:: C0SP1 semantic presentation

REAL is non empty V30() V177() V178() V179() V183() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V177() V178() V179() V180() V181() V182() V183() Element of bool REAL
bool REAL is non empty set
omega is non empty epsilon-transitive epsilon-connected ordinal V177() V178() V179() V180() V181() V182() V183() set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty V150() V151() V152() set
bool [:NAT,REAL:] is non empty set
K207() is non empty set
[:K207(),K207():] is non empty set
[:[:K207(),K207():],K207():] is non empty set
bool [:[:K207(),K207():],K207():] is non empty set
[:REAL,K207():] is non empty set
[:[:REAL,K207():],K207():] is non empty set
bool [:[:REAL,K207():],K207():] is non empty set
K213() is RLSStruct
the carrier of K213() is set
bool the carrier of K213() is non empty set
K217() is Element of bool the carrier of K213()
[:K217(),K217():] is set
[:[:K217(),K217():],REAL:] is V150() V151() V152() set
bool [:[:K217(),K217():],REAL:] is non empty set
the_set_of_l1RealSequences is Element of bool the carrier of K213()
[:the_set_of_l1RealSequences,REAL:] is V150() V151() V152() set
bool [:the_set_of_l1RealSequences,REAL:] is non empty set
COMPLEX is non empty V30() V177() V183() set
RAT is non empty V30() V177() V178() V179() V180() V183() set
INT is non empty V30() V177() V178() V179() V180() V181() V183() set
[:COMPLEX,COMPLEX:] is non empty V150() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V150() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty V150() V151() V152() set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty V150() V151() V152() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty RAT -valued V150() V151() V152() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty RAT -valued V150() V151() V152() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty RAT -valued INT -valued V150() V151() V152() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty RAT -valued INT -valued V150() V151() V152() set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
[:[:NAT,NAT:],NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:[:NAT,NAT:],NAT:] is non empty set
[:COMPLEX,REAL:] is non empty V150() V151() V152() set
bool [:COMPLEX,REAL:] is non empty set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V177() V178() V179() V180() V181() V182() V183() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real positive non negative V177() V178() V179() V180() V181() V182() Element of NAT
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real V111() V112() ext-real non positive non negative V177() V178() V179() V180() V181() V182() V183() Element of NAT
1r is V11() Element of COMPLEX
|.1r.| is V11() real ext-real Element of REAL
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real positive non negative V177() V178() V179() V180() V181() V182() Element of NAT
- 1 is V11() real ext-real non positive Element of REAL
X is non empty addLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty addLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty addLoopStr
[#] X is non empty non proper Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of the carrier of X
B is Element of the carrier of X
B + B is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (B,B) is Element of the carrier of X
[B,B] is set
{B,B} is non empty set
{B} is non empty set
{{B,B},{B}} is non empty set
the addF of X . [B,B] is set
X is non empty addLoopStr
[#] X is non empty non proper Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of the carrier of X
- B is Element of the carrier of X
X is non empty addLoopStr
[#] X is non empty non proper Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty doubleLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
B is Element of bool the carrier of X
X is non empty addLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
[#] X is non empty non proper add-closed (X) Element of bool the carrier of X
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V100() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
the carrier of X is non empty set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
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
0. X is V49(X) 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
the addF of X || the carrier of X is Relation-like Function-like set
the addF of X | [: the carrier of X, the carrier of X:] is Relation-like set
the multF of X || the carrier of X is Relation-like Function-like set
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like set
X is non empty set
[:X,X:] is non empty set
[:[:X,X:],X:] is non empty set
bool [:[:X,X:],X:] is non empty set
B is Element of X
B is Element of X
ONE is non empty Relation-like [:X,X:] -defined X -valued Function-like V23([:X,X:]) quasi_total Element of bool [:[:X,X:],X:]
f is non empty Relation-like [:X,X:] -defined X -valued Function-like V23([:X,X:]) quasi_total Element of bool [:[:X,X:],X:]
doubleLoopStr(# X,ONE,f,B,B #) is non empty strict doubleLoopStr
g is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V100() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
the carrier of g is non empty set
bool the carrier of g is non empty set
the addF of g is non empty Relation-like [: the carrier of g, the carrier of g:] -defined the carrier of g -valued Function-like V23([: the carrier of g, the carrier of g:]) quasi_total Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
[: the carrier of g, the carrier of g:] is non empty set
[:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
bool [:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
the multF of g is non empty Relation-like [: the carrier of g, the carrier of g:] -defined the carrier of g -valued Function-like V23([: the carrier of g, the carrier of g:]) quasi_total Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
1. g is left_complementable right_complementable complementable Element of the carrier of g
the OneF of g is left_complementable right_complementable complementable Element of the carrier of g
0. g is V49(g) left_complementable right_complementable complementable Element of the carrier of g
the ZeroF of g is left_complementable right_complementable complementable Element of the carrier of g
h is Element of bool the carrier of g
the addF of g || h is Relation-like Function-like set
[:h,h:] is set
the addF of g | [:h,h:] is Relation-like set
the multF of g || h is Relation-like Function-like set
the multF of g | [:h,h:] is Relation-like set
f1 is non empty doubleLoopStr
the carrier of f1 is non empty set
h1 is Element of the carrier of f1
g1 is Element of the carrier of f1
h1 * g1 is Element of the carrier of f1
the multF of f1 is non empty Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like V23([: the carrier of f1, the carrier of f1:]) quasi_total Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
[: the carrier of f1, the carrier of f1:] is non empty set
[:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:] is non empty set
the multF of f1 . (h1,g1) is Element of the carrier of f1
[h1,g1] is set
{h1,g1} is non empty set
{h1} is non empty set
{{h1,g1},{h1}} is non empty set
the multF of f1 . [h1,g1] is set
[h1,g1] is Element of [: the carrier of f1, the carrier of f1:]
the multF of g . [h1,g1] is set
the multF of g . (h1,g1) is set
the multF of g . [h1,g1] is set
h1 is Element of the carrier of f1
g1 is Element of the carrier of f1
h1 + g1 is Element of the carrier of f1
the addF of f1 is non empty Relation-like [: the carrier of f1, the carrier of f1:] -defined the carrier of f1 -valued Function-like V23([: the carrier of f1, the carrier of f1:]) quasi_total Element of bool [:[: the carrier of f1, the carrier of f1:], the carrier of f1:]
the addF of f1 . (h1,g1) is Element of the carrier of f1
[h1,g1] is set
{h1,g1} is non empty set
{h1} is non empty set
{{h1,g1},{h1}} is non empty set
the addF of f1 . [h1,g1] is set
[h1,g1] is Element of [: the carrier of f1, the carrier of f1:]
the addF of g . [h1,g1] is set
the addF of g . (h1,g1) is set
the addF of g . [h1,g1] is set
F1 is Element of the carrier of f1
hf1 is Element of the carrier of f1
F1 + hf1 is Element of the carrier of f1
the addF of f1 . (F1,hf1) is Element of the carrier of f1
[F1,hf1] is set
{F1,hf1} is non empty set
{F1} is non empty set
{{F1,hf1},{F1}} is non empty set
the addF of f1 . [F1,hf1] is set
gf1 is left_complementable right_complementable complementable Element of the carrier of g
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
gf1 + gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gf1,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the addF of g . [gf1,gPh1] is set
hf1 + F1 is Element of the carrier of f1
the addF of f1 . (hf1,F1) is Element of the carrier of f1
[hf1,F1] is set
{hf1,F1} is non empty set
{hf1} is non empty set
{{hf1,F1},{hf1}} is non empty set
the addF of f1 . [hf1,F1] is set
gPh1 + gf1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,gf1) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,gf1] is set
{gPh1,gf1} is non empty set
{gPh1} is non empty set
{{gPh1,gf1},{gPh1}} is non empty set
the addF of g . [gPh1,gf1] is set
F1 is Element of the carrier of f1
hf1 is Element of the carrier of f1
gf1 is Element of the carrier of f1
hf1 + gf1 is Element of the carrier of f1
the addF of f1 . (hf1,gf1) is Element of the carrier of f1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the addF of f1 . [hf1,gf1] is set
F1 + (hf1 + gf1) is Element of the carrier of f1
the addF of f1 . (F1,(hf1 + gf1)) is Element of the carrier of f1
[F1,(hf1 + gf1)] is set
{F1,(hf1 + gf1)} is non empty set
{F1} is non empty set
{{F1,(hf1 + gf1)},{F1}} is non empty set
the addF of f1 . [F1,(hf1 + gf1)] is set
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,(hf1 + gf1)) is set
[gPh1,(hf1 + gf1)] is set
{gPh1,(hf1 + gf1)} is non empty set
{gPh1} is non empty set
{{gPh1,(hf1 + gf1)},{gPh1}} is non empty set
the addF of g . [gPh1,(hf1 + gf1)] is set
x is left_complementable right_complementable complementable Element of the carrier of g
a is left_complementable right_complementable complementable Element of the carrier of g
x + a is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{x} is non empty set
{{x,a},{x}} is non empty set
the addF of g . [x,a] is set
gPh1 + (x + a) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,(x + a)) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,(x + a)] is set
{gPh1,(x + a)} is non empty set
{{gPh1,(x + a)},{gPh1}} is non empty set
the addF of g . [gPh1,(x + a)] is set
F1 + hf1 is Element of the carrier of f1
the addF of f1 . (F1,hf1) is Element of the carrier of f1
[F1,hf1] is set
{F1,hf1} is non empty set
{{F1,hf1},{F1}} is non empty set
the addF of f1 . [F1,hf1] is set
(F1 + hf1) + gf1 is Element of the carrier of f1
the addF of f1 . ((F1 + hf1),gf1) is Element of the carrier of f1
[(F1 + hf1),gf1] is set
{(F1 + hf1),gf1} is non empty set
{(F1 + hf1)} is non empty set
{{(F1 + hf1),gf1},{(F1 + hf1)}} is non empty set
the addF of f1 . [(F1 + hf1),gf1] is set
the addF of g . ((F1 + hf1),a) is set
[(F1 + hf1),a] is set
{(F1 + hf1),a} is non empty set
{{(F1 + hf1),a},{(F1 + hf1)}} is non empty set
the addF of g . [(F1 + hf1),a] is set
gPh1 + x is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,x) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,x] is set
{gPh1,x} is non empty set
{{gPh1,x},{gPh1}} is non empty set
the addF of g . [gPh1,x] is set
(gPh1 + x) + a is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((gPh1 + x),a) is left_complementable right_complementable complementable Element of the carrier of g
[(gPh1 + x),a] is set
{(gPh1 + x),a} is non empty set
{(gPh1 + x)} is non empty set
{{(gPh1 + x),a},{(gPh1 + x)}} is non empty set
the addF of g . [(gPh1 + x),a] is set
F1 is Element of the carrier of f1
0. f1 is V49(f1) Element of the carrier of f1
the ZeroF of f1 is Element of the carrier of f1
F1 + (0. f1) is Element of the carrier of f1
the addF of f1 . (F1,(0. f1)) is Element of the carrier of f1
[F1,(0. f1)] is set
{F1,(0. f1)} is non empty set
{F1} is non empty set
{{F1,(0. f1)},{F1}} is non empty set
the addF of f1 . [F1,(0. f1)] is set
hf1 is left_complementable right_complementable complementable Element of the carrier of g
hf1 + (0. g) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (hf1,(0. g)) is left_complementable right_complementable complementable Element of the carrier of g
[hf1,(0. g)] is set
{hf1,(0. g)} is non empty set
{hf1} is non empty set
{{hf1,(0. g)},{hf1}} is non empty set
the addF of g . [hf1,(0. g)] is set
F1 is Element of the carrier of f1
hf1 is left_complementable right_complementable complementable Element of the carrier of g
gf1 is left_complementable right_complementable complementable Element of the carrier of g
hf1 + gf1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (hf1,gf1) is left_complementable right_complementable complementable Element of the carrier of g
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the addF of g . [hf1,gf1] is set
- hf1 is left_complementable right_complementable complementable Element of the carrier of g
gPh1 is Element of the carrier of f1
F1 + gPh1 is Element of the carrier of f1
the addF of f1 . (F1,gPh1) is Element of the carrier of f1
[F1,gPh1] is set
{F1,gPh1} is non empty set
{F1} is non empty set
{{F1,gPh1},{F1}} is non empty set
the addF of f1 . [F1,gPh1] is set
gf1 is Element of the carrier of f1
F1 is Element of the carrier of f1
hf1 is Element of the carrier of f1
hf1 * gf1 is Element of the carrier of f1
the multF of f1 . (hf1,gf1) is Element of the carrier of f1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the multF of f1 . [hf1,gf1] is set
F1 * (hf1 * gf1) is Element of the carrier of f1
the multF of f1 . (F1,(hf1 * gf1)) is Element of the carrier of f1
[F1,(hf1 * gf1)] is set
{F1,(hf1 * gf1)} is non empty set
{F1} is non empty set
{{F1,(hf1 * gf1)},{F1}} is non empty set
the multF of f1 . [F1,(hf1 * gf1)] is set
the multF of g . (F1,(hf1 * gf1)) is set
the multF of g . [F1,(hf1 * gf1)] is set
x is left_complementable right_complementable complementable Element of the carrier of g
a is left_complementable right_complementable complementable Element of the carrier of g
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
a * gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[a,gPh1] is set
{a,gPh1} is non empty set
{a} is non empty set
{{a,gPh1},{a}} is non empty set
the multF of g . [a,gPh1] is set
x * (a * gPh1) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,(a * gPh1)) is left_complementable right_complementable complementable Element of the carrier of g
[x,(a * gPh1)] is set
{x,(a * gPh1)} is non empty set
{x} is non empty set
{{x,(a * gPh1)},{x}} is non empty set
the multF of g . [x,(a * gPh1)] is set
F1 * hf1 is Element of the carrier of f1
the multF of f1 . (F1,hf1) is Element of the carrier of f1
[F1,hf1] is set
{F1,hf1} is non empty set
{{F1,hf1},{F1}} is non empty set
the multF of f1 . [F1,hf1] is set
x * a is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{{x,a},{x}} is non empty set
the multF of g . [x,a] is set
(F1 * hf1) * gf1 is Element of the carrier of f1
the multF of f1 . ((F1 * hf1),gf1) is Element of the carrier of f1
[(F1 * hf1),gf1] is set
{(F1 * hf1),gf1} is non empty set
{(F1 * hf1)} is non empty set
{{(F1 * hf1),gf1},{(F1 * hf1)}} is non empty set
the multF of f1 . [(F1 * hf1),gf1] is set
(x * a) * gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . ((x * a),gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[(x * a),gPh1] is set
{(x * a),gPh1} is non empty set
{(x * a)} is non empty set
{{(x * a),gPh1},{(x * a)}} is non empty set
the multF of g . [(x * a),gPh1] is set
F1 is Element of the carrier of f1
1. f1 is Element of the carrier of f1
the OneF of f1 is Element of the carrier of f1
F1 * (1. f1) is Element of the carrier of f1
the multF of f1 . (F1,(1. f1)) is Element of the carrier of f1
[F1,(1. f1)] is set
{F1,(1. f1)} is non empty set
{F1} is non empty set
{{F1,(1. f1)},{F1}} is non empty set
the multF of f1 . [F1,(1. f1)] is set
hf1 is left_complementable right_complementable complementable Element of the carrier of g
hf1 * (1. g) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (hf1,(1. g)) is left_complementable right_complementable complementable Element of the carrier of g
[hf1,(1. g)] is set
{hf1,(1. g)} is non empty set
{hf1} is non empty set
{{hf1,(1. g)},{hf1}} is non empty set
the multF of g . [hf1,(1. g)] is set
(1. f1) * F1 is Element of the carrier of f1
the multF of f1 . ((1. f1),F1) is Element of the carrier of f1
[(1. f1),F1] is set
{(1. f1),F1} is non empty set
{(1. f1)} is non empty set
{{(1. f1),F1},{(1. f1)}} is non empty set
the multF of f1 . [(1. f1),F1] is set
(1. g) * hf1 is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . ((1. g),hf1) is left_complementable right_complementable complementable Element of the carrier of g
[(1. g),hf1] is set
{(1. g),hf1} is non empty set
{(1. g)} is non empty set
{{(1. g),hf1},{(1. g)}} is non empty set
the multF of g . [(1. g),hf1] is set
hf1 is Element of the carrier of f1
gf1 is Element of the carrier of f1
F1 is Element of the carrier of f1
hf1 + gf1 is Element of the carrier of f1
the addF of f1 . (hf1,gf1) is Element of the carrier of f1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the addF of f1 . [hf1,gf1] is set
(hf1 + gf1) * F1 is Element of the carrier of f1
the multF of f1 . ((hf1 + gf1),F1) is Element of the carrier of f1
[(hf1 + gf1),F1] is set
{(hf1 + gf1),F1} is non empty set
{(hf1 + gf1)} is non empty set
{{(hf1 + gf1),F1},{(hf1 + gf1)}} is non empty set
the multF of f1 . [(hf1 + gf1),F1] is set
the multF of g . ((hf1 + gf1),F1) is set
the multF of g . [(hf1 + gf1),F1] is set
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
x is left_complementable right_complementable complementable Element of the carrier of g
gPh1 + x is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,x) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,x] is set
{gPh1,x} is non empty set
{gPh1} is non empty set
{{gPh1,x},{gPh1}} is non empty set
the addF of g . [gPh1,x] is set
a is left_complementable right_complementable complementable Element of the carrier of g
(gPh1 + x) * a is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . ((gPh1 + x),a) is left_complementable right_complementable complementable Element of the carrier of g
[(gPh1 + x),a] is set
{(gPh1 + x),a} is non empty set
{(gPh1 + x)} is non empty set
{{(gPh1 + x),a},{(gPh1 + x)}} is non empty set
the multF of g . [(gPh1 + x),a] is set
gPh1 * a is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (gPh1,a) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,a] is set
{gPh1,a} is non empty set
{{gPh1,a},{gPh1}} is non empty set
the multF of g . [gPh1,a] is set
x * a is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{x} is non empty set
{{x,a},{x}} is non empty set
the multF of g . [x,a] is set
(gPh1 * a) + (x * a) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((gPh1 * a),(x * a)) is left_complementable right_complementable complementable Element of the carrier of g
[(gPh1 * a),(x * a)] is set
{(gPh1 * a),(x * a)} is non empty set
{(gPh1 * a)} is non empty set
{{(gPh1 * a),(x * a)},{(gPh1 * a)}} is non empty set
the addF of g . [(gPh1 * a),(x * a)] is set
gf1 * F1 is Element of the carrier of f1
the multF of f1 . (gf1,F1) is Element of the carrier of f1
[gf1,F1] is set
{gf1,F1} is non empty set
{gf1} is non empty set
{{gf1,F1},{gf1}} is non empty set
the multF of f1 . [gf1,F1] is set
the addF of g . (( the multF of g . (gPh1,a)),(gf1 * F1)) is set
[( the multF of g . (gPh1,a)),(gf1 * F1)] is set
{( the multF of g . (gPh1,a)),(gf1 * F1)} is non empty set
{( the multF of g . (gPh1,a))} is non empty set
{{( the multF of g . (gPh1,a)),(gf1 * F1)},{( the multF of g . (gPh1,a))}} is non empty set
the addF of g . [( the multF of g . (gPh1,a)),(gf1 * F1)] is set
hf1 * F1 is Element of the carrier of f1
the multF of f1 . (hf1,F1) is Element of the carrier of f1
[hf1,F1] is set
{hf1,F1} is non empty set
{{hf1,F1},{hf1}} is non empty set
the multF of f1 . [hf1,F1] is set
the addF of g . ((hf1 * F1),(gf1 * F1)) is set
[(hf1 * F1),(gf1 * F1)] is set
{(hf1 * F1),(gf1 * F1)} is non empty set
{(hf1 * F1)} is non empty set
{{(hf1 * F1),(gf1 * F1)},{(hf1 * F1)}} is non empty set
the addF of g . [(hf1 * F1),(gf1 * F1)] is set
F1 * (hf1 + gf1) is Element of the carrier of f1
the multF of f1 . (F1,(hf1 + gf1)) is Element of the carrier of f1
[F1,(hf1 + gf1)] is set
{F1,(hf1 + gf1)} is non empty set
{F1} is non empty set
{{F1,(hf1 + gf1)},{F1}} is non empty set
the multF of f1 . [F1,(hf1 + gf1)] is set
the multF of g . (F1,(hf1 + gf1)) is set
the multF of g . [F1,(hf1 + gf1)] is set
a * (gPh1 + x) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,(gPh1 + x)) is left_complementable right_complementable complementable Element of the carrier of g
