REAL is set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal V41() V46() V47() Element of bool REAL
bool REAL is non empty set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal V41() V46() V47() set
bool omega is non empty non trivial V41() set
bool NAT is non empty non trivial V41() set
COMPLEX is set
K356() is set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() Element of NAT
{{},1} is non empty V41() V45() set
K434() is set
bool K434() is non empty set
K435() is Element of bool K434()
K475() is non empty V171() L13()
the carrier of K475() is non empty set
K438( the carrier of K475()) is non empty M34( the carrier of K475())
K474(K475()) is Element of bool K438( the carrier of K475())
bool K438( the carrier of K475()) is non empty set
[:K474(K475()),NAT:] is Relation-like set
bool [:K474(K475()),NAT:] is non empty set
[:NAT,K474(K475()):] is Relation-like set
bool [:NAT,K474(K475()):] is non empty set
2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() Element of NAT
K347(2,K356()) is M18(K356())
[:K347(2,K356()),K356():] is Relation-like set
bool [:K347(2,K356()),K356():] is non empty set
3 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() Element of NAT
K347(3,K356()) is M18(K356())
[:K347(3,K356()),K356():] is Relation-like set
bool [:K347(3,K356()),K356():] is non empty set
K531() is Relation-like K347(2,K356()) -defined K356() -valued Function-like quasi_total Element of bool [:K347(2,K356()),K356():]
[:NAT,NAT:] is non empty non trivial Relation-like V41() set
[:[:NAT,NAT:],NAT:] is non empty non trivial Relation-like V41() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial V41() set
Nat_Lattice is non empty V239() V246() V249() L23()
the carrier of Nat_Lattice is non empty set
NATPLUS is non empty Element of bool NAT
[:NATPLUS,NATPLUS:] is non empty Relation-like set
[:[:NATPLUS,NATPLUS:],NATPLUS:] is non empty Relation-like set
bool [:[:NATPLUS,NATPLUS:],NATPLUS:] is non empty set
Seg 1 is Element of bool NAT
{1} is non empty trivial V41() V45() V48(1) set
Seg 2 is Element of bool NAT
{1,2} is non empty V41() V45() set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of NAT
{{}} is non empty trivial functional V41() V45() V48(1) set
meet {} is set
[:{},{}:] is Relation-like V41() set
bool [:{},{}:] is non empty V41() V45() set
X is set
[:NAT,X:] is Relation-like set
bool [:NAT,X:] is non empty set
Y is Relation-like NAT -defined X -valued Function-like quasi_total Element of bool [:NAT,X:]
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Y | f is Relation-like NAT -defined f -defined NAT -defined X -valued Function-like V41() Element of bool [:NAT,X:]
dom Y is Element of bool NAT
dom (Y | f) is V41() Element of bool f
bool f is non empty V41() V45() set
Y is set
X is set
<%> X is T-Sequence-like Relation-like X -valued Function-like V41() set
X is non empty set
X ^omega is set
[:(X ^omega),X:] is Relation-like set
bool [:(X ^omega),X:] is non empty set
Y is Relation-like X ^omega -defined X -valued Function-like total quasi_total Element of bool [:(X ^omega),X:]
f is T-Sequence-like Relation-like X -valued Function-like V41() set
Y . f is set
Y is set
X is set
the_universe_of X is set
f is set
[:Y,f:] is Relation-like set
the_transitive-closure_of X is set
Tarski-Class (the_transitive-closure_of X) is set
F2() is Relation-like Function-like set
proj1 F2() is set
F3() is Relation-like Function-like set
proj1 F3() is set
X is T-Sequence-like Relation-like Function-like set
proj1 X is epsilon-transitive epsilon-connected ordinal set
f is epsilon-transitive epsilon-connected ordinal set
y is T-Sequence-like Relation-like Function-like set
X | f is T-Sequence-like Relation-like proj2 X -valued Function-like set
proj2 X is set
X . f is set
F1(y) is set
Y is T-Sequence-like Relation-like Function-like set
proj1 Y is epsilon-transitive epsilon-connected ordinal set
f is epsilon-transitive epsilon-connected ordinal set
y is T-Sequence-like Relation-like Function-like set
Y | f is T-Sequence-like Relation-like proj2 Y -valued Function-like set
proj2 Y is set
Y . f is set
F1(y) is set
X is T-Sequence-like Relation-like Function-like set
proj1 X is epsilon-transitive epsilon-connected ordinal set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X . Y is set
X | Y is T-Sequence-like Relation-like proj2 X -valued Function-like V41() set
proj2 X is set
F1((X | Y)) is set
f is epsilon-transitive epsilon-connected ordinal set
X . f is set
X | f is T-Sequence-like Relation-like proj2 X -valued Function-like set
F1((X | f)) is set
F1() is non empty set
[:NAT,F1():] is non empty non trivial Relation-like V41() set
bool [:NAT,F1():] is non empty non trivial V41() set
F3() is non empty Relation-like NAT -defined F1() -valued Function-like total quasi_total Element of bool [:NAT,F1():]
F4() is non empty Relation-like NAT -defined F1() -valued Function-like total quasi_total Element of bool [:NAT,F1():]
X is Relation-like Function-like set
proj1 X is set
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X . f is set
X | f is Relation-like Function-like V41() set
(F1(),(X | f)) is T-Sequence-like Relation-like F1() -valued Function-like V41() set
F2((F1(),(X | f))) is Element of F1()
