 1)) is set
[(x,0),(n + 1)] is set
{(x,0),(n + 1)} is non empty V192() set
{{(x,0),(n + 1)},{(x,0)}} is non empty set
(power F_Complex) . [(x,0),(n + 1)] is set
((x to_power n),0) * (x,0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
the multF of F_Complex . (((x to_power n),0),(x,0)) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
[((x to_power n),0),(x,0)] is set
{((x to_power n),0),(x,0)} is non empty V192() set
{((x to_power n),0)} is non empty V192() set
{{((x to_power n),0),(x,0)},{((x to_power n),0)}} is non empty set
the multF of F_Complex . [((x to_power n),0),(x,0)] is set
K147(((x to_power n),0),(x,0)) is complex Element of COMPLEX
x to_power 1 is complex real ext-real Element of REAL
(x to_power n) * (x to_power 1) is complex real ext-real Element of REAL
(((x to_power n) * (x to_power 1)),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
((x to_power n) * (x to_power 1)) + (0 * <i>) is complex set
x to_power (n + 1) is complex real ext-real Element of REAL
((x to_power (n + 1)),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x to_power (n + 1)) + (0 * <i>) is complex set
x to_power (n + 1) is complex real ext-real set
((x to_power (n + 1)),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x to_power (n + 1)) + (0 * <i>) is complex set
(power F_Complex) . ((x,0),0) is set
[(x,0),0] is set
{(x,0),0} is non empty V192() set
{{(x,0),0},{(x,0)}} is non empty set
(power F_Complex) . [(x,0),0] is set
1r + (0 * <i>) is complex set
x to_power 0 is complex real ext-real Element of REAL
((x to_power 0),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x to_power 0) + (0 * <i>) is complex set
x to_power 0 is complex real ext-real set
((x to_power 0),0) is complex left_complementable right_complementable complementable Element of the carrier of F_Complex
(x to_power 0) + (0 * <i>) is complex set