[a,(gPh1 + x)] is set
{a,(gPh1 + x)} is non empty set
{a} is non empty set
{{a,(gPh1 + x)},{a}} is non empty set
the multF of g . [a,(gPh1 + x)] is set
a * gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[a,gPh1] is set
{a,gPh1} is non empty set
{{a,gPh1},{a}} is non empty set
the multF of g . [a,gPh1] is set
a * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,x) is left_complementable right_complementable complementable Element of the carrier of g
[a,x] is set
{a,x} is non empty set
{{a,x},{a}} is non empty set
the multF of g . [a,x] is set
(a * gPh1) + (a * x) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((a * gPh1),(a * x)) is left_complementable right_complementable complementable Element of the carrier of g
[(a * gPh1),(a * x)] is set
{(a * gPh1),(a * x)} is non empty set
{(a * gPh1)} is non empty set
{{(a * gPh1),(a * x)},{(a * gPh1)}} is non empty set
the addF of g . [(a * gPh1),(a * x)] is set
the multF of g . (F1,gPh1) is set
[F1,gPh1] is set
{F1,gPh1} is non empty set
{{F1,gPh1},{F1}} is non empty set
the multF of g . [F1,gPh1] is set
F1 * gf1 is Element of the carrier of f1
the multF of f1 . (F1,gf1) is Element of the carrier of f1
[F1,gf1] is set
{F1,gf1} is non empty set
{{F1,gf1},{F1}} is non empty set
the multF of f1 . [F1,gf1] is set
the addF of g . (( the multF of g . (F1,gPh1)),(F1 * gf1)) is set
[( the multF of g . (F1,gPh1)),(F1 * gf1)] is set
{( the multF of g . (F1,gPh1)),(F1 * gf1)} is non empty set
{( the multF of g . (F1,gPh1))} is non empty set
{{( the multF of g . (F1,gPh1)),(F1 * gf1)},{( the multF of g . (F1,gPh1))}} is non empty set
the addF of g . [( the multF of g . (F1,gPh1)),(F1 * gf1)] is set
F1 * hf1 is Element of the carrier of f1
the multF of f1 . (F1,hf1) is Element of the carrier of f1
[F1,hf1] is set
{F1,hf1} is non empty set
{{F1,hf1},{F1}} is non empty set
the multF of f1 . [F1,hf1] is set
the addF of g . ((F1 * hf1),(F1 * gf1)) is set
[(F1 * hf1),(F1 * gf1)] is set
{(F1 * hf1),(F1 * gf1)} is non empty set
{(F1 * hf1)} is non empty set
{{(F1 * hf1),(F1 * gf1)},{(F1 * hf1)}} is non empty set
the addF of g . [(F1 * hf1),(F1 * gf1)] is set
(F1 * hf1) + (F1 * gf1) is Element of the carrier of f1
the addF of f1 . ((F1 * hf1),(F1 * gf1)) is Element of the carrier of f1
the addF of f1 . [(F1 * hf1),(F1 * gf1)] is set
(hf1 * F1) + (gf1 * F1) is Element of the carrier of f1
the addF of f1 . ((hf1 * F1),(gf1 * F1)) is Element of the carrier of f1
the addF of f1 . [(hf1 * F1),(gf1 * F1)] is set
0. f1 is V49(f1) Element of the carrier of f1
the ZeroF of f1 is Element of the carrier of f1
1. f1 is Element of the carrier of f1
the OneF of f1 is Element of the carrier of f1
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V100() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
[#] X is non empty non proper add-closed (X) (X) Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X || ([#] X) is Relation-like Function-like set
[:([#] X),([#] X):] is non empty set
the addF of X | [:([#] X),([#] X):] is Relation-like set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || ([#] X) is Relation-like Function-like set
the multF of X | [:([#] X),([#] X):] is Relation-like 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
0. X is V49(X) 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
doubleLoopStr(# the carrier of X, the addF of X, the multF of X,(1. X),(0. X) #) is non empty strict doubleLoopStr
X is non empty multLoopStr_0
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty addLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X || B is Relation-like Function-like set
[:B,B:] is set
the addF of X | [:B,B:] is Relation-like set
[:[:B,B:],B:] is set
bool [:[:B,B:],B:] is non empty set
dom the addF of X is Relation-like the carrier of X -defined the carrier of X -valued Element of bool [: the carrier of X, the carrier of X:]
bool [: the carrier of X, the carrier of X:] is non empty set
B is set
( the addF of X || B) . B is set
ONE is set
f is set
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
dom ( the addF of X || B) is set
h is Element of the carrier of X
g is Element of the carrier of X
h + g is Element of the carrier of X
the addF of X . (h,g) is Element of the carrier of X
[h,g] is set
{h,g} is non empty set
{h} is non empty set
{{h,g},{h}} is non empty set
the addF of X . [h,g] is set
dom ( the addF of X || B) is set
X is non empty multLoopStr_0
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || B is Relation-like Function-like set
[:B,B:] is set
the multF of X | [:B,B:] is Relation-like set
[:[:B,B:],B:] is set
bool [:[:B,B:],B:] is non empty set
dom the multF of X is Relation-like the carrier of X -defined the carrier of X -valued Element of bool [: the carrier of X, the carrier of X:]
bool [: the carrier of X, the carrier of X:] is non empty set
B is set
( the multF of X || B) . B is set
ONE is set
f is set
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
dom ( the multF of X || B) is set
h is Element of the carrier of X
g is Element of the carrier of X
h * g is Element of the carrier of X
the multF of X . (h,g) is Element of the carrier of X
[h,g] is set
{h,g} is non empty set
{h} is non empty set
{{h,g},{h}} is non empty set
the multF of X . [h,g] is set
dom ( the multF of X || B) is set
X is non empty right_complementable add-associative right_zeroed doubleLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
0. X is V49(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
the Element of B is Element of B
ONE is right_complementable Element of the carrier of X
- ONE is right_complementable Element of the carrier of X
ONE + (- ONE) is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (ONE,(- ONE)) is right_complementable Element of the carrier of X
[ONE,(- ONE)] is set
{ONE,(- ONE)} is non empty set
{ONE} is non empty set
{{ONE,(- ONE)},{ONE}} is non empty set
the addF of X . [ONE,(- ONE)] is set
X is non empty multLoopStr_0
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
1. X is Element of the carrier of X
the OneF of X is Element of the carrier of X
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed V100() unital associative right-distributive left-distributive right_unital well-unital distributive left_unital doubleLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
(X,B) is Relation-like [:B,B:] -defined B -valued Function-like quasi_total Element of bool [:[:B,B:],B:]
[:B,B:] is set
[:[:B,B:],B:] is set
bool [:[:B,B:],B:] is non empty set
(X,B) is Relation-like [:B,B:] -defined B -valued Function-like quasi_total Element of bool [:[:B,B:],B:]
(X,B) is Element of B
(X,B) is Element of B
doubleLoopStr(# B,(X,B),(X,B),(X,B),(X,B) #) is strict doubleLoopStr
1_ X is left_complementable right_complementable complementable Element of the carrier of X
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
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || B is Relation-like Function-like set
the multF of X | [:B,B:] is Relation-like set
0. X is V49(X) 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
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the addF of X || B is Relation-like Function-like set
the addF of X | [:B,B:] is Relation-like set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
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
0. X is V49(X) 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
the addF of X || the carrier of X is Relation-like Function-like set
the addF of X | [: the carrier of X, the carrier of X:] is Relation-like set
the multF of X || the carrier of X is Relation-like Function-like set
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like set
the Mult of X | [:REAL, the carrier of X:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
X is non empty set
[:X,X:] is non empty set
[:[:X,X:],X:] is non empty set
bool [:[:X,X:],X:] is non empty set
[:REAL,X:] is non empty set
[:[:REAL,X:],X:] is non empty set
bool [:[:REAL,X:],X:] is non empty set
B is Element of X
B is Element of X
ONE is non empty Relation-like [:X,X:] -defined X -valued Function-like V23([:X,X:]) quasi_total Element of bool [:[:X,X:],X:]
f is non empty Relation-like [:X,X:] -defined X -valued Function-like V23([:X,X:]) quasi_total Element of bool [:[:X,X:],X:]
g is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of g is non empty set
bool the carrier of g is non empty set
0. g is V49(g) left_complementable right_complementable complementable Element of the carrier of g
the ZeroF of g is left_complementable right_complementable complementable Element of the carrier of g
1. g is left_complementable right_complementable complementable Element of the carrier of g
the OneF of g is left_complementable right_complementable complementable Element of the carrier of g
the addF of g is non empty Relation-like [: the carrier of g, the carrier of g:] -defined the carrier of g -valued Function-like V23([: the carrier of g, the carrier of g:]) quasi_total Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
[: the carrier of g, the carrier of g:] is non empty set
[:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
bool [:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
the multF of g is non empty Relation-like [: the carrier of g, the carrier of g:] -defined the carrier of g -valued Function-like V23([: the carrier of g, the carrier of g:]) quasi_total Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
[:REAL, the carrier of g:] is non empty set
the Mult of g is non empty Relation-like [:REAL, the carrier of g:] -defined the carrier of g -valued Function-like V23([:REAL, the carrier of g:]) quasi_total Element of bool [:[:REAL, the carrier of g:], the carrier of g:]
[:[:REAL, the carrier of g:], the carrier of g:] is non empty set
bool [:[:REAL, the carrier of g:], the carrier of g:] is non empty set
h is Element of bool the carrier of g
the addF of g || h is Relation-like Function-like set
[:h,h:] is set
the addF of g | [:h,h:] is Relation-like set
the multF of g || h is Relation-like Function-like set
the multF of g | [:h,h:] is Relation-like set
[:REAL,h:] is set
the Mult of g | [:REAL,h:] is Relation-like [:REAL, the carrier of g:] -defined the carrier of g -valued Function-like Element of bool [:[:REAL, the carrier of g:], the carrier of g:]
f1 is non empty Relation-like [:REAL,X:] -defined X -valued Function-like V23([:REAL,X:]) quasi_total Element of bool [:[:REAL,X:],X:]
AlgebraStr(# X,f,ONE,f1,B,B #) is strict AlgebraStr
h1 is non empty AlgebraStr
the carrier of h1 is non empty set
g1 is Element of the carrier of h1
F1 is Element of the carrier of h1
g1 + F1 is Element of the carrier of h1
the addF of h1 is non empty Relation-like [: the carrier of h1, the carrier of h1:] -defined the carrier of h1 -valued Function-like V23([: the carrier of h1, the carrier of h1:]) quasi_total Element of bool [:[: the carrier of h1, the carrier of h1:], the carrier of h1:]
[: the carrier of h1, the carrier of h1:] is non empty set
[:[: the carrier of h1, the carrier of h1:], the carrier of h1:] is non empty set
bool [:[: the carrier of h1, the carrier of h1:], the carrier of h1:] is non empty set
the addF of h1 . (g1,F1) is Element of the carrier of h1
[g1,F1] is set
{g1,F1} is non empty set
{g1} is non empty set
{{g1,F1},{g1}} is non empty set
the addF of h1 . [g1,F1] is set
[g1,F1] is Element of [: the carrier of h1, the carrier of h1:]
the addF of g . [g1,F1] is set
the addF of g . (g1,F1) is set
the addF of g . [g1,F1] is set
g1 is Element of the carrier of h1
F1 is Element of the carrier of h1
g1 * F1 is Element of the carrier of h1
the multF of h1 is non empty Relation-like [: the carrier of h1, the carrier of h1:] -defined the carrier of h1 -valued Function-like V23([: the carrier of h1, the carrier of h1:]) quasi_total Element of bool [:[: the carrier of h1, the carrier of h1:], the carrier of h1:]
the multF of h1 . (g1,F1) is Element of the carrier of h1
[g1,F1] is set
{g1,F1} is non empty set
{g1} is non empty set
{{g1,F1},{g1}} is non empty set
the multF of h1 . [g1,F1] is set
[g1,F1] is Element of [: the carrier of h1, the carrier of h1:]
the multF of g . [g1,F1] is set
the multF of g . (g1,F1) is set
the multF of g . [g1,F1] is set
F1 is Element of the carrier of h1
g1 is V11() real ext-real Element of REAL
g1 * F1 is Element of the carrier of h1
the Mult of h1 is non empty Relation-like [:REAL, the carrier of h1:] -defined the carrier of h1 -valued Function-like V23([:REAL, the carrier of h1:]) quasi_total Element of bool [:[:REAL, the carrier of h1:], the carrier of h1:]
[:REAL, the carrier of h1:] is non empty set
[:[:REAL, the carrier of h1:], the carrier of h1:] is non empty set
bool [:[:REAL, the carrier of h1:], the carrier of h1:] is non empty set
the Mult of h1 . (g1,F1) is set
[g1,F1] is set
{g1,F1} is non empty set
{g1} is non empty V177() V178() V179() set
{{g1,F1},{g1}} is non empty set
the Mult of h1 . [g1,F1] is set
[g1,F1] is Element of [:REAL, the carrier of h1:]
the Mult of g . [g1,F1] is set
the Mult of g . (g1,F1) is set
the Mult of g . [g1,F1] is set
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
hf1 + gf1 is Element of the carrier of h1
the addF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the addF of h1 . [hf1,gf1] is set
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
x is left_complementable right_complementable complementable Element of the carrier of g
gPh1 + x is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gPh1,x) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,x] is set
{gPh1,x} is non empty set
{gPh1} is non empty set
{{gPh1,x},{gPh1}} is non empty set
the addF of g . [gPh1,x] is set
gf1 + hf1 is Element of the carrier of h1
the addF of h1 . (gf1,hf1) is Element of the carrier of h1
[gf1,hf1] is set
{gf1,hf1} is non empty set
{gf1} is non empty set
{{gf1,hf1},{gf1}} is non empty set
the addF of h1 . [gf1,hf1] is set
x + gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[x,gPh1] is set
{x,gPh1} is non empty set
{x} is non empty set
{{x,gPh1},{x}} is non empty set
the addF of g . [x,gPh1] is set
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
gPh1 is Element of the carrier of h1
gf1 + gPh1 is Element of the carrier of h1
the addF of h1 . (gf1,gPh1) is Element of the carrier of h1
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the addF of h1 . [gf1,gPh1] is set
hf1 + (gf1 + gPh1) is Element of the carrier of h1
the addF of h1 . (hf1,(gf1 + gPh1)) is Element of the carrier of h1
[hf1,(gf1 + gPh1)] is set
{hf1,(gf1 + gPh1)} is non empty set
{hf1} is non empty set
{{hf1,(gf1 + gPh1)},{hf1}} is non empty set
the addF of h1 . [hf1,(gf1 + gPh1)] is set
x is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,(gf1 + gPh1)) is set
[x,(gf1 + gPh1)] is set
{x,(gf1 + gPh1)} is non empty set
{x} is non empty set
{{x,(gf1 + gPh1)},{x}} is non empty set
the addF of g . [x,(gf1 + gPh1)] is set
a is left_complementable right_complementable complementable Element of the carrier of g
bb is left_complementable right_complementable complementable Element of the carrier of g
a + bb is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (a,bb) is left_complementable right_complementable complementable Element of the carrier of g
[a,bb] is set
{a,bb} is non empty set
{a} is non empty set
{{a,bb},{a}} is non empty set
the addF of g . [a,bb] is set
x + (a + bb) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,(a + bb)) is left_complementable right_complementable complementable Element of the carrier of g
[x,(a + bb)] is set
{x,(a + bb)} is non empty set
{{x,(a + bb)},{x}} is non empty set
the addF of g . [x,(a + bb)] is set
hf1 + gf1 is Element of the carrier of h1
the addF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the addF of h1 . [hf1,gf1] is set
(hf1 + gf1) + gPh1 is Element of the carrier of h1
the addF of h1 . ((hf1 + gf1),gPh1) is Element of the carrier of h1
[(hf1 + gf1),gPh1] is set
{(hf1 + gf1),gPh1} is non empty set
{(hf1 + gf1)} is non empty set
{{(hf1 + gf1),gPh1},{(hf1 + gf1)}} is non empty set
the addF of h1 . [(hf1 + gf1),gPh1] is set
the addF of g . ((hf1 + gf1),bb) is set
[(hf1 + gf1),bb] is set
{(hf1 + gf1),bb} is non empty set
{{(hf1 + gf1),bb},{(hf1 + gf1)}} is non empty set
the addF of g . [(hf1 + gf1),bb] is set
x + a is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{{x,a},{x}} is non empty set
the addF of g . [x,a] is set
(x + a) + bb is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((x + a),bb) is left_complementable right_complementable complementable Element of the carrier of g
[(x + a),bb] is set
{(x + a),bb} is non empty set
{(x + a)} is non empty set
{{(x + a),bb},{(x + a)}} is non empty set
the addF of g . [(x + a),bb] is set
hf1 is Element of the carrier of h1
0. h1 is V49(h1) Element of the carrier of h1
the ZeroF of h1 is Element of the carrier of h1
hf1 + (0. h1) is Element of the carrier of h1
the addF of h1 . (hf1,(0. h1)) is Element of the carrier of h1
[hf1,(0. h1)] is set
{hf1,(0. h1)} is non empty set
{hf1} is non empty set
{{hf1,(0. h1)},{hf1}} is non empty set
the addF of h1 . [hf1,(0. h1)] is set
gf1 is left_complementable right_complementable complementable Element of the carrier of g
gf1 + (0. g) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gf1,(0. g)) is left_complementable right_complementable complementable Element of the carrier of g
[gf1,(0. g)] is set
{gf1,(0. g)} is non empty set
{gf1} is non empty set
{{gf1,(0. g)},{gf1}} is non empty set
the addF of g . [gf1,(0. g)] is set
hf1 is Element of the carrier of h1
gf1 is left_complementable right_complementable complementable Element of the carrier of g
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
gf1 + gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (gf1,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the addF of g . [gf1,gPh1] is set
- gf1 is left_complementable right_complementable complementable Element of the carrier of g
x is Element of the carrier of h1
hf1 + x is Element of the carrier of h1
the addF of h1 . (hf1,x) is Element of the carrier of h1
[hf1,x] is set
{hf1,x} is non empty set
{hf1} is non empty set
{{hf1,x},{hf1}} is non empty set
the addF of h1 . [hf1,x] is set
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
hf1 * gf1 is Element of the carrier of h1
the multF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the multF of h1 . [hf1,gf1] is set
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
x is left_complementable right_complementable complementable Element of the carrier of g
gPh1 * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (gPh1,x) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,x] is set
{gPh1,x} is non empty set
{gPh1} is non empty set
{{gPh1,x},{gPh1}} is non empty set
the multF of g . [gPh1,x] is set
gf1 * hf1 is Element of the carrier of h1
the multF of h1 . (gf1,hf1) is Element of the carrier of h1
[gf1,hf1] is set
{gf1,hf1} is non empty set
{gf1} is non empty set
{{gf1,hf1},{gf1}} is non empty set
the multF of h1 . [gf1,hf1] is set
x * gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,gPh1) is left_complementable right_complementable complementable Element of the carrier of g
[x,gPh1] is set
{x,gPh1} is non empty set
{x} is non empty set
{{x,gPh1},{x}} is non empty set
the multF of g . [x,gPh1] is set
gPh1 is Element of the carrier of h1
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
gf1 * gPh1 is Element of the carrier of h1
the multF of h1 . (gf1,gPh1) is Element of the carrier of h1
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the multF of h1 . [gf1,gPh1] is set
hf1 * (gf1 * gPh1) is Element of the carrier of h1
the multF of h1 . (hf1,(gf1 * gPh1)) is Element of the carrier of h1
[hf1,(gf1 * gPh1)] is set
{hf1,(gf1 * gPh1)} is non empty set
{hf1} is non empty set
{{hf1,(gf1 * gPh1)},{hf1}} is non empty set
the multF of h1 . [hf1,(gf1 * gPh1)] is set
the multF of g . (hf1,(gf1 * gPh1)) is set
the multF of g . [hf1,(gf1 * gPh1)] is set
a is left_complementable right_complementable complementable Element of the carrier of g
bb is left_complementable right_complementable complementable Element of the carrier of g
x is left_complementable right_complementable complementable Element of the carrier of g
bb * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (bb,x) is left_complementable right_complementable complementable Element of the carrier of g
[bb,x] is set
{bb,x} is non empty set
{bb} is non empty set
{{bb,x},{bb}} is non empty set
the multF of g . [bb,x] is set
a * (bb * x) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,(bb * x)) is left_complementable right_complementable complementable Element of the carrier of g
[a,(bb * x)] is set
{a,(bb * x)} is non empty set
{a} is non empty set
{{a,(bb * x)},{a}} is non empty set
the multF of g . [a,(bb * x)] is set
hf1 * gf1 is Element of the carrier of h1
the multF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the multF of h1 . [hf1,gf1] is set