F3() | f is T-Sequence-like Relation-like NAT -defined f -defined NAT -defined F1() -valued Function-like V41() Element of bool [:NAT,F1():]
F2((F3() | f)) is Element of F1()
Y is Relation-like Function-like set
proj1 Y is set
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Y . f is set
Y | f is Relation-like Function-like V41() set
(F1(),(Y | f)) is T-Sequence-like Relation-like F1() -valued Function-like V41() set
F2((F1(),(Y | f))) is Element of F1()
F4() | f is T-Sequence-like Relation-like NAT -defined f -defined NAT -defined F1() -valued Function-like V41() Element of bool [:NAT,F1():]
F2((F4() | f)) is Element of F1()
F1() is non empty set
[:NAT,F1():] is non empty non trivial Relation-like V41() set
bool [:NAT,F1():] is non empty non trivial V41() set
X is Relation-like Function-like set
proj1 X is set
proj2 X is set
Y is set
f is set
X . f is set
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X | y is Relation-like Function-like V41() set
(F1(),(X | y)) is T-Sequence-like Relation-like F1() -valued Function-like V41() set
F2((F1(),(X | y))) is Element of F1()
Y is non empty Relation-like NAT -defined F1() -valued Function-like total quasi_total Element of bool [:NAT,F1():]
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Y . f is set
Y | f is T-Sequence-like Relation-like NAT -defined f -defined NAT -defined F1() -valued Function-like V41() Element of bool [:NAT,F1():]
F2((Y | f)) is Element of F1()
X . f is set
(F1(),(Y | f)) is T-Sequence-like Relation-like F1() -valued Function-like V41() set
F2((F1(),(Y | f))) is Element of F1()
[:[:{},{}:],{}:] is Relation-like V41() set
bool [:[:{},{}:],{}:] is non empty V41() V45() set
the empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:{},{}:] -defined {} -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:{},{}:],{}:] is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:{},{}:] -defined {} -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:{},{}:],{}:]
multMagma(# {}, the empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:{},{}:] -defined {} -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:{},{}:],{}:] #) is strict multMagma
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
X is multMagma
the carrier of X is set
nabla the carrier of X is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V258() V260() V265() Element of bool [: the carrier of X, the carrier of X:]
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
f is Element of the carrier of X
y is Element of the carrier of X
Class ((nabla the carrier of X),y) is Element of bool the carrier of X
bool the carrier of X is non empty set
x is Element of the carrier of X
c6 is Element of the carrier of X
Class ((nabla the carrier of X),c6) is Element of bool the carrier of X
f * x is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,x) is Element of the carrier of X
y * c6 is Element of the carrier of X
the multF of X . (y,c6) is Element of the carrier of X
Class ((nabla the carrier of X),(y * c6)) is Element of bool the carrier of X
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
nabla the carrier of X is Relation-like the carrier of X -defined the carrier of X -defined the carrier of X -valued the carrier of X -valued total total quasi_total quasi_total V258() V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() Element of bool [: the carrier of X, the carrier of X:]
f is Element of the carrier of X
Class (Y,f) is Element of bool the carrier of X
bool the carrier of X is non empty set
y is Element of the carrier of X
Class (Y,y) is Element of bool the carrier of X
x is Element of the carrier of X
Class (Y,x) is Element of bool the carrier of X
c6 is Element of the carrier of X
Class (Y,c6) is Element of bool the carrier of X
f * x is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,x) is Element of the carrier of X
Class (Y,(f * x)) is Element of bool the carrier of X
y * c6 is Element of the carrier of X
the multF of X . (y,c6) is Element of the carrier of X
Class (Y,(y * c6)) is Element of bool the carrier of X
f * x is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,x) is Element of the carrier of X
y * c6 is Element of the carrier of X
the multF of X . (y,c6) is Element of the carrier of X
Class (Y,(y * c6)) is Element of bool the carrier of X
Class (Y,(f * x)) is Element of bool the carrier of X
f is Element of the carrier of X
y is Element of the carrier of X
Class (Y,y) is Element of bool the carrier of X
x is Element of the carrier of X
c6 is Element of the carrier of X
Class (Y,c6) is Element of bool the carrier of X
f * x is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,x) is Element of the carrier of X
y * c6 is Element of the carrier of X
the multF of X . (y,c6) is Element of the carrier of X
Class (Y,(y * c6)) is Element of bool the carrier of X
Class (Y,f) is Element of bool the carrier of X
Class (Y,x) is Element of bool the carrier of X
Class (Y,(f * x)) is Element of bool the carrier of X
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
Class Y is with_non-empty_elements a_partition of the carrier of X
[:(Class Y),(Class Y):] is Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
f is non empty set
y is Element of f
x is Element of f
c6 is set
Class (Y,c6) is Element of bool the carrier of X
bool the carrier of X is non empty set