a * bb is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (a,bb) is left_complementable right_complementable complementable Element of the carrier of g
[a,bb] is set
{a,bb} is non empty set
{{a,bb},{a}} is non empty set
the multF of g . [a,bb] is set
(hf1 * gf1) * gPh1 is Element of the carrier of h1
the multF of h1 . ((hf1 * gf1),gPh1) is Element of the carrier of h1
[(hf1 * gf1),gPh1] is set
{(hf1 * gf1),gPh1} is non empty set
{(hf1 * gf1)} is non empty set
{{(hf1 * gf1),gPh1},{(hf1 * gf1)}} is non empty set
the multF of h1 . [(hf1 * gf1),gPh1] is set
(a * bb) * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . ((a * bb),x) is left_complementable right_complementable complementable Element of the carrier of g
[(a * bb),x] is set
{(a * bb),x} is non empty set
{(a * bb)} is non empty set
{{(a * bb),x},{(a * bb)}} is non empty set
the multF of g . [(a * bb),x] is set
hf1 is Element of the carrier of h1
1. h1 is Element of the carrier of h1
the OneF of h1 is Element of the carrier of h1
hf1 * (1. h1) is Element of the carrier of h1
the multF of h1 . (hf1,(1. h1)) is Element of the carrier of h1
[hf1,(1. h1)] is set
{hf1,(1. h1)} is non empty set
{hf1} is non empty set
{{hf1,(1. h1)},{hf1}} is non empty set
the multF of h1 . [hf1,(1. h1)] is set
gf1 is left_complementable right_complementable complementable Element of the carrier of g
gf1 * (1. g) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (gf1,(1. g)) is left_complementable right_complementable complementable Element of the carrier of g
[gf1,(1. g)] is set
{gf1,(1. g)} is non empty set
{gf1} is non empty set
{{gf1,(1. g)},{gf1}} is non empty set
the multF of g . [gf1,(1. g)] is set
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
gPh1 is Element of the carrier of h1
gf1 + gPh1 is Element of the carrier of h1
the addF of h1 . (gf1,gPh1) is Element of the carrier of h1
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the addF of h1 . [gf1,gPh1] is set
a is left_complementable right_complementable complementable Element of the carrier of g
bb is left_complementable right_complementable complementable Element of the carrier of g
a + bb is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (a,bb) is left_complementable right_complementable complementable Element of the carrier of g
[a,bb] is set
{a,bb} is non empty set
{a} is non empty set
{{a,bb},{a}} is non empty set
the addF of g . [a,bb] is set
hf1 * (gf1 + gPh1) is Element of the carrier of h1
the multF of h1 . (hf1,(gf1 + gPh1)) is Element of the carrier of h1
[hf1,(gf1 + gPh1)] is set
{hf1,(gf1 + gPh1)} is non empty set
{hf1} is non empty set
{{hf1,(gf1 + gPh1)},{hf1}} is non empty set
the multF of h1 . [hf1,(gf1 + gPh1)] is set
x is left_complementable right_complementable complementable Element of the carrier of g
x * (a + bb) is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,(a + bb)) is left_complementable right_complementable complementable Element of the carrier of g
[x,(a + bb)] is set
{x,(a + bb)} is non empty set
{x} is non empty set
{{x,(a + bb)},{x}} is non empty set
the multF of g . [x,(a + bb)] is set
x * a is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{{x,a},{x}} is non empty set
the multF of g . [x,a] is set
x * bb is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (x,bb) is left_complementable right_complementable complementable Element of the carrier of g
[x,bb] is set
{x,bb} is non empty set
{{x,bb},{x}} is non empty set
the multF of g . [x,bb] is set
(x * a) + (x * bb) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((x * a),(x * bb)) is left_complementable right_complementable complementable Element of the carrier of g
[(x * a),(x * bb)] is set
{(x * a),(x * bb)} is non empty set
{(x * a)} is non empty set
{{(x * a),(x * bb)},{(x * a)}} is non empty set
the addF of g . [(x * a),(x * bb)] is set
hf1 * gf1 is Element of the carrier of h1
the multF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the multF of h1 . [hf1,gf1] is set
hf1 * gPh1 is Element of the carrier of h1
the multF of h1 . (hf1,gPh1) is Element of the carrier of h1
[hf1,gPh1] is set
{hf1,gPh1} is non empty set
{{hf1,gPh1},{hf1}} is non empty set
the multF of h1 . [hf1,gPh1] is set
(hf1 * gf1) + (hf1 * gPh1) is Element of the carrier of h1
the addF of h1 . ((hf1 * gf1),(hf1 * gPh1)) is Element of the carrier of h1
[(hf1 * gf1),(hf1 * gPh1)] is set
{(hf1 * gf1),(hf1 * gPh1)} is non empty set
{(hf1 * gf1)} is non empty set
{{(hf1 * gf1),(hf1 * gPh1)},{(hf1 * gf1)}} is non empty set
the addF of h1 . [(hf1 * gf1),(hf1 * gPh1)] is set
hf1 is V11() real ext-real set
gf1 is Element of the carrier of h1
gPh1 is Element of the carrier of h1
gf1 + gPh1 is Element of the carrier of h1
the addF of h1 . (gf1,gPh1) is Element of the carrier of h1
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty set
{{gf1,gPh1},{gf1}} is non empty set
the addF of h1 . [gf1,gPh1] is set
hf1 * (gf1 + gPh1) is Element of the carrier of h1
the Mult of h1 . (hf1,(gf1 + gPh1)) is set
[hf1,(gf1 + gPh1)] is set
{hf1,(gf1 + gPh1)} is non empty set
{hf1} is non empty V177() V178() V179() set
{{hf1,(gf1 + gPh1)},{hf1}} is non empty set
the Mult of h1 . [hf1,(gf1 + gPh1)] is set
hf1 * gf1 is Element of the carrier of h1
the Mult of h1 . (hf1,gf1) is set
[hf1,gf1] is set
{hf1,gf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the Mult of h1 . [hf1,gf1] is set
hf1 * gPh1 is Element of the carrier of h1
the Mult of h1 . (hf1,gPh1) is set
[hf1,gPh1] is set
{hf1,gPh1} is non empty set
{{hf1,gPh1},{hf1}} is non empty set
the Mult of h1 . [hf1,gPh1] is set
(hf1 * gf1) + (hf1 * gPh1) is Element of the carrier of h1
the addF of h1 . ((hf1 * gf1),(hf1 * gPh1)) is Element of the carrier of h1
[(hf1 * gf1),(hf1 * gPh1)] is set
{(hf1 * gf1),(hf1 * gPh1)} is non empty set
{(hf1 * gf1)} is non empty set
{{(hf1 * gf1),(hf1 * gPh1)},{(hf1 * gf1)}} is non empty set
the addF of h1 . [(hf1 * gf1),(hf1 * gPh1)] is set
bb is V11() real ext-real Element of REAL
bb * gf1 is Element of the carrier of h1
the Mult of h1 . (bb,gf1) is set
[bb,gf1] is set
{bb,gf1} is non empty set
{bb} is non empty V177() V178() V179() set
{{bb,gf1},{bb}} is non empty set
the Mult of h1 . [bb,gf1] is set
x is left_complementable right_complementable complementable Element of the carrier of g
bb * x is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (bb,x) is set
[bb,x] is set
{bb,x} is non empty set
{{bb,x},{bb}} is non empty set
the Mult of g . [bb,x] is set
a is left_complementable right_complementable complementable Element of the carrier of g
x + a is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . (x,a) is left_complementable right_complementable complementable Element of the carrier of g
[x,a] is set
{x,a} is non empty set
{x} is non empty set
{{x,a},{x}} is non empty set
the addF of g . [x,a] is set
bb * (gf1 + gPh1) is Element of the carrier of h1
the Mult of h1 . (bb,(gf1 + gPh1)) is set
[bb,(gf1 + gPh1)] is set
{bb,(gf1 + gPh1)} is non empty set
{{bb,(gf1 + gPh1)},{bb}} is non empty set
the Mult of h1 . [bb,(gf1 + gPh1)] is set
bb * (x + a) is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (bb,(x + a)) is set
[bb,(x + a)] is set
{bb,(x + a)} is non empty set
{{bb,(x + a)},{bb}} is non empty set
the Mult of g . [bb,(x + a)] is set
hf1 * x is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (hf1,x) is set
[hf1,x] is set
{hf1,x} is non empty set
{{hf1,x},{hf1}} is non empty set
the Mult of g . [hf1,x] is set
hf1 * a is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (hf1,a) is set
[hf1,a] is set
{hf1,a} is non empty set
{{hf1,a},{hf1}} is non empty set
the Mult of g . [hf1,a] is set
(hf1 * x) + (hf1 * a) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((hf1 * x),(hf1 * a)) is left_complementable right_complementable complementable Element of the carrier of g
[(hf1 * x),(hf1 * a)] is set
{(hf1 * x),(hf1 * a)} is non empty set
{(hf1 * x)} is non empty set
{{(hf1 * x),(hf1 * a)},{(hf1 * x)}} is non empty set
the addF of g . [(hf1 * x),(hf1 * a)] is set
bb * gPh1 is Element of the carrier of h1
the Mult of h1 . (bb,gPh1) is set
[bb,gPh1] is set
{bb,gPh1} is non empty set
{{bb,gPh1},{bb}} is non empty set
the Mult of h1 . [bb,gPh1] is set
bb * a is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (bb,a) is set
[bb,a] is set
{bb,a} is non empty set
{{bb,a},{bb}} is non empty set
the Mult of g . [bb,a] is set
hf1 is V11() real ext-real set
gf1 is V11() real ext-real set
hf1 + gf1 is V11() real ext-real set
a is Element of the carrier of h1
(hf1 + gf1) * a is Element of the carrier of h1
the Mult of h1 . ((hf1 + gf1),a) is set
[(hf1 + gf1),a] is set
{(hf1 + gf1),a} is non empty set
{(hf1 + gf1)} is non empty V177() V178() V179() set
{{(hf1 + gf1),a},{(hf1 + gf1)}} is non empty set
the Mult of h1 . [(hf1 + gf1),a] is set
hf1 * a is Element of the carrier of h1
the Mult of h1 . (hf1,a) is set
[hf1,a] is set
{hf1,a} is non empty set
{hf1} is non empty V177() V178() V179() set
{{hf1,a},{hf1}} is non empty set
the Mult of h1 . [hf1,a] is set
gf1 * a is Element of the carrier of h1
the Mult of h1 . (gf1,a) is set
[gf1,a] is set
{gf1,a} is non empty set
{gf1} is non empty V177() V178() V179() set
{{gf1,a},{gf1}} is non empty set
the Mult of h1 . [gf1,a] is set
(hf1 * a) + (gf1 * a) is Element of the carrier of h1
the addF of h1 . ((hf1 * a),(gf1 * a)) is Element of the carrier of h1
[(hf1 * a),(gf1 * a)] is set
{(hf1 * a),(gf1 * a)} is non empty set
{(hf1 * a)} is non empty set
{{(hf1 * a),(gf1 * a)},{(hf1 * a)}} is non empty set
the addF of h1 . [(hf1 * a),(gf1 * a)] is set
gPh1 is V11() real ext-real Element of REAL
gPh1 * a is Element of the carrier of h1
the Mult of h1 . (gPh1,a) is set
[gPh1,a] is set
{gPh1,a} is non empty set
{gPh1} is non empty V177() V178() V179() set
{{gPh1,a},{gPh1}} is non empty set
the Mult of h1 . [gPh1,a] is set
bb is left_complementable right_complementable complementable Element of the carrier of g
gPh1 * bb is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (gPh1,bb) is set
[gPh1,bb] is set
{gPh1,bb} is non empty set
{{gPh1,bb},{gPh1}} is non empty set
the Mult of g . [gPh1,bb] is set
x is V11() real ext-real Element of REAL
x * a is Element of the carrier of h1
the Mult of h1 . (x,a) is set
[x,a] is set
{x,a} is non empty set
{x} is non empty V177() V178() V179() set
{{x,a},{x}} is non empty set
the Mult of h1 . [x,a] is set
x * bb is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (x,bb) is set
[x,bb] is set
{x,bb} is non empty set
{{x,bb},{x}} is non empty set
the Mult of g . [x,bb] is set
gPh1 + x is V11() real ext-real Element of REAL
(gPh1 + x) * a is Element of the carrier of h1
the Mult of h1 . ((gPh1 + x),a) is set
[(gPh1 + x),a] is set
{(gPh1 + x),a} is non empty set
{(gPh1 + x)} is non empty V177() V178() V179() set
{{(gPh1 + x),a},{(gPh1 + x)}} is non empty set
the Mult of h1 . [(gPh1 + x),a] is set
(hf1 + gf1) * bb is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . ((hf1 + gf1),bb) is set
[(hf1 + gf1),bb] is set
{(hf1 + gf1),bb} is non empty set
{{(hf1 + gf1),bb},{(hf1 + gf1)}} is non empty set
the Mult of g . [(hf1 + gf1),bb] is set
hf1 * bb is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (hf1,bb) is set
[hf1,bb] is set
{hf1,bb} is non empty set
{{hf1,bb},{hf1}} is non empty set
the Mult of g . [hf1,bb] is set
gf1 * bb is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (gf1,bb) is set
[gf1,bb] is set
{gf1,bb} is non empty set
{{gf1,bb},{gf1}} is non empty set
the Mult of g . [gf1,bb] is set
(hf1 * bb) + (gf1 * bb) is left_complementable right_complementable complementable Element of the carrier of g
the addF of g . ((hf1 * bb),(gf1 * bb)) is left_complementable right_complementable complementable Element of the carrier of g
[(hf1 * bb),(gf1 * bb)] is set
{(hf1 * bb),(gf1 * bb)} is non empty set
{(hf1 * bb)} is non empty set
{{(hf1 * bb),(gf1 * bb)},{(hf1 * bb)}} is non empty set
the addF of g . [(hf1 * bb),(gf1 * bb)] is set
hf1 is V11() real ext-real set
gf1 is V11() real ext-real set
hf1 * gf1 is V11() real ext-real set
gPh1 is Element of the carrier of h1
(hf1 * gf1) * gPh1 is Element of the carrier of h1
the Mult of h1 . ((hf1 * gf1),gPh1) is set
[(hf1 * gf1),gPh1] is set
{(hf1 * gf1),gPh1} is non empty set
{(hf1 * gf1)} is non empty V177() V178() V179() set
{{(hf1 * gf1),gPh1},{(hf1 * gf1)}} is non empty set
the Mult of h1 . [(hf1 * gf1),gPh1] is set
gf1 * gPh1 is Element of the carrier of h1
the Mult of h1 . (gf1,gPh1) is set
[gf1,gPh1] is set
{gf1,gPh1} is non empty set
{gf1} is non empty V177() V178() V179() set
{{gf1,gPh1},{gf1}} is non empty set
the Mult of h1 . [gf1,gPh1] is set
hf1 * (gf1 * gPh1) is Element of the carrier of h1
the Mult of h1 . (hf1,(gf1 * gPh1)) is set
[hf1,(gf1 * gPh1)] is set
{hf1,(gf1 * gPh1)} is non empty set
{hf1} is non empty V177() V178() V179() set
{{hf1,(gf1 * gPh1)},{hf1}} is non empty set
the Mult of h1 . [hf1,(gf1 * gPh1)] is set
bb is V11() real ext-real Element of REAL
bb * gPh1 is Element of the carrier of h1
the Mult of h1 . (bb,gPh1) is set
[bb,gPh1] is set
{bb,gPh1} is non empty set
{bb} is non empty V177() V178() V179() set
{{bb,gPh1},{bb}} is non empty set
the Mult of h1 . [bb,gPh1] is set
x is left_complementable right_complementable complementable Element of the carrier of g
bb * x is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (bb,x) is set
[bb,x] is set
{bb,x} is non empty set
{{bb,x},{bb}} is non empty set
the Mult of g . [bb,x] is set
a is V11() real ext-real Element of REAL
a * bb is V11() real ext-real Element of REAL
(a * bb) * gPh1 is Element of the carrier of h1
the Mult of h1 . ((a * bb),gPh1) is set
[(a * bb),gPh1] is set
{(a * bb),gPh1} is non empty set
{(a * bb)} is non empty V177() V178() V179() set
{{(a * bb),gPh1},{(a * bb)}} is non empty set
the Mult of h1 . [(a * bb),gPh1] is set
(hf1 * gf1) * x is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . ((hf1 * gf1),x) is set
[(hf1 * gf1),x] is set
{(hf1 * gf1),x} is non empty set
{{(hf1 * gf1),x},{(hf1 * gf1)}} is non empty set
the Mult of g . [(hf1 * gf1),x] is set
hf1 * (bb * x) is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (hf1,(bb * x)) is set
[hf1,(bb * x)] is set
{hf1,(bb * x)} is non empty set
{{hf1,(bb * x)},{hf1}} is non empty set
the Mult of g . [hf1,(bb * x)] is set
hf1 is Element of the carrier of h1
gf1 is Element of the carrier of h1
hf1 * gf1 is Element of the carrier of h1
the multF of h1 . (hf1,gf1) is Element of the carrier of h1
[hf1,gf1] is set
{hf1,gf1} is non empty set
{hf1} is non empty set
{{hf1,gf1},{hf1}} is non empty set
the multF of h1 . [hf1,gf1] is set
a is V11() real ext-real Element of REAL
a * (hf1 * gf1) is Element of the carrier of h1
the Mult of h1 . (a,(hf1 * gf1)) is set
[a,(hf1 * gf1)] is set
{a,(hf1 * gf1)} is non empty set
{a} is non empty V177() V178() V179() set
{{a,(hf1 * gf1)},{a}} is non empty set
the Mult of h1 . [a,(hf1 * gf1)] is set
a * hf1 is Element of the carrier of h1
the Mult of h1 . (a,hf1) is set
[a,hf1] is set
{a,hf1} is non empty set
{{a,hf1},{a}} is non empty set
the Mult of h1 . [a,hf1] is set
(a * hf1) * gf1 is Element of the carrier of h1
the multF of h1 . ((a * hf1),gf1) is Element of the carrier of h1
[(a * hf1),gf1] is set
{(a * hf1),gf1} is non empty set
{(a * hf1)} is non empty set
{{(a * hf1),gf1},{(a * hf1)}} is non empty set
the multF of h1 . [(a * hf1),gf1] is set
gPh1 is left_complementable right_complementable complementable Element of the carrier of g
a * gPh1 is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (a,gPh1) is set
[a,gPh1] is set
{a,gPh1} is non empty set
{{a,gPh1},{a}} is non empty set
the Mult of g . [a,gPh1] is set
x is left_complementable right_complementable complementable Element of the carrier of g
gPh1 * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . (gPh1,x) is left_complementable right_complementable complementable Element of the carrier of g
[gPh1,x] is set
{gPh1,x} is non empty set
{gPh1} is non empty set
{{gPh1,x},{gPh1}} is non empty set
the multF of g . [gPh1,x] is set
a * (gPh1 * x) is left_complementable right_complementable complementable Element of the carrier of g
the Mult of g . (a,(gPh1 * x)) is set
[a,(gPh1 * x)] is set
{a,(gPh1 * x)} is non empty set
{{a,(gPh1 * x)},{a}} is non empty set
the Mult of g . [a,(gPh1 * x)] is set
(a * gPh1) * x is left_complementable right_complementable complementable Element of the carrier of g
the multF of g . ((a * gPh1),x) is left_complementable right_complementable complementable Element of the carrier of g
[(a * gPh1),x] is set
{(a * gPh1),x} is non empty set
{(a * gPh1)} is non empty set
{{(a * gPh1),x},{(a * gPh1)}} is non empty set
the multF of g . [(a * gPh1),x] is set
0. h1 is V49(h1) Element of the carrier of h1
the ZeroF of h1 is Element of the carrier of h1
1. h1 is Element of the carrier of h1
the OneF of h1 is Element of the carrier of h1
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
[#] X is non empty non proper add-closed (X) (X) Element of bool the carrier of X
the carrier of X is non empty set
bool the carrier of X is non empty set
[:REAL, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
[:REAL,([#] X):] is non empty set
the Mult of X | [:REAL,([#] X):] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X || ([#] X) is Relation-like Function-like set
[:([#] X),([#] X):] is non empty set
the addF of X | [:([#] X),([#] X):] is Relation-like set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || ([#] X) is Relation-like Function-like set
the multF of X | [:([#] X),([#] X):] is Relation-like 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
0. X is V49(X) 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
AlgebraStr(# the carrier of X, the multF of X, the addF of X, the Mult of X,(1. X),(0. X) #) is strict AlgebraStr
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
bool the carrier of X is non empty set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
B is Element of bool the carrier of X
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
[:REAL, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
[:REAL,B:] is set
the Mult of X | [:REAL,B:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL,B:],B:] is set
bool [:[:REAL,B:],B:] is non empty set
dom the Mult of X is Relation-like REAL -defined the carrier of X -valued Element of bool [:REAL, the carrier of X:]
bool [:REAL, the carrier of X:] is non empty set
B is set
( the Mult of X | [:REAL,B:]) . B is set
ONE is set
f is set
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
g is V11() real ext-real Element of REAL
[g,f] is set
{g,f} is non empty set
{g} is non empty V177() V178() V179() set
{{g,f},{g}} is non empty set
dom ( the Mult of X | [:REAL,B:]) is Relation-like REAL -defined the carrier of X -valued Element of bool [:REAL, the carrier of X:]
h is left_complementable right_complementable complementable Element of the carrier of X
g * h is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (g,h) is set
[g,h] is set
{g,h} is non empty set
{{g,h},{g}} is non empty set
the Mult of X . [g,h] is set
dom ( the Mult of X | [:REAL,B:]) is Relation-like REAL -defined the carrier of X -valued Element of bool [:REAL, the carrier of X:]
X is non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative vector-associative AlgebraStr
0. X is V49(X) right_complementable Element of the carrier of X
the carrier of X is non empty set
the ZeroF of X is right_complementable Element of the carrier of X
B is V11() real ext-real Element of REAL
B * (0. X) is right_complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (B,(0. X)) is set
[B,(0. X)] is set
{B,(0. X)} is non empty set
{B} is non empty V177() V178() V179() set
{{B,(0. X)},{B}} is non empty set
the Mult of X . [B,(0. X)] is set
(B * (0. X)) + (B * (0. X)) is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((B * (0. X)),(B * (0. X))) is right_complementable Element of the carrier of X
[(B * (0. X)),(B * (0. X))] is set
{(B * (0. X)),(B * (0. X))} is non empty set
{(B * (0. X))} is non empty set
{{(B * (0. X)),(B * (0. X))},{(B * (0. X))}} is non empty set
the addF of X . [(B * (0. X)),(B * (0. X))] is set
(0. X) + (0. X) is right_complementable Element of the carrier of X
the addF of X . ((0. X),(0. X)) is right_complementable Element of the carrier of X
[(0. X),(0. X)] is set
{(0. X),(0. X)} is non empty set
{(0. X)} is non empty set
{{(0. X),(0. X)},{(0. X)}} is non empty set
the addF of X . [(0. X),(0. X)] is set
B * ((0. X) + (0. X)) is right_complementable Element of the carrier of X
the Mult of X . (B,((0. X) + (0. X))) is set
[B,((0. X) + (0. X))] is set
{B,((0. X) + (0. X))} is non empty set
{{B,((0. X) + (0. X))},{B}} is non empty set
the Mult of X . [B,((0. X) + (0. X))] is set
(B * (0. X)) + (0. X) is right_complementable Element of the carrier of X
the addF of X . ((B * (0. X)),(0. X)) is right_complementable Element of the carrier of X
[(B * (0. X)),(0. X)] is set