w2 is set
Class (Y,w2) is Element of bool the carrier of X
w1 is Element of the carrier of X
v1 is Element of the carrier of X
w1 * v1 is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (w1,v1) is Element of the carrier of X
Class (Y,(w1 * v1)) is Element of bool the carrier of X
v2 is Element of f
c11 is Element of Class Y
w9 is Element of Class Y
z2 is Element of the carrier of X
Class (Y,z2) is Element of bool the carrier of X
z1 is Element of the carrier of X
Class (Y,z1) is Element of bool the carrier of X
z2 * z1 is Element of the carrier of X
the multF of X . (z2,z1) is Element of the carrier of X
Class (Y,(z2 * z1)) is Element of bool the carrier of X
[:f,f:] is non empty Relation-like set
[:[:f,f:],f:] is non empty Relation-like set
bool [:[:f,f:],f:] is non empty set
y is non empty Relation-like [:f,f:] -defined f -valued Function-like total quasi_total Element of bool [:[:f,f:],f:]
x is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
c6 is Element of Class Y
w2 is Element of the carrier of X
Class (Y,w2) is Element of bool the carrier of X
bool the carrier of X is non empty set
w1 is Element of Class Y
v1 is Element of the carrier of X
Class (Y,v1) is Element of bool the carrier of X
x . (c6,w1) is Element of Class Y
w2 * v1 is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (w2,v1) is Element of the carrier of X
Class (Y,(w2 * v1)) is Element of bool the carrier of X
the Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):] is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
f is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
y is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
x is set
c6 is set
f . (x,c6) is set
y . (x,c6) is set
w1 is Element of Class Y
v1 is set
Class (Y,v1) is Element of bool the carrier of X
bool the carrier of X is non empty set
w2 is Element of Class Y
v2 is set
Class (Y,v2) is Element of bool the carrier of X
f . (w1,w2) is Element of Class Y
c11 is Element of the carrier of X
w9 is Element of the carrier of X
c11 * w9 is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (c11,w9) is Element of the carrier of X
Class (Y,(c11 * w9)) is Element of bool the carrier of X
y is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
x is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
c6 is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
w1 is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
w2 is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
v1 is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
Class Y is with_non-empty_elements a_partition of the carrier of X
(X,Y) is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,Y) is multMagma
Class Y is with_non-empty_elements a_partition of the carrier of X
(X,Y) is Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
X is non empty multMagma
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,Y) is strict multMagma
Class Y is non empty with_non-empty_elements a_partition of the carrier of X
(X,Y) is non empty Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like total quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is non empty Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is non empty Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
X is non empty multMagma
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,Y) is non empty strict multMagma
Class Y is non empty with_non-empty_elements a_partition of the carrier of X
(X,Y) is non empty Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like total quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is non empty Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is non empty Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
the carrier of (X,Y) is non empty set
[: the carrier of X, the carrier of (X,Y):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (X,Y):] is non empty set
f is set
y is Element of the carrier of X
Class (Y,y) is Element of bool the carrier of X
bool the carrier of X is non empty set
x is set
f is Relation-like Function-like set
proj1 f is set
proj2 f is set
y is set
x is set
f . x is set
c6 is Element of the carrier of X
Class (Y,c6) is Element of bool the carrier of X
bool the carrier of X is non empty set
y is non empty Relation-like the carrier of X -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of (X,Y):]
x is Element of the carrier of X
y . x is Element of the carrier of (X,Y)
Class (Y,x) is Element of bool the carrier of X
bool the carrier of X is non empty set
c6 is Element of the carrier of X
Class (Y,c6) is Element of bool the carrier of X
f is non empty Relation-like the carrier of X -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of (X,Y):]
y is non empty Relation-like the carrier of X -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of (X,Y):]
x is Element of the carrier of X
f . x is Element of the carrier of (X,Y)
Class (Y,x) is Element of bool the carrier of X
bool the carrier of X is non empty set
y . x is Element of the carrier of (X,Y)
X is non empty multMagma
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,Y) is non empty strict multMagma
Class Y is non empty with_non-empty_elements a_partition of the carrier of X