{(B * (0. X)),(0. X)} is non empty set
{{(B * (0. X)),(0. X)},{(B * (0. X))}} is non empty set
the addF of X . [(B * (0. X)),(0. X)] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative AlgebraStr
the carrier of X is non empty set
B is left_complementable right_complementable complementable Element of the carrier of X
1 * B is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (1,B) is set
[1,B] is set
{1,B} is non empty set
{1} is non empty V177() V178() V179() V180() V181() V182() set
{{1,B},{1}} is non empty set
the Mult of X . [1,B] is set
- B is left_complementable right_complementable complementable Element of the carrier of X
1 * (- B) is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (1,(- B)) is set
[1,(- B)] is set
{1,(- B)} is non empty set
{{1,(- B)},{1}} is non empty set
the Mult of X . [1,(- B)] is set
(1 * B) + (1 * (- B)) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((1 * B),(1 * (- B))) is left_complementable right_complementable complementable Element of the carrier of X
[(1 * B),(1 * (- B))] is set
{(1 * B),(1 * (- B))} is non empty set
{(1 * B)} is non empty set
{{(1 * B),(1 * (- B))},{(1 * B)}} is non empty set
the addF of X . [(1 * B),(1 * (- B))] is set
B + (- B) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X . (B,(- B)) is left_complementable right_complementable complementable Element of the carrier of X
[B,(- B)] is set
{B,(- B)} is non empty set
{B} is non empty set
{{B,(- B)},{B}} is non empty set
the addF of X . [B,(- B)] is set
1 * (B + (- B)) is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (1,(B + (- B))) is set
[1,(B + (- B))] is set
{1,(B + (- B))} is non empty set
{{1,(B + (- B))},{1}} is non empty set
the Mult of X . [1,(B + (- B))] is set
(1 * B) - (1 * B) is left_complementable right_complementable complementable Element of the carrier of X
- (1 * B) is left_complementable right_complementable complementable Element of the carrier of X
(1 * B) + (- (1 * B)) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X . ((1 * B),(- (1 * B))) is left_complementable right_complementable complementable Element of the carrier of X
[(1 * B),(- (1 * B))] is set
{(1 * B),(- (1 * B))} is non empty set
{{(1 * B),(- (1 * B))},{(1 * B)}} is non empty set
the addF of X . [(1 * B),(- (1 * B))] is set
0. X is V49(X) 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
1 * (0. X) is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (1,(0. X)) is set
[1,(0. X)] is set
{1,(0. X)} is non empty set
{{1,(0. X)},{1}} is non empty set
the Mult of X . [1,(0. X)] is set
1 * 1 is V11() real ext-real non negative Element of REAL
(1 * 1) * B is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . ((1 * 1),B) is set
[(1 * 1),B] is set
{(1 * 1),B} is non empty set
{(1 * 1)} is non empty V177() V178() V179() set
{{(1 * 1),B},{(1 * 1)}} is non empty set
the Mult of X . [(1 * 1),B] is set
1 * (1 * B) is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (1,(1 * B)) is set
[1,(1 * B)] is set
{1,(1 * B)} is non empty set
{{1,(1 * B)},{1}} is non empty set
the Mult of X . [1,(1 * B)] is set
(1 * (1 * B)) - (1 * B) is left_complementable right_complementable complementable Element of the carrier of X
(1 * (1 * B)) + (- (1 * B)) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X . ((1 * (1 * B)),(- (1 * B))) is left_complementable right_complementable complementable Element of the carrier of X
[(1 * (1 * B)),(- (1 * B))] is set
{(1 * (1 * B)),(- (1 * B))} is non empty set
{(1 * (1 * B))} is non empty set
{{(1 * (1 * B)),(- (1 * B))},{(1 * (1 * B))}} is non empty set
the addF of X . [(1 * (1 * B)),(- (1 * B))] is set
(1 * B) - B is left_complementable right_complementable complementable Element of the carrier of X
(1 * B) + (- B) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X . ((1 * B),(- B)) is left_complementable right_complementable complementable Element of the carrier of X
[(1 * B),(- B)] is set
{(1 * B),(- B)} is non empty set
{{(1 * B),(- B)},{(1 * B)}} is non empty set
the addF of X . [(1 * B),(- B)] is set
1 * ((1 * B) - B) is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (1,((1 * B) - B)) is set
[1,((1 * B) - B)] is set
{1,((1 * B) - B)} is non empty set
{{1,((1 * B) - B)},{1}} is non empty set
the Mult of X . [1,((1 * B) - B)] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative AlgebraStr
the carrier of X is non empty set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
bool the carrier of X is non empty set
B is Element of bool the carrier of X
(X,B) is Relation-like [:B,B:] -defined B -valued Function-like quasi_total Element of bool [:[:B,B:],B:]
[:B,B:] is set
[:[:B,B:],B:] is set
bool [:[:B,B:],B:] is non empty set
(X,B) is Relation-like [:B,B:] -defined B -valued Function-like quasi_total Element of bool [:[:B,B:],B:]
(X,B) is Relation-like [:REAL,B:] -defined B -valued Function-like quasi_total Element of bool [:[:REAL,B:],B:]
[:REAL,B:] is set
[:[:REAL,B:],B:] is set
bool [:[:REAL,B:],B:] is non empty set
(X,B) is Element of B
(X,B) is Element of B
AlgebraStr(# B,(X,B),(X,B),(X,B),(X,B),(X,B) #) is strict AlgebraStr
[:REAL, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:REAL,B:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
1_ X is left_complementable right_complementable complementable Element of the carrier of X
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
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || B is Relation-like Function-like set
the multF of X | [:B,B:] is Relation-like set
0. X is V49(X) 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
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the addF of X || B is Relation-like Function-like set
the addF of X | [:B,B:] is Relation-like set
X is non empty set
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
X is non empty set
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
B is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
1 * B is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:REAL, the carrier of (RAlgebra X):] is non empty set
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) . (1,B) is set
[1,B] is set
{1,B} is non empty set
{1} is non empty V177() V178() V179() V180() V181() V182() set
{{1,B},{1}} is non empty set
the Mult of (RAlgebra X) . [1,B] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
the carrier of X is non empty set
1_ X is left_complementable right_complementable complementable Element of the carrier of X
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
0. X is V49(X) 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
B is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital (X)
the carrier of B is non empty set
1_ B is left_complementable right_complementable complementable Element of the carrier of B
1. B is left_complementable right_complementable complementable Element of the carrier of B
the OneF of B is left_complementable right_complementable complementable Element of the carrier of B
0. B is V49(B) left_complementable right_complementable complementable Element of the carrier of B
the ZeroF of B is left_complementable right_complementable complementable Element of the carrier of B
B is left_complementable right_complementable complementable Element of the carrier of B
f is left_complementable right_complementable complementable Element of the carrier of X
ONE is left_complementable right_complementable complementable Element of the carrier of B
g is left_complementable right_complementable complementable Element of the carrier of X
B + ONE is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (B,ONE) is left_complementable right_complementable complementable Element of the carrier of B
[B,ONE] is set
{B,ONE} is non empty set
{B} is non empty set
{{B,ONE},{B}} is non empty set
the addF of B . [B,ONE] is set
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X || the carrier of B is Relation-like Function-like set
the addF of X | [: the carrier of B, the carrier of B:] is Relation-like set
[B,ONE] is Element of [: the carrier of B, the carrier of B:]
( the addF of X || the carrier of B) . [B,ONE] is set
f + g is left_complementable right_complementable complementable Element of the carrier of X
the addF of X . (f,g) is left_complementable right_complementable complementable Element of the carrier of X
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of X . [f,g] is set
B is left_complementable right_complementable complementable Element of the carrier of B
f is left_complementable right_complementable complementable Element of the carrier of X
ONE is left_complementable right_complementable complementable Element of the carrier of B
g is left_complementable right_complementable complementable Element of the carrier of X
B * ONE is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
the multF of B . (B,ONE) is left_complementable right_complementable complementable Element of the carrier of B
[B,ONE] is set
{B,ONE} is non empty set
{B} is non empty set
{{B,ONE},{B}} is non empty set
the multF of B . [B,ONE] is set
the multF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V23([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of B is Relation-like Function-like set
the multF of X | [: the carrier of B, the carrier of B:] is Relation-like set
[B,ONE] is Element of [: the carrier of B, the carrier of B:]
( the multF of X || the carrier of B) . [B,ONE] is set
f * g is left_complementable right_complementable complementable Element of the carrier of X
the multF of X . (f,g) is left_complementable right_complementable complementable Element of the carrier of X
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the multF of X . [f,g] is set
B is left_complementable right_complementable complementable Element of the carrier of B
ONE is left_complementable right_complementable complementable Element of the carrier of X
f is V11() real ext-real Element of REAL
f * B is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (f,B) is set
[f,B] is set
{f,B} is non empty set
{f} is non empty V177() V178() V179() set
{{f,B},{f}} is non empty set
the Mult of B . [f,B] is set
[:REAL, the carrier of X:] is non empty set
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V23([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:[:REAL, the carrier of X:], the carrier of X:] is non empty set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X | [:REAL, the carrier of B:] is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[f,B] is Element of [:REAL, the carrier of B:]
( the Mult of X | [:REAL, the carrier of B:]) . [f,B] is set
f * ONE is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (f,ONE) is set
[f,ONE] is set
{f,ONE} is non empty set
{{f,ONE},{f}} is non empty set
the Mult of X . [f,ONE] is set
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
B is set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
X is non empty set
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
(X) is non empty Element of bool the carrier of (RAlgebra X)
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
ONE is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
ONE * f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) is non empty Relation-like [: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([: the carrier of (RAlgebra X), the carrier of (RAlgebra X):]) quasi_total Element of bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] is non empty set
[:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the multF of (RAlgebra X) . (ONE,f) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the multF of (RAlgebra X) . [ONE,f] is set
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f1 (#) g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
dom (f1 (#) g) is Element of bool X
bool X is non empty set
X /\ X is set
(f1 (#) g) | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
dom f1 is Element of bool X
dom g is Element of bool X
(dom f1) /\ (dom g) is Element of bool X
X /\ (dom g) is Element of bool X
h is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
dom h is Element of bool X
h1 is set
h . h1 is V11() real ext-real Element of REAL
f1 . h1 is V11() real ext-real Element of REAL
g . h1 is V11() real ext-real Element of REAL
(f1 . h1) * (g . h1) is V11() real ext-real Element of REAL
h1 is Relation-like Function-like set
dom h1 is set
rng h1 is set
f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
g is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f + g is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the addF of (RAlgebra X) is non empty Relation-like [: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([: the carrier of (RAlgebra X), the carrier of (RAlgebra X):]) quasi_total Element of bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] is non empty set
[:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the addF of (RAlgebra X) . (f,g) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of (RAlgebra X) . [f,g] is set
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h1 + h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
dom (h1 + h) is Element of bool X
bool X is non empty set
X /\ X is set
(h1 + h) | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
dom h1 is Element of bool X
dom h is Element of bool X
(dom h1) /\ (dom h) is Element of bool X
X /\ (dom h) is Element of bool X
f1 is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
dom f1 is Element of bool X
g1 is set
f1 . g1 is V11() real ext-real Element of REAL
h1 . g1 is V11() real ext-real Element of REAL
h . g1 is V11() real ext-real Element of REAL
(h1 . g1) + (h . g1) is V11() real ext-real Element of REAL
g1 is Relation-like Function-like set
dom g1 is set
rng g1 is set
f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
- f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
- g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
- 1 is V11() real ext-real non positive set
(- 1) (#) g is Relation-like X -defined Function-like V23(X) V150() V151() V152() set
(- g) | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
(- 1) * f is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:REAL, the carrier of (RAlgebra X):] is non empty set
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) . ((- 1),f) is set
[(- 1),f] is set
{(- 1),f} is non empty set
{(- 1)} is non empty V177() V178() V179() set
{{(- 1),f},{(- 1)}} is non empty set
the Mult of (RAlgebra X) . [(- 1),f] is set
h1 is set
dom h is Element of bool X
bool X is non empty set
h . h1 is V11() real ext-real Element of REAL
f1 is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
g1 is Element of X
f1 . g1 is V11() real ext-real Element of REAL
(- 1) * (f1 . g1) is V11() real ext-real Element of REAL
g . h1 is V11() real ext-real Element of REAL
- (g . h1) is V11() real ext-real Element of REAL
dom g is Element of bool X
h1 is Relation-like Function-like set
dom h1 is set
rng h1 is set
f is V11() real ext-real Element of REAL
g is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f * g is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:REAL, the carrier of (RAlgebra X):] is non empty set
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) . (f,g) is set
[f,g] is set
{f,g} is non empty set
{f} is non empty V177() V178() V179() set
{{f,g},{f}} is non empty set
the Mult of (RAlgebra X) . [f,g] is set
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f (#) h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(f (#) h) | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f1 is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
dom h is Element of bool X
bool X is non empty set
dom f1 is Element of bool X
h1 is set
f1 . h1 is V11() real ext-real Element of REAL
h . h1 is V11() real ext-real Element of REAL
f * (h . h1) is V11() real ext-real Element of REAL
h1 is Relation-like Function-like set
dom h1 is set
rng h1 is set
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
1. (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the OneF of (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
X is non empty set
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
(X) is non empty Element of bool the carrier of (RAlgebra X)
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
X is non empty set
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
B is left_complementable right_complementable complementable Element of the carrier of (X)
1 * B is left_complementable right_complementable complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (1,B) is set
[1,B] is set
{1,B} is non empty set
{1} is non empty V177() V178() V179() V180() V181() V182() set
{{1,B},{1}} is non empty set
the Mult of (X) . [1,B] is set
B is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
1 * B is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:REAL, the carrier of (RAlgebra X):] is non empty set
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) . (1,B) is set
[1,B] is set
{1,B} is non empty set
{{1,B},{1}} is non empty set
the Mult of (RAlgebra X) . [1,B] is set
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
B is left_complementable right_complementable complementable Element of the carrier of (X)
B is left_complementable right_complementable complementable Element of the carrier of (X)
B + B is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (B,B) is left_complementable right_complementable complementable Element of the carrier of (X)
[B,B] is set
{B,B} is non empty set
{B} is non empty set
{{B,B},{B}} is non empty set
the addF of (X) . [B,B] is set
ONE is left_complementable right_complementable complementable Element of the carrier of (X)
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
h1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f1 + h1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the addF of (RAlgebra X) is non empty Relation-like [: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([: the carrier of (RAlgebra X), the carrier of (RAlgebra X):]) quasi_total Element of bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] is non empty set
[:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the addF of (RAlgebra X) . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the addF of (RAlgebra X) . [f1,h1] is set
F1 is Element of X
h . F1 is V11() real ext-real Element of REAL
f . F1 is V11() real ext-real Element of REAL
g . F1 is V11() real ext-real Element of REAL
(f . F1) + (g . F1) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
B is V11() real ext-real Element of REAL
B is left_complementable right_complementable complementable Element of the carrier of (X)
B * B is left_complementable right_complementable complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (B,B) is set
[B,B] is set
{B,B} is non empty set
{B} is non empty V177() V178() V179() set
{{B,B},{B}} is non empty set
the Mult of (X) . [B,B] is set
ONE is left_complementable right_complementable complementable Element of the carrier of (X)
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
h is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
B * h is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:REAL, the carrier of (RAlgebra X):] is non empty set
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) . (B,h) is set
[B,h] is set
{B,h} is non empty set
{{B,h},{B}} is non empty set
the Mult of (RAlgebra X) . [B,h] is set
h1 is Element of X
g . h1 is V11() real ext-real Element of REAL
f . h1 is V11() real ext-real Element of REAL
B * (f . h1) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
B is left_complementable right_complementable complementable Element of the carrier of (X)
B is left_complementable right_complementable complementable Element of the carrier of (X)
B * B is left_complementable right_complementable complementable Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (B,B) is left_complementable right_complementable complementable Element of the carrier of (X)
[B,B] is set
{B,B} is non empty set
{B} is non empty set
{{B,B},{B}} is non empty set
the multF of (X) . [B,B] is set
ONE is left_complementable right_complementable complementable Element of the carrier of (X)
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
h1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
f1 * h1 is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) is non empty Relation-like [: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([: the carrier of (RAlgebra X), the carrier of (RAlgebra X):]) quasi_total Element of bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] is non empty set
[:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the multF of (RAlgebra X) . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the multF of (RAlgebra X) . [f1,h1] is set
F1 is Element of X
h . F1 is V11() real ext-real Element of REAL
f . F1 is V11() real ext-real Element of REAL
g . F1 is V11() real ext-real Element of REAL
(f . F1) * (g . F1) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
0. (X) is V49((X)) left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is left_complementable right_complementable complementable Element of the carrier of (X)
0. (RAlgebra X) is V49( RAlgebra X) left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the ZeroF of (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
1_ (X) is left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty 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_ (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
1. (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the OneF of (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
B is set
X is non empty set
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
the Element of X is Element of X
ONE is set
f is Element of X
B . f is V11() real ext-real Element of REAL
abs (B . f) is V11() real ext-real Element of REAL
B . the Element of X is V11() real ext-real Element of REAL
abs (B . the Element of X) is V11() real ext-real Element of REAL
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,B) is non empty V177() V178() V179() Element of bool REAL
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
dom B is Element of bool X
bool X is non empty set
X /\ (dom B) is Element of bool X
B is V11() real ext-real set
X /\ X is set
ONE is ext-real set
f is Element of X
B . f is V11() real ext-real Element of REAL
abs (B . f) is V11() real ext-real Element of REAL
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,B) is non empty V177() V178() V179() Element of bool REAL
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,B) is V11() real ext-real Element of REAL
B is V11() real ext-real Element of REAL
f is set
dom B is Element of bool X
bool X is non empty set
X /\ (dom B) is Element of bool X
B . f is V11() real ext-real Element of REAL
abs (B . f) is V11() real ext-real Element of REAL
ONE is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
B is set
(X,B) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X,(X,B)) is non empty V177() V178() V179() Element of bool REAL
{ (abs ((X,B) . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,(X,B)) is V11() real ext-real Element of REAL
B is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
B is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
B is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
ONE is set
B . ONE is V11() real ext-real Element of REAL
B . ONE is V11() real ext-real Element of REAL
(X,ONE) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X,(X,ONE)) is non empty V177() V178() V179() Element of bool REAL
{ (abs ((X,ONE) . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,(X,ONE)) is V11() real ext-real Element of REAL
dom B is Element of bool (X)
bool (X) is non empty set
dom B is Element of bool (X)
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,B) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X) . B is V11() real ext-real Element of REAL
(X,B) is non empty V177() V178() V179() Element of bool REAL
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,B) is V11() real ext-real Element of REAL
B is set
(X,B) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X,(X,B)) is non empty V177() V178() V179() Element of bool REAL
{ (abs ((X,B) . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,(X,B)) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
X is non empty set
(X) is Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
X is non empty set
(X) is non empty Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of the carrier of (X)
ONE is Element of the carrier of (X)
B * ONE is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (B,ONE) is Element of the carrier of (X)
[B,ONE] is set
{B,ONE} is non empty set
{B} is non empty set
{{B,ONE},{B}} is non empty set
the multF of (X) . [B,ONE] is set
g is Element of (X)
1_ (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
1. (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the OneF of (RAlgebra X) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
((RAlgebra X),(X)) . (g,(1_ (RAlgebra X))) is set
[g,(1_ (RAlgebra X))] is set
{g,(1_ (RAlgebra X))} is non empty set
{g} is non empty set
{{g,(1_ (RAlgebra X))},{g}} is non empty set
((RAlgebra X),(X)) . [g,(1_ (RAlgebra X))] is set
the multF of (RAlgebra X) is non empty Relation-like [: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([: the carrier of (RAlgebra X), the carrier of (RAlgebra X):]) quasi_total Element of bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):] is non empty set
[:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[: the carrier of (RAlgebra X), the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the multF of (RAlgebra X) || (X) is Relation-like Function-like set
the multF of (RAlgebra X) | [:(X),(X):] is Relation-like set
( the multF of (RAlgebra X) || (X)) . (g,(1_ (RAlgebra X))) is set
( the multF of (RAlgebra X) || (X)) . [g,(1_ (RAlgebra X))] is set
ONE * B is Element of the carrier of (X)
the multF of (X) . (ONE,B) is Element of the carrier of (X)
[ONE,B] is set
{ONE,B} is non empty set
{ONE} is non empty set
{{ONE,B},{ONE}} is non empty set
the multF of (X) . [ONE,B] is set
((RAlgebra X),(X)) . ((1_ (RAlgebra X)),g) is set
[(1_ (RAlgebra X)),g] is set
{(1_ (RAlgebra X)),g} is non empty set
{(1_ (RAlgebra X))} is non empty set
{{(1_ (RAlgebra X)),g},{(1_ (RAlgebra X))}} is non empty set
((RAlgebra X),(X)) . [(1_ (RAlgebra X)),g] is set
( the multF of (RAlgebra X) || (X)) . ((1_ (RAlgebra X)),g) is set
( the multF of (RAlgebra X) || (X)) . [(1_ (RAlgebra X)),g] is set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
1_ (X) is left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty 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)
[g,(1_ (RAlgebra X))] is Element of [:(X), the carrier of (RAlgebra X):]
[:(X), the carrier of (RAlgebra X):] is non empty set
g * (1_ (RAlgebra X)) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) . (g,(1_ (RAlgebra X))) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) . [g,(1_ (RAlgebra X))] is set
[(1_ (RAlgebra X)),g] is Element of [: the carrier of (RAlgebra X),(X):]
[: the carrier of (RAlgebra X),(X):] is non empty set
(1_ (RAlgebra X)) * g is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) . ((1_ (RAlgebra X)),g) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the multF of (RAlgebra X) . [(1_ (RAlgebra X)),g] is set
X is non empty set
(X) is non empty Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of the carrier of (X)
B is Element of the carrier of (X)
B * B is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (B,B) is Element of the carrier of (X)
[B,B] is set
{B,B} is non empty set
{B} is non empty set
{{B,B},{B}} is non empty set
the multF of (X) . [B,B] is set
ONE is Element of the carrier of (X)
B * ONE is Element of the carrier of (X)
the multF of (X) . (B,ONE) is Element of the carrier of (X)
[B,ONE] is set
{B,ONE} is non empty set
{B} is non empty set
{{B,ONE},{B}} is non empty set
the multF of (X) . [B,ONE] is set
X is Normed_AlgebraStr
the carrier of X is set
the multF of X is Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like quasi_total 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 non empty set
the addF of X is Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the Mult of X is Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like quasi_total 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 non empty set
the OneF of X is Element of the carrier of X
the ZeroF of X is Element of the carrier of X
AlgebraStr(# the carrier of X, the multF of X, the addF of X, the Mult of X, the OneF of X, the ZeroF of X #) is strict AlgebraStr
B is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
B is non empty AlgebraStr
the carrier of B is non empty set
ONE is Element of the carrier of B
1. B is Element of the carrier of B
the OneF of B is Element of the carrier of B
ONE * (1. B) is Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (ONE,(1. B)) is Element of the carrier of B
[ONE,(1. B)] is set
{ONE,(1. B)} is non empty set
{ONE} is non empty set
{{ONE,(1. B)},{ONE}} is non empty set
the multF of B . [ONE,(1. B)] is set
the carrier of B is non empty set
f is left_complementable right_complementable complementable Element of the carrier of B
1. B is left_complementable right_complementable complementable Element of the carrier of B
the OneF of B is left_complementable right_complementable complementable Element of the carrier of B
f * (1. B) is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (f,(1. B)) is left_complementable right_complementable complementable Element of the carrier of B
[f,(1. B)] is set
{f,(1. B)} is non empty set
{f} is non empty set
{{f,(1. B)},{f}} is non empty set
the multF of B . [f,(1. B)] is set
the carrier of B is non empty set
ONE is Element of the carrier of B
f is Element of the carrier of B
g is Element of the carrier of B
f + g is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (f,g) is Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of B . [f,g] is set
ONE * (f + g) is Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
the multF of B . (ONE,(f + g)) is Element of the carrier of B
[ONE,(f + g)] is set
{ONE,(f + g)} is non empty set
{ONE} is non empty set
{{ONE,(f + g)},{ONE}} is non empty set
the multF of B . [ONE,(f + g)] is set
ONE * f is Element of the carrier of B
the multF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{{ONE,f},{ONE}} is non empty set
the multF of B . [ONE,f] is set
ONE * g is Element of the carrier of B
the multF of B . (ONE,g) is Element of the carrier of B
[ONE,g] is set
{ONE,g} is non empty set
{{ONE,g},{ONE}} is non empty set
the multF of B . [ONE,g] is set
(ONE * f) + (ONE * g) is Element of the carrier of B
the addF of B . ((ONE * f),(ONE * g)) is Element of the carrier of B
[(ONE * f),(ONE * g)] is set
{(ONE * f),(ONE * g)} is non empty set
{(ONE * f)} is non empty set
{{(ONE * f),(ONE * g)},{(ONE * f)}} is non empty set
the addF of B . [(ONE * f),(ONE * g)] is set
the carrier of B is non empty set
h is left_complementable right_complementable complementable Element of the carrier of B
f1 is left_complementable right_complementable complementable Element of the carrier of B
h1 is left_complementable right_complementable complementable Element of the carrier of B
f1 + h1 is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of B
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the addF of B . [f1,h1] is set
h * (f1 + h1) is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
the multF of B . (h,(f1 + h1)) is left_complementable right_complementable complementable Element of the carrier of B
[h,(f1 + h1)] is set
{h,(f1 + h1)} is non empty set
{h} is non empty set
{{h,(f1 + h1)},{h}} is non empty set
the multF of B . [h,(f1 + h1)] is set
h * f1 is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (h,f1) is left_complementable right_complementable complementable Element of the carrier of B
[h,f1] is set
{h,f1} is non empty set
{{h,f1},{h}} is non empty set
the multF of B . [h,f1] is set
h * h1 is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (h,h1) is left_complementable right_complementable complementable Element of the carrier of B
[h,h1] is set
{h,h1} is non empty set
{{h,h1},{h}} is non empty set
the multF of B . [h,h1] is set
(h * f1) + (h * h1) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . ((h * f1),(h * h1)) is left_complementable right_complementable complementable Element of the carrier of B
[(h * f1),(h * h1)] is set
{(h * f1),(h * h1)} is non empty set
{(h * f1)} is non empty set
{{(h * f1),(h * h1)},{(h * f1)}} is non empty set
the addF of B . [(h * f1),(h * h1)] is set
the carrier of B is non empty set
ONE is Element of the carrier of B
the carrier of B is non empty set
f is left_complementable right_complementable complementable Element of the carrier of B
0. B is V49(B) left_complementable right_complementable complementable Element of the carrier of B
the ZeroF of B is left_complementable right_complementable complementable Element of the carrier of B
g is left_complementable right_complementable complementable Element of the carrier of B
f + g is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (f,g) is left_complementable right_complementable complementable Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of B . [f,g] is set
h is Element of the carrier of B
ONE + h is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (ONE,h) is Element of the carrier of B
[ONE,h] is set
{ONE,h} is non empty set
{ONE} is non empty set
{{ONE,h},{ONE}} is non empty set
the addF of B . [ONE,h] is set
0. B is V49(B) Element of the carrier of B
the ZeroF of B is Element of the carrier of B
ONE is Element of the carrier of B
f is Element of the carrier of B
ONE * f is Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the multF of B . [ONE,f] is set
g is Element of the carrier of B
(ONE * f) * g is Element of the carrier of B
the multF of B . ((ONE * f),g) is Element of the carrier of B
[(ONE * f),g] is set
{(ONE * f),g} is non empty set
{(ONE * f)} is non empty set
{{(ONE * f),g},{(ONE * f)}} is non empty set
the multF of B . [(ONE * f),g] is set
f * g is Element of the carrier of B
the multF of B . (f,g) is Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the multF of B . [f,g] is set
ONE * (f * g) is Element of the carrier of B
the multF of B . (ONE,(f * g)) is Element of the carrier of B
[ONE,(f * g)] is set
{ONE,(f * g)} is non empty set
{{ONE,(f * g)},{ONE}} is non empty set
the multF of B . [ONE,(f * g)] is set
the carrier of B is non empty set
f1 is left_complementable right_complementable complementable Element of the carrier of B
h1 is left_complementable right_complementable complementable Element of the carrier of B
f1 * h1 is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of B
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the multF of B . [f1,h1] is set
h is left_complementable right_complementable complementable Element of the carrier of B
(f1 * h1) * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . ((f1 * h1),h) is left_complementable right_complementable complementable Element of the carrier of B
[(f1 * h1),h] is set
{(f1 * h1),h} is non empty set
{(f1 * h1)} is non empty set
{{(f1 * h1),h},{(f1 * h1)}} is non empty set
the multF of B . [(f1 * h1),h] is set
h1 * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (h1,h) is left_complementable right_complementable complementable Element of the carrier of B
[h1,h] is set
{h1,h} is non empty set
{h1} is non empty set
{{h1,h},{h1}} is non empty set
the multF of B . [h1,h] is set
f1 * (h1 * h) is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (f1,(h1 * h)) is left_complementable right_complementable complementable Element of the carrier of B
[f1,(h1 * h)] is set
{f1,(h1 * h)} is non empty set
{{f1,(h1 * h)},{f1}} is non empty set
the multF of B . [f1,(h1 * h)] is set
ONE is Element of the carrier of B
f is Element of the carrier of B
ONE * f is Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the multF of B . [ONE,f] is set
f * ONE is Element of the carrier of B
the multF of B . (f,ONE) is Element of the carrier of B
[f,ONE] is set
{f,ONE} is non empty set
{f} is non empty set
{{f,ONE},{f}} is non empty set
the multF of B . [f,ONE] is set
the carrier of B is non empty set
g is left_complementable right_complementable complementable Element of the carrier of B
h is left_complementable right_complementable complementable Element of the carrier of B
g * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (g,h) is left_complementable right_complementable complementable Element of the carrier of B
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the multF of B . [g,h] is set
h * g is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (h,g) is left_complementable right_complementable complementable Element of the carrier of B
[h,g] is set
{h,g} is non empty set
{h} is non empty set
{{h,g},{h}} is non empty set
the multF of B . [h,g] is set
ONE is Element of the carrier of B
f is Element of the carrier of B
ONE + f is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the addF of B . [ONE,f] is set
g is Element of the carrier of B
(ONE + f) + g is Element of the carrier of B
the addF of B . ((ONE + f),g) is Element of the carrier of B
[(ONE + f),g] is set
{(ONE + f),g} is non empty set
{(ONE + f)} is non empty set
{{(ONE + f),g},{(ONE + f)}} is non empty set
the addF of B . [(ONE + f),g] is set
f + g is Element of the carrier of B
the addF of B . (f,g) is Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of B . [f,g] is set
ONE + (f + g) is Element of the carrier of B
the addF of B . (ONE,(f + g)) is Element of the carrier of B
[ONE,(f + g)] is set
{ONE,(f + g)} is non empty set
{{ONE,(f + g)},{ONE}} is non empty set
the addF of B . [ONE,(f + g)] is set
the carrier of B is non empty set
h is left_complementable right_complementable complementable Element of the carrier of B
f1 is left_complementable right_complementable complementable Element of the carrier of B
h + f1 is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (h,f1) is left_complementable right_complementable complementable Element of the carrier of B
[h,f1] is set
{h,f1} is non empty set
{h} is non empty set
{{h,f1},{h}} is non empty set
the addF of B . [h,f1] is set
h1 is left_complementable right_complementable complementable Element of the carrier of B
(h + f1) + h1 is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . ((h + f1),h1) is left_complementable right_complementable complementable Element of the carrier of B
[(h + f1),h1] is set
{(h + f1),h1} is non empty set
{(h + f1)} is non empty set
{{(h + f1),h1},{(h + f1)}} is non empty set
the addF of B . [(h + f1),h1] is set
f1 + h1 is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of B
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the addF of B . [f1,h1] is set
h + (f1 + h1) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . (h,(f1 + h1)) is left_complementable right_complementable complementable Element of the carrier of B
[h,(f1 + h1)] is set
{h,(f1 + h1)} is non empty set
{{h,(f1 + h1)},{h}} is non empty set
the addF of B . [h,(f1 + h1)] is set
ONE is Element of the carrier of B
f is Element of the carrier of B
ONE + f is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the addF of B . [ONE,f] is set
f + ONE is Element of the carrier of B
the addF of B . (f,ONE) is Element of the carrier of B
[f,ONE] is set
{f,ONE} is non empty set
{f} is non empty set
{{f,ONE},{f}} is non empty set
the addF of B . [f,ONE] is set
the carrier of B is non empty set
g is left_complementable right_complementable complementable Element of the carrier of B
h is left_complementable right_complementable complementable Element of the carrier of B
g + h is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (g,h) is left_complementable right_complementable complementable Element of the carrier of B
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the addF of B . [g,h] is set
h + g is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . (h,g) is left_complementable right_complementable complementable Element of the carrier of B
[h,g] is set
{h,g} is non empty set
{h} is non empty set
{{h,g},{h}} is non empty set
the addF of B . [h,g] is set
ONE is V11() real ext-real set
f is Element of the carrier of B
g is Element of the carrier of B
f + g is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (f,g) is Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of B . [f,g] is set
ONE * (f + g) is Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (ONE,(f + g)) is set
[ONE,(f + g)] is set
{ONE,(f + g)} is non empty set
{ONE} is non empty V177() V178() V179() set
{{ONE,(f + g)},{ONE}} is non empty set
the Mult of B . [ONE,(f + g)] is set
ONE * f is Element of the carrier of B
the Mult of B . (ONE,f) is set
[ONE,f] is set
{ONE,f} is non empty set
{{ONE,f},{ONE}} is non empty set
the Mult of B . [ONE,f] is set
ONE * g is Element of the carrier of B
the Mult of B . (ONE,g) is set
[ONE,g] is set
{ONE,g} is non empty set
{{ONE,g},{ONE}} is non empty set
the Mult of B . [ONE,g] is set
(ONE * f) + (ONE * g) is Element of the carrier of B
the addF of B . ((ONE * f),(ONE * g)) is Element of the carrier of B
[(ONE * f),(ONE * g)] is set
{(ONE * f),(ONE * g)} is non empty set
{(ONE * f)} is non empty set
{{(ONE * f),(ONE * g)},{(ONE * f)}} is non empty set
the addF of B . [(ONE * f),(ONE * g)] is set
the carrier of B is non empty set
h is left_complementable right_complementable complementable Element of the carrier of B
f1 is left_complementable right_complementable complementable Element of the carrier of B
h + f1 is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (h,f1) is left_complementable right_complementable complementable Element of the carrier of B
[h,f1] is set
{h,f1} is non empty set
{h} is non empty set
{{h,f1},{h}} is non empty set
the addF of B . [h,f1] is set
ONE * (h + f1) is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (ONE,(h + f1)) is set
[ONE,(h + f1)] is set
{ONE,(h + f1)} is non empty set
{{ONE,(h + f1)},{ONE}} is non empty set
the Mult of B . [ONE,(h + f1)] is set
ONE * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (ONE,h) is set
[ONE,h] is set
{ONE,h} is non empty set
{{ONE,h},{ONE}} is non empty set
the Mult of B . [ONE,h] is set
ONE * f1 is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (ONE,f1) is set
[ONE,f1] is set
{ONE,f1} is non empty set
{{ONE,f1},{ONE}} is non empty set
the Mult of B . [ONE,f1] is set
(ONE * h) + (ONE * f1) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . ((ONE * h),(ONE * f1)) is left_complementable right_complementable complementable Element of the carrier of B
[(ONE * h),(ONE * f1)] is set
{(ONE * h),(ONE * f1)} is non empty set
{(ONE * h)} is non empty set
{{(ONE * h),(ONE * f1)},{(ONE * h)}} is non empty set
the addF of B . [(ONE * h),(ONE * f1)] is set
ONE is V11() real ext-real set
f is V11() real ext-real set
ONE + f is V11() real ext-real set
g is Element of the carrier of B
(ONE + f) * g is Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . ((ONE + f),g) is set
[(ONE + f),g] is set
{(ONE + f),g} is non empty set
{(ONE + f)} is non empty V177() V178() V179() set
{{(ONE + f),g},{(ONE + f)}} is non empty set
the Mult of B . [(ONE + f),g] is set
ONE * g is Element of the carrier of B
the Mult of B . (ONE,g) is set
[ONE,g] is set
{ONE,g} is non empty set
{ONE} is non empty V177() V178() V179() set
{{ONE,g},{ONE}} is non empty set
the Mult of B . [ONE,g] is set
f * g is Element of the carrier of B
the Mult of B . (f,g) is set
[f,g] is set
{f,g} is non empty set
{f} is non empty V177() V178() V179() set
{{f,g},{f}} is non empty set
the Mult of B . [f,g] is set
(ONE * g) + (f * g) is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . ((ONE * g),(f * g)) is Element of the carrier of B
[(ONE * g),(f * g)] is set
{(ONE * g),(f * g)} is non empty set
{(ONE * g)} is non empty set
{{(ONE * g),(f * g)},{(ONE * g)}} is non empty set
the addF of B . [(ONE * g),(f * g)] is set
the carrier of B is non empty set
h is left_complementable right_complementable complementable Element of the carrier of B
(ONE + f) * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . ((ONE + f),h) is set
[(ONE + f),h] is set
{(ONE + f),h} is non empty set
{{(ONE + f),h},{(ONE + f)}} is non empty set
the Mult of B . [(ONE + f),h] is set
ONE * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (ONE,h) is set
[ONE,h] is set
{ONE,h} is non empty set
{{ONE,h},{ONE}} is non empty set
the Mult of B . [ONE,h] is set
f * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (f,h) is set
[f,h] is set
{f,h} is non empty set
{{f,h},{f}} is non empty set
the Mult of B . [f,h] is set
(ONE * h) + (f * h) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . ((ONE * h),(f * h)) is left_complementable right_complementable complementable Element of the carrier of B
[(ONE * h),(f * h)] is set
{(ONE * h),(f * h)} is non empty set
{(ONE * h)} is non empty set
{{(ONE * h),(f * h)},{(ONE * h)}} is non empty set
the addF of B . [(ONE * h),(f * h)] is set
ONE is V11() real ext-real set
f is V11() real ext-real set
ONE * f is V11() real ext-real set
g is Element of the carrier of B
(ONE * f) * g is Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . ((ONE * f),g) is set
[(ONE * f),g] is set
{(ONE * f),g} is non empty set
{(ONE * f)} is non empty V177() V178() V179() set
{{(ONE * f),g},{(ONE * f)}} is non empty set
the Mult of B . [(ONE * f),g] is set
f * g is Element of the carrier of B
the Mult of B . (f,g) is set
[f,g] is set
{f,g} is non empty set
{f} is non empty V177() V178() V179() set
{{f,g},{f}} is non empty set
the Mult of B . [f,g] is set
ONE * (f * g) is Element of the carrier of B
the Mult of B . (ONE,(f * g)) is set
[ONE,(f * g)] is set
{ONE,(f * g)} is non empty set
{ONE} is non empty V177() V178() V179() set
{{ONE,(f * g)},{ONE}} is non empty set
the Mult of B . [ONE,(f * g)] is set
the carrier of B is non empty set
h is left_complementable right_complementable complementable Element of the carrier of B
(ONE * f) * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . ((ONE * f),h) is set
[(ONE * f),h] is set
{(ONE * f),h} is non empty set
{{(ONE * f),h},{(ONE * f)}} is non empty set
the Mult of B . [(ONE * f),h] is set
f * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (f,h) is set
[f,h] is set
{f,h} is non empty set
{{f,h},{f}} is non empty set
the Mult of B . [f,h] is set
ONE * (f * h) is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (ONE,(f * h)) is set
[ONE,(f * h)] is set
{ONE,(f * h)} is non empty set
{{ONE,(f * h)},{ONE}} is non empty set
the Mult of B . [ONE,(f * h)] is set
ONE is Element of the carrier of B
f is Element of the carrier of B
ONE * f is Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (ONE,f) is Element of the carrier of B
[ONE,f] is set
{ONE,f} is non empty set
{ONE} is non empty set
{{ONE,f},{ONE}} is non empty set
the multF of B . [ONE,f] is set
the carrier of B is non empty set
f1 is V11() real ext-real Element of REAL
f1 * (ONE * f) is Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (f1,(ONE * f)) is set
[f1,(ONE * f)] is set
{f1,(ONE * f)} is non empty set
{f1} is non empty V177() V178() V179() set
{{f1,(ONE * f)},{f1}} is non empty set
the Mult of B . [f1,(ONE * f)] is set
f1 * ONE is Element of the carrier of B
the Mult of B . (f1,ONE) is set
[f1,ONE] is set
{f1,ONE} is non empty set
{{f1,ONE},{f1}} is non empty set
the Mult of B . [f1,ONE] is set
(f1 * ONE) * f is Element of the carrier of B
the multF of B . ((f1 * ONE),f) is Element of the carrier of B
[(f1 * ONE),f] is set
{(f1 * ONE),f} is non empty set
{(f1 * ONE)} is non empty set
{{(f1 * ONE),f},{(f1 * ONE)}} is non empty set
the multF of B . [(f1 * ONE),f] is set
g is left_complementable right_complementable complementable Element of the carrier of B
h is left_complementable right_complementable complementable Element of the carrier of B
g * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (g,h) is left_complementable right_complementable complementable Element of the carrier of B
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the multF of B . [g,h] is set
f1 * (g * h) is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (f1,(g * h)) is set
[f1,(g * h)] is set
{f1,(g * h)} is non empty set
{{f1,(g * h)},{f1}} is non empty set
the Mult of B . [f1,(g * h)] is set
f1 * g is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (f1,g) is set
[f1,g] is set
{f1,g} is non empty set
{{f1,g},{f1}} is non empty set
the Mult of B . [f1,g] is set
(f1 * g) * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . ((f1 * g),h) is left_complementable right_complementable complementable Element of the carrier of B
[(f1 * g),h] is set
{(f1 * g),h} is non empty set
{(f1 * g)} is non empty set
{{(f1 * g),h},{(f1 * g)}} is non empty set
the multF of B . [(f1 * g),h] is set
0. B is V49(B) Element of the carrier of B
the ZeroF of B is Element of the carrier of B
ONE is Element of the carrier of B
ONE + (0. B) is Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (ONE,(0. B)) is Element of the carrier of B
[ONE,(0. B)] is set
{ONE,(0. B)} is non empty set
{ONE} is non empty set
{{ONE,(0. B)},{ONE}} is non empty set
the addF of B . [ONE,(0. B)] is set
the carrier of B is non empty set
f is left_complementable right_complementable complementable Element of the carrier of B
0. B is V49(B) left_complementable right_complementable complementable Element of the carrier of B
the ZeroF of B is left_complementable right_complementable complementable Element of the carrier of B
f + (0. B) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the addF of B . (f,(0. B)) is left_complementable right_complementable complementable Element of the carrier of B
[f,(0. B)] is set
{f,(0. B)} is non empty set
{f} is non empty set
{{f,(0. B)},{f}} is non empty set
the addF of B . [f,(0. B)] is set
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
1. (X) is Element of the carrier of (X)
the carrier of (X) is non empty set
the OneF of (X) is Element of the carrier of (X)
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
1_ (X) is left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty 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)
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of the carrier of (X)
((RAlgebra X),(X)) . (1,B) is set
[1,B] is set
{1,B} is non empty set
{1} is non empty V177() V178() V179() V180() V181() V182() set
{{1,B},{1}} is non empty set
((RAlgebra X),(X)) . [1,B] is set
ONE is Element of (X)
[1,ONE] is Element of [:NAT,(X):]
[:NAT,(X):] is non empty set
{1,ONE} is non empty set
{{1,ONE},{1}} is non empty set
[:REAL, the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) is non empty Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like V23([:REAL, the carrier of (RAlgebra X):]) quasi_total Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
[:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):] is non empty set
the Mult of (RAlgebra X) | [:REAL,(X):] is Relation-like [:REAL, the carrier of (RAlgebra X):] -defined the carrier of (RAlgebra X) -valued Function-like Element of bool [:[:REAL, the carrier of (RAlgebra X):], the carrier of (RAlgebra X):]
( the Mult of (RAlgebra X) | [:REAL,(X):]) . (1,ONE) is set
[1,ONE] is set
( the Mult of (RAlgebra X) | [:REAL,(X):]) . [1,ONE] is set
the Mult of (RAlgebra X) . (1,ONE) is left_complementable right_complementable complementable Element of the carrier of (RAlgebra X)
the Mult of (RAlgebra X) . [1,ONE] is set
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of the carrier of (X)
1 * B is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (1,B) is set
[1,B] is set
{1,B} is non empty set
{1} is non empty V177() V178() V179() V180() V181() V182() set
{{1,B},{1}} is non empty set
the Mult of (X) . [1,B] is set
X is non empty set
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
0. (X) is V49((X)) Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
0. (X) is V49((X)) left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is left_complementable right_complementable complementable Element of the carrier of (X)
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of X
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
B . B is V11() real ext-real Element of REAL
abs (B . B) is V11() real ext-real Element of REAL
ONE is Element of the carrier of (X)
||.ONE.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . ONE is V11() real ext-real Element of REAL
(X,B) is non empty V177() V178() V179() Element of bool REAL
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
upper_bound (X,B) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is Element of the carrier of (X)
||.B.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . B is V11() real ext-real Element of REAL
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,B) is non empty V177() V178() V179() Element of bool REAL
{ (abs (B . b₁)) where b₁ is Element of X : verum } is set
ONE is set
g is V11() real ext-real Element of REAL
h is Element of X
B . h is V11() real ext-real Element of REAL
abs (B . h) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
upper_bound (X,B) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
0. (X) is V49((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
B is Element of the carrier of (X)
||.B.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . B is V11() real ext-real Element of REAL
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,f) is non empty V177() V178() V179() Element of bool REAL
{ (abs (f . b₁)) where b₁ is Element of X : verum } is set
g is set
upper_bound (X,f) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
h1 is Element of X
f . h1 is V11() real ext-real Element of REAL
abs (f . h1) is V11() real ext-real Element of REAL
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
abs 0 is V11() real V112() ext-real Element of REAL
h is V11() real ext-real Element of REAL
f1 is V11() real ext-real set
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f is Element of the carrier of (X)
g is Element of the carrier of (X)
f + g is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (f,g) is Element of the carrier of (X)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the addF of (X) . [f,g] is set
h is Element of the carrier of (X)
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
g1 is left_complementable right_complementable complementable Element of the carrier of (X)
f1 is left_complementable right_complementable complementable Element of the carrier of (X)
h1 is left_complementable right_complementable complementable Element of the carrier of (X)
f1 + h1 is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of (X)
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the addF of (X) . [f1,h1] is set
F1 is Element of X
ONE . F1 is V11() real ext-real Element of REAL
B . F1 is V11() real ext-real Element of REAL
B . F1 is V11() real ext-real Element of REAL
(B . F1) + (B . F1) is V11() real ext-real Element of REAL
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is V11() real ext-real Element of REAL
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f is Element of the carrier of (X)
B * f is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (B,f) is set
[B,f] is set
{B,f} is non empty set
{B} is non empty V177() V178() V179() set
{{B,f},{B}} is non empty set
the Mult of (X) . [B,f] is set
g is Element of the carrier of (X)
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
f1 is left_complementable right_complementable complementable Element of the carrier of (X)
h is left_complementable right_complementable complementable Element of the carrier of (X)
B * h is left_complementable right_complementable complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (B,h) is set
[B,h] is set
{B,h} is non empty set
{{B,h},{B}} is non empty set
the Mult of (X) . [B,h] is set
h1 is Element of X
ONE . h1 is V11() real ext-real Element of REAL
B . h1 is V11() real ext-real Element of REAL
B * (B . h1) is V11() real ext-real Element of REAL
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f is Element of the carrier of (X)
g is Element of the carrier of (X)
f * g is Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (f,g) is Element of the carrier of (X)
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the multF of (X) . [f,g] is set
h is Element of the carrier of (X)
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
g1 is left_complementable right_complementable complementable Element of the carrier of (X)
f1 is left_complementable right_complementable complementable Element of the carrier of (X)
h1 is left_complementable right_complementable complementable Element of the carrier of (X)
f1 * h1 is left_complementable right_complementable complementable Element of the carrier of (X)
the multF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (f1,h1) is left_complementable right_complementable complementable Element of the carrier of (X)
[f1,h1] is set
{f1,h1} is non empty set
{f1} is non empty set
{{f1,h1},{f1}} is non empty set
the multF of (X) . [f1,h1] is set
F1 is Element of X
ONE . F1 is V11() real ext-real Element of REAL
B . F1 is V11() real ext-real Element of REAL
B . F1 is V11() real ext-real Element of REAL
(B . F1) * (B . F1) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
0. (X) is V49((X)) Element of the carrier of (X)
the ZeroF of (X) is Element of the carrier of (X)
B is V11() real ext-real Element of REAL
abs B is V11() real ext-real Element of REAL
B is Element of the carrier of (X)
||.B.|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . B is V11() real ext-real Element of REAL
B * B is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (B,B) is set
[B,B] is set
{B,B} is non empty set
{B} is non empty V177() V178() V179() set
{{B,B},{B}} is non empty set
the Mult of (X) . [B,B] is set
||.(B * B).|| is V11() real ext-real Element of REAL
the U8 of (X) . (B * B) is V11() real ext-real Element of REAL
(abs B) * ||.B.|| is V11() real ext-real Element of REAL
ONE is Element of the carrier of (X)
B + ONE is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (B,ONE) is Element of the carrier of (X)
[B,ONE] is set
{B,ONE} is non empty set
{B} is non empty set
{{B,ONE},{B}} is non empty set
the addF of (X) . [B,ONE] is set
||.(B + ONE).|| is V11() real ext-real Element of REAL
the U8 of (X) . (B + ONE) is V11() real ext-real Element of REAL
||.ONE.|| is V11() real ext-real Element of REAL
the U8 of (X) . ONE is V11() real ext-real Element of REAL
||.B.|| + ||.ONE.|| is V11() real ext-real Element of REAL
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,h) is non empty V177() V178() V179() Element of bool REAL
{ (abs (h . b₁)) where b₁ is Element of X : verum } is set
f1 is set
upper_bound (X,h) is V11() real ext-real Element of REAL
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 is V11() real ext-real Element of REAL
F1 is Element of X
h . F1 is V11() real ext-real Element of REAL
abs (h . F1) is V11() real ext-real Element of REAL
abs 0 is V11() real V112() ext-real Element of REAL
h1 is V11() real ext-real Element of REAL
g1 is V11() real ext-real set
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f1 is Element of X
h . f1 is V11() real ext-real Element of REAL
abs (h . f1) is V11() real ext-real Element of REAL
g . f1 is V11() real ext-real Element of REAL
abs (g . f1) is V11() real ext-real Element of REAL
(abs (h . f1)) + (abs (g . f1)) is V11() real ext-real Element of REAL
f . f1 is V11() real ext-real Element of REAL
abs (f . f1) is V11() real ext-real Element of REAL
(h . f1) + (g . f1) is V11() real ext-real Element of REAL
abs ((h . f1) + (g . f1)) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
(X,f) is non empty V177() V178() V179() Element of bool REAL
{ (abs (f . b₁)) where b₁ is Element of X : verum } is set
h1 is Element of X
f . h1 is V11() real ext-real Element of REAL
abs (f . h1) is V11() real ext-real Element of REAL
upper_bound (X,f) is V11() real ext-real Element of REAL
f1 is V11() real ext-real set
f is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h is Element of X
g . h is V11() real ext-real Element of REAL
abs (g . h) is V11() real ext-real Element of REAL
f . h is V11() real ext-real Element of REAL
B * (f . h) is V11() real ext-real Element of REAL
abs (B * (f . h)) is V11() real ext-real Element of REAL
abs (f . h) is V11() real ext-real Element of REAL
(abs B) * (abs (f . h)) is V11() real ext-real Element of REAL
h is V11() real ext-real Element of REAL
(X,g) is non empty V177() V178() V179() Element of bool REAL
{ (abs (g . b₁)) where b₁ is Element of X : verum } is set
f1 is Element of X
g . f1 is V11() real ext-real Element of REAL
abs (g . f1) is V11() real ext-real Element of REAL
upper_bound (X,g) is V11() real ext-real Element of REAL
h is V11() real ext-real set
B " is V11() real ext-real Element of REAL
abs (B ") is V11() real ext-real Element of REAL
1 / B is V11() real ext-real Element of REAL
B " is V11() real ext-real set
1 * (B ") is V11() real ext-real set
abs (1 / B) is V11() real ext-real Element of REAL
1 / (abs B) is V11() real ext-real Element of REAL
(abs B) " is V11() real ext-real set
1 * ((abs B) ") is V11() real ext-real set
h is Element of X
g . h is V11() real ext-real Element of REAL
f . h is V11() real ext-real Element of REAL
B * (f . h) is V11() real ext-real Element of REAL
(B ") * (g . h) is V11() real ext-real Element of REAL
(B ") * B is V11() real ext-real Element of REAL
((B ") * B) * (f . h) is V11() real ext-real Element of REAL
1 * (f . h) is V11() real ext-real Element of REAL
abs ((B ") * (g . h)) is V11() real ext-real Element of REAL
abs (g . h) is V11() real ext-real Element of REAL
(abs (B ")) * (abs (g . h)) is V11() real ext-real Element of REAL
abs (f . h) is V11() real ext-real Element of REAL
(abs B) " is V11() real ext-real Element of REAL
((abs B) ") * ||.(B * B).|| is V11() real ext-real Element of REAL
h is V11() real ext-real Element of REAL
(X,f) is non empty V177() V178() V179() Element of bool REAL
{ (abs (f . b₁)) where b₁ is Element of X : verum } is set
f1 is Element of X
f . f1 is V11() real ext-real Element of REAL
abs (f . f1) is V11() real ext-real Element of REAL
upper_bound (X,f) is V11() real ext-real Element of REAL
h is V11() real ext-real set
(abs B) * (((abs B) ") * ||.(B * B).||) is V11() real ext-real Element of REAL
(abs B) * ((abs B) ") is V11() real ext-real Element of REAL
((abs B) * ((abs B) ")) * ||.(B * B).|| is V11() real ext-real Element of REAL
1 * ||.(B * B).|| is V11() real ext-real Element of REAL
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
the carrier of (X) is non empty set
h is left_complementable right_complementable complementable Element of the carrier of (X)
B * h is left_complementable right_complementable complementable Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (B,h) is set
[B,h] is set
{B,h} is non empty set
{{B,h},{B}} is non empty set
the Mult of (X) . [B,h] is set
B + B is V11() real ext-real Element of REAL
(B + B) * h is left_complementable right_complementable complementable Element of the carrier of (X)
the Mult of (X) . ((B + B),h) is set
[(B + B),h] is set
{(B + B),h} is non empty set
{(B + B)} is non empty V177() V178() V179() set
{{(B + B),h},{(B + B)}} is non empty set
the Mult of (X) . [(B + B),h] is set
(B * h) + (B * h) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . ((B * h),(B * h)) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B * h),(B * h)] is set
{(B * h),(B * h)} is non empty set
{(B * h)} is non empty set
{{(B * h),(B * h)},{(B * h)}} is non empty set
the addF of (X) . [(B * h),(B * h)] is set
0. (X) is V49((X)) 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)
- (B * h) is left_complementable right_complementable complementable Element of the carrier of (X)
(- (B * h)) + ((B * h) + (B * h)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((- (B * h)),((B * h) + (B * h))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(- (B * h)),((B * h) + (B * h))] is set
{(- (B * h)),((B * h) + (B * h))} is non empty set
{(- (B * h))} is non empty set
{{(- (B * h)),((B * h) + (B * h))},{(- (B * h))}} is non empty set
the addF of (X) . [(- (B * h)),((B * h) + (B * h))] is set
(- (B * h)) + (B * h) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((- (B * h)),(B * h)) is left_complementable right_complementable complementable Element of the carrier of (X)
[(- (B * h)),(B * h)] is set
{(- (B * h)),(B * h)} is non empty set
{{(- (B * h)),(B * h)},{(- (B * h))}} is non empty set
the addF of (X) . [(- (B * h)),(B * h)] is set
((- (B * h)) + (B * h)) + (B * h) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (((- (B * h)) + (B * h)),(B * h)) is left_complementable right_complementable complementable Element of the carrier of (X)
[((- (B * h)) + (B * h)),(B * h)] is set
{((- (B * h)) + (B * h)),(B * h)} is non empty set
{((- (B * h)) + (B * h))} is non empty set
{{((- (B * h)) + (B * h)),(B * h)},{((- (B * h)) + (B * h))}} is non empty set
the addF of (X) . [((- (B * h)) + (B * h)),(B * h)] is set
(0. (X)) + (B * h) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((0. (X)),(B * h)) is left_complementable right_complementable complementable Element of the carrier of (X)
[(0. (X)),(B * h)] is set
{(0. (X)),(B * h)} is non empty set
{(0. (X))} is non empty set
{{(0. (X)),(B * h)},{(0. (X))}} is non empty set
the addF of (X) . [(0. (X)),(B * h)] is set
0 * ||.B.|| is V11() real ext-real Element of REAL
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f1 is Element of X
h . f1 is V11() real ext-real Element of REAL
abs (h . f1) is V11() real ext-real Element of REAL
g is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g . f1 is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
0. (X) is V49((X)) Element of the carrier of (X)
the carrier of (X) is non empty set
the ZeroF of (X) is Element of the carrier of (X)
||.(0. (X)).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . (0. (X)) is V11() real ext-real Element of REAL
B is Element of the carrier of (X)
||.B.|| is V11() real ext-real Element of REAL
the U8 of (X) . B is V11() real ext-real Element of REAL
B is Element of the carrier of (X)
||.B.|| is V11() real ext-real Element of REAL
the U8 of (X) . B is V11() real ext-real Element of REAL
ONE is Element of the carrier of (X)
f is V11() real ext-real Element of REAL
f * ONE is Element of the carrier of (X)
the Mult of (X) is non empty Relation-like [:REAL, the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([:REAL, the carrier of (X):]) quasi_total Element of bool [:[:REAL, the carrier of (X):], the carrier of (X):]
[:REAL, the carrier of (X):] is non empty set
[:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
bool [:[:REAL, the carrier of (X):], the carrier of (X):] is non empty set
the Mult of (X) . (f,ONE) is set
[f,ONE] is set
{f,ONE} is non empty set
{f} is non empty V177() V178() V179() set
{{f,ONE},{f}} is non empty set
the Mult of (X) . [f,ONE] is set
||.(f * ONE).|| is V11() real ext-real Element of REAL
the U8 of (X) . (f * ONE) is V11() real ext-real Element of REAL
abs f is V11() real ext-real Element of REAL
||.ONE.|| is V11() real ext-real Element of REAL
the U8 of (X) . ONE is V11() real ext-real Element of REAL
(abs f) * ||.ONE.|| is V11() real ext-real Element of REAL
g is Element of the carrier of (X)
h is Element of the carrier of (X)
g + h is Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (g,h) is Element of the carrier of (X)
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the addF of (X) . [g,h] is set
||.(g + h).|| is V11() real ext-real Element of REAL
the U8 of (X) . (g + h) is V11() real ext-real Element of REAL
||.g.|| is V11() real ext-real Element of REAL
the U8 of (X) . g is V11() real ext-real Element of REAL
||.h.|| is V11() real ext-real Element of REAL
the U8 of (X) . h is V11() real ext-real Element of REAL
||.g.|| + ||.h.|| is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
X is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f is left_complementable right_complementable complementable Element of the carrier of (X)
g is left_complementable right_complementable complementable Element of the carrier of (X)
f - g is left_complementable right_complementable complementable Element of the carrier of (X)
- g is left_complementable right_complementable complementable Element of the carrier of (X)
f + (- g) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . (f,(- g)) is left_complementable right_complementable complementable Element of the carrier of (X)
[f,(- g)] is set
{f,(- g)} is non empty set
{f} is non empty set
{{f,(- g)},{f}} is non empty set
the addF of (X) . [f,(- g)] is set
h is left_complementable right_complementable complementable Element of the carrier of (X)
f1 is Element of X
ONE . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
(B . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
(B . f1) + (- (B . f1)) is V11() real ext-real set
(ONE . f1) + (B . f1) is V11() real ext-real Element of REAL
h + g is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (h,g) is left_complementable right_complementable complementable Element of the carrier of (X)
[h,g] is set
{h,g} is non empty set
{h} is non empty set
{{h,g},{h}} is non empty set
the addF of (X) . [h,g] is set
g - g is left_complementable right_complementable complementable Element of the carrier of (X)
g + (- g) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (g,(- g)) is left_complementable right_complementable complementable Element of the carrier of (X)
[g,(- g)] is set
{g,(- g)} is non empty set
{g} is non empty set
{{g,(- g)},{g}} is non empty set
the addF of (X) . [g,(- g)] is set
h + (g - g) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (h,(g - g)) is left_complementable right_complementable complementable Element of the carrier of (X)
[h,(g - g)] is set
{h,(g - g)} is non empty set
{{h,(g - g)},{h}} is non empty set
the addF of (X) . [h,(g - g)] is set
0. (X) is V49((X)) 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)
h + (0. (X)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (h,(0. (X))) is left_complementable right_complementable complementable Element of the carrier of (X)
[h,(0. (X))] is set
{h,(0. (X))} is non empty set
{{h,(0. (X))},{h}} is non empty set
the addF of (X) . [h,(0. (X))] is set
f - (g - g) is left_complementable right_complementable complementable Element of the carrier of (X)
- (g - g) is left_complementable right_complementable complementable Element of the carrier of (X)
f + (- (g - g)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (f,(- (g - g))) is left_complementable right_complementable complementable Element of the carrier of (X)
[f,(- (g - g))] is set
{f,(- (g - g))} is non empty set
{{f,(- (g - g))},{f}} is non empty set
the addF of (X) . [f,(- (g - g))] is set
f - (0. (X)) is left_complementable right_complementable complementable Element of the carrier of (X)
- (0. (X)) is left_complementable right_complementable complementable Element of the carrier of (X)
f + (- (0. (X))) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . (f,(- (0. (X)))) is left_complementable right_complementable complementable Element of the carrier of (X)
[f,(- (0. (X)))] is set
{f,(- (0. (X)))} is non empty set
{{f,(- (0. (X)))},{f}} is non empty set
the addF of (X) . [f,(- (0. (X)))] is set
f1 is Element of X
B . f1 is V11() real ext-real Element of REAL
ONE . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
(ONE . f1) + (B . f1) is V11() real ext-real Element of REAL
(B . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
(B . f1) + (- (B . f1)) is V11() real ext-real set
f1 is Element of X
ONE . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
(B . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
(B . f1) + (- (B . f1)) is V11() real ext-real set
f1 is Element of X
ONE . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
(B . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
(B . f1) + (- (B . f1)) is V11() real ext-real set
h1 is Element of X
ONE . h1 is V11() real ext-real Element of REAL
B . h1 is V11() real ext-real Element of REAL
B . h1 is V11() real ext-real Element of REAL
(B . h1) - (B . h1) is V11() real ext-real Element of REAL
- (B . h1) is V11() real ext-real set
(B . h1) + (- (B . h1)) is V11() real ext-real set
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
[:NAT, the carrier of (X):] is non empty set
bool [:NAT, the carrier of (X):] is non empty set
B is non empty Relation-like NAT -defined the carrier of (X) -valued Function-like V23( NAT ) quasi_total Element of bool [:NAT, the carrier of (X):]
B is Element of X
ONE is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
lim ONE is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
ONE . f is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
ONE . g is V11() real ext-real Element of REAL
(ONE . f) - (ONE . g) is V11() real ext-real Element of REAL
- (ONE . g) is V11() real ext-real set
(ONE . f) + (- (ONE . g)) is V11() real ext-real set
abs ((ONE . f) - (ONE . g)) is V11() real ext-real Element of REAL
B . f is left_complementable right_complementable complementable Element of the carrier of (X)
B . g is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f) - (B . g) is left_complementable right_complementable complementable Element of the carrier of (X)
- (B . g) is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f) + (- (B . g)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . ((B . f),(- (B . g))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . f),(- (B . g))] is set
{(B . f),(- (B . g))} is non empty set
{(B . f)} is non empty set
{{(B . f),(- (B . g))},{(B . f)}} is non empty set
the addF of (X) . [(B . f),(- (B . g))] is set
||.((B . f) - (B . g)).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . ((B . f) - (B . g)) is V11() real ext-real Element of REAL
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . f)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . g)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . f)) . B is V11() real ext-real Element of REAL
(X,(B . g)) . B is V11() real ext-real Element of REAL
h . B is V11() real ext-real Element of REAL
f is V11() real ext-real set
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . f1 is left_complementable right_complementable complementable Element of the carrier of (X)
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f1) - (B . h) is left_complementable right_complementable complementable Element of the carrier of (X)
- (B . h) is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f1) + (- (B . h)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . ((B . f1),(- (B . h))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . f1),(- (B . h))] is set
{(B . f1),(- (B . h))} is non empty set
{(B . f1)} is non empty set
{{(B . f1),(- (B . h))},{(B . f1)}} is non empty set
the addF of (X) . [(B . f1),(- (B . h))] is set
||.((B . f1) - (B . h)).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . ((B . f1) - (B . h)) is V11() real ext-real Element of REAL
ONE . f1 is V11() real ext-real Element of REAL
ONE . h is V11() real ext-real Element of REAL
(ONE . f1) - (ONE . h) is V11() real ext-real Element of REAL
- (ONE . h) is V11() real ext-real set
(ONE . f1) + (- (ONE . h)) is V11() real ext-real set
abs ((ONE . f1) - (ONE . h)) is V11() real ext-real Element of REAL
B is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE is V11() real ext-real set
f is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . f1 is left_complementable right_complementable complementable Element of the carrier of (X)
||.(B . f1).|| is V11() real ext-real Element of REAL
the U8 of (X) is non empty Relation-like the carrier of (X) -defined REAL -valued Function-like V23( the carrier of (X)) quasi_total V150() V151() V152() Element of bool [: the carrier of (X),REAL:]
[: the carrier of (X),REAL:] is non empty V150() V151() V152() set
bool [: the carrier of (X),REAL:] is non empty set
the U8 of (X) . (B . f1) is V11() real ext-real Element of REAL
||.B.|| is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
||.B.|| . f1 is V11() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f1) - (B . h) is left_complementable right_complementable complementable Element of the carrier of (X)
- (B . h) is left_complementable right_complementable complementable Element of the carrier of (X)
(B . f1) + (- (B . h)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) is non empty Relation-like [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like V23([: the carrier of (X), the carrier of (X):]) quasi_total Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the addF of (X) . ((B . f1),(- (B . h))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . f1),(- (B . h))] is set
{(B . f1),(- (B . h))} is non empty set
{(B . f1)} is non empty set
{{(B . f1),(- (B . h))},{(B . f1)}} is non empty set
the addF of (X) . [(B . f1),(- (B . h))] is set
||.((B . f1) - (B . h)).|| is V11() real ext-real Element of REAL
the U8 of (X) . ((B . f1) - (B . h)) is V11() real ext-real Element of REAL
||.(B . h).|| is V11() real ext-real Element of REAL
the U8 of (X) . (B . h) is V11() real ext-real Element of REAL
||.(B . f1).|| - ||.(B . h).|| is V11() real ext-real Element of REAL
- ||.(B . h).|| is V11() real ext-real set
||.(B . f1).|| + (- ||.(B . h).||) is V11() real ext-real set
abs (||.(B . f1).|| - ||.(B . h).||) is V11() real ext-real Element of REAL
||.B.|| . h is V11() real ext-real Element of REAL
(||.B.|| . f1) - (||.B.|| . h) is V11() real ext-real Element of REAL
- (||.B.|| . h) is V11() real ext-real set
(||.B.|| . f1) + (- (||.B.|| . h)) is V11() real ext-real set
abs ((||.B.|| . f1) - (||.B.|| . h)) is V11() real ext-real Element of REAL
f1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
||.B.|| . f1 is V11() real ext-real Element of REAL
(||.B.|| . f1) - (||.B.|| . h) is V11() real ext-real Element of REAL
(||.B.|| . f1) + (- (||.B.|| . h)) is V11() real ext-real set
abs ((||.B.|| . f1) - (||.B.|| . h)) is V11() real ext-real Element of REAL
ONE is set
dom B is Element of bool X
bool X is non empty set
X /\ (dom B) is Element of bool X
B . ONE is V11() real ext-real Element of REAL
f is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
lim f is V11() real ext-real Element of REAL
abs f is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() bounded_below Element of bool [:NAT,REAL:]
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
(abs f) . g is V11() real ext-real Element of REAL
||.B.|| . g is V11() real ext-real Element of REAL
f . g is V11() real ext-real Element of REAL
B . g is left_complementable right_complementable complementable Element of the carrier of (X)
(X,(B . g)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . g)) . ONE is V11() real ext-real Element of REAL
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
abs (f . g) is V11() real ext-real Element of REAL
||.(B . g).|| is V11() real ext-real Element of REAL
the U8 of (X) . (B . g) is V11() real ext-real Element of REAL
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
abs (B . ONE) is V11() real ext-real Element of REAL
lim (abs f) is V11() real ext-real Element of REAL
lim ||.B.|| is V11() real ext-real Element of REAL
B | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
f1 is Element of X
B . f1 is V11() real ext-real Element of REAL
h1 is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
lim h1 is V11() real ext-real Element of REAL
h1 . h is V11() real ext-real Element of REAL
NAT --> (h1 . h) is non empty Relation-like NAT -defined REAL -valued Function-like constant V23( NAT ) quasi_total V150() V151() V152() bounded convergent Element of bool [:NAT,REAL:]
g1 is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
h1 - g1 is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
- g1 is Relation-like NAT -defined Function-like V23( NAT ) V150() V151() V152() set
- 1 is V11() real ext-real non positive set
(- 1) (#) g1 is Relation-like NAT -defined Function-like V23( NAT ) V150() V151() V152() set
h1 + (- g1) is Relation-like NAT -defined Function-like V23( NAT ) V150() V151() V152() set
hf1 is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
gf1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h1 . gf1 is V11() real ext-real Element of REAL
gPh1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h1 . gPh1 is V11() real ext-real Element of REAL
(h1 . gf1) - (h1 . gPh1) is V11() real ext-real Element of REAL
- (h1 . gPh1) is V11() real ext-real set
(h1 . gf1) + (- (h1 . gPh1)) is V11() real ext-real set
abs ((h1 . gf1) - (h1 . gPh1)) is V11() real ext-real Element of REAL
B . gf1 is left_complementable right_complementable complementable Element of the carrier of (X)
B . gPh1 is left_complementable right_complementable complementable Element of the carrier of (X)
(B . gf1) - (B . gPh1) is left_complementable right_complementable complementable Element of the carrier of (X)
- (B . gPh1) is left_complementable right_complementable complementable Element of the carrier of (X)
(B . gf1) + (- (B . gPh1)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((B . gf1),(- (B . gPh1))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . gf1),(- (B . gPh1))] is set
{(B . gf1),(- (B . gPh1))} is non empty set
{(B . gf1)} is non empty set
{{(B . gf1),(- (B . gPh1))},{(B . gf1)}} is non empty set
the addF of (X) . [(B . gf1),(- (B . gPh1))] is set
||.((B . gf1) - (B . gPh1)).|| is V11() real ext-real Element of REAL
the U8 of (X) . ((B . gf1) - (B . gPh1)) is V11() real ext-real Element of REAL
x is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
x | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . gf1)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
a is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
a | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . gPh1)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
a is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
a | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . gf1)) . f1 is V11() real ext-real Element of REAL
(X,(B . gPh1)) . f1 is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
gf1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
hf1 . gf1 is V11() real ext-real Element of REAL
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
B . gf1 is left_complementable right_complementable complementable Element of the carrier of (X)
(B . h) - (B . gf1) is left_complementable right_complementable complementable Element of the carrier of (X)
- (B . gf1) is left_complementable right_complementable complementable Element of the carrier of (X)
(B . h) + (- (B . gf1)) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((B . h),(- (B . gf1))) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . h),(- (B . gf1))] is set
{(B . h),(- (B . gf1))} is non empty set
{(B . h)} is non empty set
{{(B . h),(- (B . gf1))},{(B . h)}} is non empty set
the addF of (X) . [(B . h),(- (B . gf1))] is set
||.((B . h) - (B . gf1)).|| is V11() real ext-real Element of REAL
the U8 of (X) . ((B . h) - (B . gf1)) is V11() real ext-real Element of REAL
h1 . gf1 is V11() real ext-real Element of REAL
(h1 . gf1) - (h1 . h) is V11() real ext-real Element of REAL
- (h1 . h) is V11() real ext-real set
(h1 . gf1) + (- (h1 . h)) is V11() real ext-real set
abs ((h1 . gf1) - (h1 . h)) is V11() real ext-real Element of REAL
(h1 . h) - (h1 . gf1) is V11() real ext-real Element of REAL
- (h1 . gf1) is V11() real ext-real set
(h1 . h) + (- (h1 . gf1)) is V11() real ext-real set
abs ((h1 . h) - (h1 . gf1)) is V11() real ext-real Element of REAL
gf1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
(h1 - g1) . gf1 is V11() real ext-real Element of REAL
h1 . gf1 is V11() real ext-real Element of REAL
- g1 is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() Element of bool [:NAT,REAL:]
(- g1) . gf1 is V11() real ext-real Element of REAL
(h1 . gf1) + ((- g1) . gf1) is V11() real ext-real Element of REAL
g1 . gf1 is V11() real ext-real Element of REAL
- (g1 . gf1) is V11() real ext-real Element of REAL
(h1 . gf1) + (- (g1 . gf1)) is V11() real ext-real Element of REAL
(h1 . gf1) - (h1 . h) is V11() real ext-real Element of REAL
- (h1 . h) is V11() real ext-real set
(h1 . gf1) + (- (h1 . h)) is V11() real ext-real set
gf1 is set
hf1 . gf1 is V11() real ext-real Element of REAL
gPh1 is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h1 . gPh1 is V11() real ext-real Element of REAL
(h1 . gPh1) - (h1 . h) is V11() real ext-real Element of REAL
(h1 . gPh1) + (- (h1 . h)) is V11() real ext-real set
abs ((h1 . gPh1) - (h1 . h)) is V11() real ext-real Element of REAL
(h1 - g1) . gPh1 is V11() real ext-real Element of REAL
abs ((h1 - g1) . gPh1) is V11() real ext-real Element of REAL
abs (h1 - g1) is non empty Relation-like NAT -defined REAL -valued Function-like V23( NAT ) quasi_total V150() V151() V152() bounded_below Element of bool [:NAT,REAL:]
(abs (h1 - g1)) . gf1 is V11() real ext-real Element of REAL
lim hf1 is V11() real ext-real Element of REAL
lim g1 is V11() real ext-real Element of REAL
g1 . 0 is V11() real ext-real Element of REAL
lim (h1 - g1) is V11() real ext-real Element of REAL
(B . f1) - (h1 . h) is V11() real ext-real Element of REAL
(B . f1) + (- (h1 . h)) is V11() real ext-real set
abs ((B . f1) - (h1 . h)) is V11() real ext-real Element of REAL
(h1 . h) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
(h1 . h) + (- (B . f1)) is V11() real ext-real set
abs ((h1 . h) - (B . f1)) is V11() real ext-real Element of REAL
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
(X,(B . h)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . h)) . f1 is V11() real ext-real Element of REAL
((X,(B . h)) . f1) - (B . f1) is V11() real ext-real Element of REAL
((X,(B . h)) . f1) + (- (B . f1)) is V11() real ext-real set
abs (((X,(B . h)) . f1) - (B . f1)) is V11() real ext-real Element of REAL
f1 is Element of X
(X,(B . h)) . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
((X,(B . h)) . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
((X,(B . h)) . f1) + (- (B . f1)) is V11() real ext-real set
abs (((X,(B . h)) . f1) - (B . f1)) is V11() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
(X,(B . h)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 is Element of X
(X,(B . h)) . f1 is V11() real ext-real Element of REAL
B . f1 is V11() real ext-real Element of REAL
((X,(B . h)) . f1) - (B . f1) is V11() real ext-real Element of REAL
- (B . f1) is V11() real ext-real set
((X,(B . h)) . f1) + (- (B . f1)) is V11() real ext-real set
abs (((X,(B . h)) . f1) - (B . f1)) is V11() real ext-real Element of REAL
ONE is left_complementable right_complementable complementable Element of the carrier of (X)
f is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(B . h) - ONE is left_complementable right_complementable complementable Element of the carrier of (X)
- ONE is left_complementable right_complementable complementable Element of the carrier of (X)
(B . h) + (- ONE) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((B . h),(- ONE)) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . h),(- ONE)] is set
{(B . h),(- ONE)} is non empty set
{(B . h)} is non empty set
{{(B . h),(- ONE)},{(B . h)}} is non empty set
the addF of (X) . [(B . h),(- ONE)] is set
h1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,(B . h)) is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 is Element of X
h1 . g1 is V11() real ext-real Element of REAL
f1 . g1 is V11() real ext-real Element of REAL
B . g1 is V11() real ext-real Element of REAL
(f1 . g1) - (B . g1) is V11() real ext-real Element of REAL
- (B . g1) is V11() real ext-real set
(f1 . g1) + (- (B . g1)) is V11() real ext-real set
abs (h1 . g1) is V11() real ext-real Element of REAL
g1 is V11() real ext-real Element of REAL
(X,h1) is non empty V177() V178() V179() Element of bool REAL
{ (abs (h1 . b₁)) where b₁ is Element of X : verum } is set
F1 is Element of X
h1 . F1 is V11() real ext-real Element of REAL
abs (h1 . F1) is V11() real ext-real Element of REAL
upper_bound (X,h1) is V11() real ext-real Element of REAL
||.((B . h) - ONE).|| is V11() real ext-real Element of REAL
the U8 of (X) . ((B . h) - ONE) is V11() real ext-real Element of REAL
g1 is V11() real ext-real set
f is V11() real ext-real Element of REAL
f / 2 is V11() real ext-real Element of REAL
2 " is non empty V11() real ext-real positive non negative set
f * (2 ") is V11() real ext-real set
g is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real V177() V178() V179() V180() V181() V182() Element of NAT
B . h is left_complementable right_complementable complementable Element of the carrier of (X)
(B . h) - ONE is left_complementable right_complementable complementable Element of the carrier of (X)
- ONE is left_complementable right_complementable complementable Element of the carrier of (X)
(B . h) + (- ONE) is left_complementable right_complementable complementable Element of the carrier of (X)
the addF of (X) . ((B . h),(- ONE)) is left_complementable right_complementable complementable Element of the carrier of (X)
[(B . h),(- ONE)] is set
{(B . h),(- ONE)} is non empty set
{(B . h)} is non empty set
{{(B . h),(- ONE)},{(B . h)}} is non empty set
the addF of (X) . [(B . h),(- ONE)] is set
||.((B . h) - ONE).|| is V11() real ext-real Element of REAL
the U8 of (X) . ((B . h) - ONE) is V11() real ext-real Element of REAL
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
the carrier of (X) is non empty set
[:NAT, the carrier of (X):] is non empty set
bool [:NAT, the carrier of (X):] is non empty set
B is non empty Relation-like NAT -defined the carrier of (X) -valued Function-like V23( NAT ) quasi_total Element of bool [:NAT, the carrier of (X):]
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like unital Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
X is non empty set
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like unital complete Normed_AlgebraStr
(X) is non empty add-closed ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) ( RAlgebra X) Element of bool the carrier of (RAlgebra X)
RAlgebra X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() strict vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
Funcs (X,REAL) is non empty functional FUNCTION_DOMAIN of X, REAL
RealFuncMult X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:(Funcs (X,REAL)),(Funcs (X,REAL)):] is non empty set
[:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncAdd X is non empty Relation-like [:(Funcs (X,REAL)),(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:(Funcs (X,REAL)),(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:(Funcs (X,REAL)),(Funcs (X,REAL)):],(Funcs (X,REAL)):]
RealFuncExtMult X is non empty Relation-like [:REAL,(Funcs (X,REAL)):] -defined Funcs (X,REAL) -valued Function-like V23([:REAL,(Funcs (X,REAL)):]) quasi_total Function-yielding V117() Element of bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):]
[:REAL,(Funcs (X,REAL)):] is non empty set
[:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
bool [:[:REAL,(Funcs (X,REAL)):],(Funcs (X,REAL)):] is non empty set
RealFuncUnit X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 1 is non empty Relation-like non-empty X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
[:X,NAT:] is non empty RAT -valued INT -valued V150() V151() V152() V153() set
bool [:X,NAT:] is non empty set
RealFuncZero X is Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of Funcs (X,REAL)
X --> 0 is non empty Relation-like X -defined NAT -valued RAT -valued INT -valued Function-like constant V23(X) quasi_total V150() V151() V152() V153() bounded Element of bool [:X,NAT:]
AlgebraStr(# (Funcs (X,REAL)),(RealFuncMult X),(RealFuncAdd X),(RealFuncExtMult X),(RealFuncUnit X),(RealFuncZero X) #) is strict AlgebraStr
the carrier of (RAlgebra X) is non empty set
bool the carrier of (RAlgebra X) is non empty set
[:X,REAL:] is non empty V150() V151() V152() set
bool [:X,REAL:] is non empty set
{ b₁ where b₁ is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:] : b₁ | X is bounded } is set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty set
[:[:(X),(X):],(X):] is non empty set
bool [:[:(X),(X):],(X):] is non empty set
((RAlgebra X),(X)) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like V23([:(X),(X):]) quasi_total Element of bool [:[:(X),(X):],(X):]
((RAlgebra X),(X)) is non empty Relation-like [:REAL,(X):] -defined (X) -valued Function-like V23([:REAL,(X):]) quasi_total Element of bool [:[:REAL,(X):],(X):]
[:REAL,(X):] is non empty set
[:[:REAL,(X):],(X):] is non empty set
bool [:[:REAL,(X):],(X):] is non empty set
((RAlgebra X),(X)) is Element of (X)
((RAlgebra X),(X)) is Element of (X)
(X) is non empty Relation-like (X) -defined REAL -valued Function-like V23((X)) quasi_total V150() V151() V152() Element of bool [:(X),REAL:]
[:(X),REAL:] is non empty V150() V151() V152() set
bool [:(X),REAL:] is non empty set
Normed_AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),(X) #) is strict Normed_AlgebraStr
B is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V100() discerning reflexive RealNormSpace-like vector-associative associative right-distributive right_unital Normed_AlgebraStr
1. B is left_complementable right_complementable complementable Element of the carrier of B
the carrier of B is non empty set
the OneF of B is left_complementable right_complementable complementable Element of the carrier of B
ONE is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
ONE | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative V100() vector-associative unital associative commutative right-distributive right_unital well-unital left_unital AlgebraStr
AlgebraStr(# (X),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)),((RAlgebra X),(X)) #) is strict AlgebraStr
1_ (X) is left_complementable right_complementable complementable Element of the carrier of (X)
the carrier of (X) is non empty 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)
(X,ONE) is non empty V177() V178() V179() Element of bool REAL
{ (abs (ONE . b₁)) where b₁ is Element of X : verum } is set
the Element of X is Element of X
g is set
(X --> 1) . the Element of X is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real Element of REAL
h is Element of X
(X --> 1) . h is epsilon-transitive epsilon-connected ordinal natural V11() real V111() V112() ext-real Element of REAL
abs ((X --> 1) . h) is V11() real V112() ext-real Element of REAL
abs ((X --> 1) . the Element of X) is V11() real V112() ext-real Element of REAL
{1} is non empty V177() V178() V179() V180() V181() V182() Element of bool REAL
upper_bound (X,ONE) is V11() real ext-real Element of REAL
||.(1. B).|| is V11() real ext-real Element of REAL
the U8 of B is non empty Relation-like the carrier of B -defined REAL -valued Function-like V23( the carrier of B) quasi_total V150() V151() V152() Element of bool [: the carrier of B,REAL:]
[: the carrier of B,REAL:] is non empty V150() V151() V152() set
bool [: the carrier of B,REAL:] is non empty set
the U8 of B . (1. B) is V11() real ext-real Element of REAL
g is left_complementable right_complementable complementable Element of the carrier of B
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h is left_complementable right_complementable complementable Element of the carrier of B
h1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g * h is left_complementable right_complementable complementable Element of the carrier of B
the multF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
the multF of B . (g,h) is left_complementable right_complementable complementable Element of the carrier of B
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the multF of B . [g,h] is set
f is V11() real ext-real Element of REAL
f * (g * h) is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B is non empty Relation-like [:REAL, the carrier of B:] -defined the carrier of B -valued Function-like V23([:REAL, the carrier of B:]) quasi_total Element of bool [:[:REAL, the carrier of B:], the carrier of B:]
[:REAL, the carrier of B:] is non empty set
[:[:REAL, the carrier of B:], the carrier of B:] is non empty set
bool [:[:REAL, the carrier of B:], the carrier of B:] is non empty set
the Mult of B . (f,(g * h)) is set
[f,(g * h)] is set
{f,(g * h)} is non empty set
{f} is non empty V177() V178() V179() set
{{f,(g * h)},{f}} is non empty set
the Mult of B . [f,(g * h)] is set
g1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
F1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
F1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
f * h is left_complementable right_complementable complementable Element of the carrier of B
the Mult of B . (f,h) is set
[f,h] is set
{f,h} is non empty set
{{f,h},{f}} is non empty set
the Mult of B . [f,h] is set
hf1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
hf1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
gf1 is Element of X
g1 . gf1 is V11() real ext-real Element of REAL
F1 . gf1 is V11() real ext-real Element of REAL
f * (F1 . gf1) is V11() real ext-real Element of REAL
f1 . gf1 is V11() real ext-real Element of REAL
h1 . gf1 is V11() real ext-real Element of REAL
(f1 . gf1) * (h1 . gf1) is V11() real ext-real Element of REAL
f * ((f1 . gf1) * (h1 . gf1)) is V11() real ext-real Element of REAL
f * (h1 . gf1) is V11() real ext-real Element of REAL
(f1 . gf1) * (f * (h1 . gf1)) is V11() real ext-real Element of REAL
hf1 . gf1 is V11() real ext-real Element of REAL
(f1 . gf1) * (hf1 . gf1) is V11() real ext-real Element of REAL
g * (f * h) is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (g,(f * h)) is left_complementable right_complementable complementable Element of the carrier of B
[g,(f * h)] is set
{g,(f * h)} is non empty set
{{g,(f * h)},{g}} is non empty set
the multF of B . [g,(f * h)] is set
f is left_complementable right_complementable complementable Element of the carrier of B
h is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g is left_complementable right_complementable complementable Element of the carrier of B
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,f1) is non empty V177() V178() V179() Element of bool REAL
{ (abs (f1 . b₁)) where b₁ is Element of X : verum } is set
f * g is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (f,g) is left_complementable right_complementable complementable Element of the carrier of B
[f,g] is set
{f,g} is non empty set
{f} is non empty set
{{f,g},{f}} is non empty set
the multF of B . [f,g] is set
h1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
(X,h) is non empty V177() V178() V179() Element of bool REAL
{ (abs (h . b₁)) where b₁ is Element of X : verum } is set
g1 is V11() real ext-real set
(X,h1) is non empty V177() V178() V179() Element of bool REAL
{ (abs (h1 . b₁)) where b₁ is Element of X : verum } is set
F1 is Element of X
h1 . F1 is V11() real ext-real Element of REAL
abs (h1 . F1) is V11() real ext-real Element of REAL
f1 . F1 is V11() real ext-real Element of REAL
abs (f1 . F1) is V11() real ext-real Element of REAL
upper_bound (X,f1) is V11() real ext-real Element of REAL
h . F1 is V11() real ext-real Element of REAL
abs (h . F1) is V11() real ext-real Element of REAL
upper_bound (X,h) is V11() real ext-real Element of REAL
(upper_bound (X,h)) * (abs (f1 . F1)) is V11() real ext-real Element of REAL
(upper_bound (X,h)) * (upper_bound (X,f1)) is V11() real ext-real Element of REAL
(abs (h . F1)) * (abs (f1 . F1)) is V11() real ext-real Element of REAL
(h . F1) * (f1 . F1) is V11() real ext-real Element of REAL
abs ((h . F1) * (f1 . F1)) is V11() real ext-real Element of REAL
upper_bound (X,h1) is V11() real ext-real Element of REAL
||.g.|| is V11() real ext-real Element of REAL
the U8 of B . g is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real Element of REAL
the U8 of B . f is V11() real ext-real Element of REAL
||.(f * g).|| is V11() real ext-real Element of REAL
the U8 of B . (f * g) is V11() real ext-real Element of REAL
||.f.|| * ||.g.|| is V11() real ext-real Element of REAL
f is left_complementable right_complementable complementable Element of the carrier of B
f1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
f1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h is left_complementable right_complementable complementable Element of the carrier of B
h1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
h1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g is left_complementable right_complementable complementable Element of the carrier of B
g1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
g1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g + h is left_complementable right_complementable complementable Element of the carrier of B
the addF of B is non empty Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V23([: the carrier of B, the carrier of B:]) quasi_total Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
the addF of B . (g,h) is left_complementable right_complementable complementable Element of the carrier of B
[g,h] is set
{g,h} is non empty set
{g} is non empty set
{{g,h},{g}} is non empty set
the addF of B . [g,h] is set
(g + h) * f is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . ((g + h),f) is left_complementable right_complementable complementable Element of the carrier of B
[(g + h),f] is set
{(g + h),f} is non empty set
{(g + h)} is non empty set
{{(g + h),f},{(g + h)}} is non empty set
the multF of B . [(g + h),f] is set
F1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
F1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
h * f is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (h,f) is left_complementable right_complementable complementable Element of the carrier of B
[h,f] is set
{h,f} is non empty set
{h} is non empty set
{{h,f},{h}} is non empty set
the multF of B . [h,f] is set
hf1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
hf1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
g * f is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . (g,f) is left_complementable right_complementable complementable Element of the carrier of B
[g,f] is set
{g,f} is non empty set
{{g,f},{g}} is non empty set
the multF of B . [g,f] is set
gf1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
gf1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
gPh1 is non empty Relation-like X -defined REAL -valued Function-like V23(X) quasi_total V150() V151() V152() Element of bool [:X,REAL:]
gPh1 | X is Relation-like X -defined REAL -valued Function-like V150() V151() V152() Element of bool [:X,REAL:]
x is Element of X
F1 . x is V11() real ext-real Element of REAL
gPh1 . x is V11() real ext-real Element of REAL
f1 . x is V11() real ext-real Element of REAL
(gPh1 . x) * (f1 . x) is V11() real ext-real Element of REAL
g1 . x is V11() real ext-real Element of REAL
h1 . x is V11() real ext-real Element of REAL
(g1 . x) + (h1 . x) is V11() real ext-real Element of REAL
((g1 . x) + (h1 . x)) * (f1 . x) is V11() real ext-real Element of REAL
(g1 . x) * (f1 . x) is V11() real ext-real Element of REAL
(h1 . x) * (f1 . x) is V11() real ext-real Element of REAL
((g1 . x) * (f1 . x)) + ((h1 . x) * (f1 . x)) is V11() real ext-real Element of REAL
gf1 . x is V11() real ext-real Element of REAL
(gf1 . x) + ((h1 . x) * (f1 . x)) is V11() real ext-real Element of REAL
hf1 . x is V11() real ext-real Element of REAL
(gf1 . x) + (hf1 . x) is V11() real ext-real Element of REAL
(g * f) + (h * f) is left_complementable right_complementable complementable Element of the carrier of B
the addF of B . ((g * f),(h * f)) is left_complementable right_complementable complementable Element of the carrier of B
[(g * f),(h * f)] is set
{(g * f),(h * f)} is non empty set
{(g * f)} is non empty set
{{(g * f),(h * f)},{(g * f)}} is non empty set
the addF of B . [(g * f),(h * f)] is set
f is left_complementable right_complementable complementable Element of the carrier of B
(1. B) * f is left_complementable right_complementable complementable Element of the carrier of B
the multF of B . ((1. B),f) is left_complementable right_complementable complementable Element of the carrier of B
[(1. B),f] is set
{(1. B),f} is non empty set
{(1. B)} is non empty set
{{(1. B),f},{(1. B)}} is non empty set
the multF of B . [(1. B),f] is set