(X,Y) is non empty Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like total quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is non empty Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is non empty Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
(X,Y) is non empty Relation-like the carrier of X -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of (X,Y):]
the carrier of (X,Y) is non empty set
[: the carrier of X, the carrier of (X,Y):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (X,Y):] is non empty set
f is Element of the carrier of X
y is Element of the carrier of X
f * y 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 total 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:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
(X,Y) . (f * y) is Element of the carrier of (X,Y)
(X,Y) . f is Element of the carrier of (X,Y)
(X,Y) . y is Element of the carrier of (X,Y)
((X,Y) . f) * ((X,Y) . y) is Element of the carrier of (X,Y)
the multF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the multF of (X,Y) . (((X,Y) . f),((X,Y) . y)) is Element of the carrier of (X,Y)
Class (Y,f) is Element of bool the carrier of X
bool the carrier of X is non empty set
Class (Y,y) is Element of bool the carrier of X
(X,Y) . (((X,Y) . f),((X,Y) . y)) is set
Class (Y,(f * y)) is Element of bool the carrier of X
X is non empty multMagma
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,Y) is non empty strict multMagma
Class Y is non empty with_non-empty_elements a_partition of the carrier of X
(X,Y) is non empty Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like total quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is non empty Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is non empty Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),(X,Y) #) is strict multMagma
the carrier of (X,Y) is non empty set
(X,Y) is non empty Relation-like the carrier of X -defined the carrier of (X,Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of X, the carrier of (X,Y):]
[: the carrier of X, the carrier of (X,Y):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (X,Y):] is non empty set
rng (X,Y) is non empty Element of bool the carrier of (X,Y)
bool the carrier of (X,Y) is non empty set
f is set
y is set
Class (Y,y) is Element of bool the carrier of X
bool the carrier of X is non empty set
dom (X,Y) is non empty Element of bool the carrier of X
x is Element of the carrier of X
(X,Y) . x is Element of the carrier of (X,Y)
Class (Y,x) is Element of bool the carrier of X
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like [: the carrier of X, the carrier of X:] -valued Function-like set
proj1 Y is set
{ b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
meet { b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
y is set
x is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
bool {} is non empty V41() V45() set
nabla the carrier of X is Relation-like the carrier of X -defined the carrier of X -defined the carrier of X -valued the carrier of X -valued total total quasi_total quasi_total V258() V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
y is set
Y . y is set
x is Element of the carrier of X
c6 is Element of the carrier of X
[x,c6] is V26() set
{x,c6} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,c6},{x}} is non empty V41() V45() set
Class ((nabla the carrier of X),c6) is Element of bool the carrier of X
bool the carrier of X is non empty set
bool [: the carrier of X, the carrier of X:] is non empty Element of bool (bool [: the carrier of X, the carrier of X:])
bool (bool [: the carrier of X, the carrier of X:]) is non empty set
y is set
x is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
y is Element of bool (bool [: the carrier of X, the carrier of X:])
x is set
c6 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is Relation-like [: the carrier of X, the carrier of X:] -valued Function-like set
proj1 Y is set
(X,Y) is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() Element of bool [: the carrier of X, the carrier of X:]
{ b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
meet { b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
f is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
y is set
X is multMagma
the carrier of X is set
[: the carrier of X, the carrier of X:] is Relation-like set
Y is Relation-like [: the carrier of X, the carrier of X:] -valued Function-like set
(X,Y) is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() Element of bool [: the carrier of X, the carrier of X:]
bool [: the carrier of X, the carrier of X:] is non empty set
proj1 Y is set
{ b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
meet { b1 where b1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:] : for b2 being set
for b3, b4 being Element of the carrier of X holds
( not b2 in proj1 Y or not Y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
y is Element of the carrier of X
x is Element of the carrier of X
Class ((X,Y),x) is Element of bool the carrier of X
bool the carrier of X is non empty set
c6 is Element of the carrier of X
w1 is Element of the carrier of X
Class ((X,Y),w1) is Element of bool the carrier of X
y * c6 is Element of the carrier of X
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:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (y,c6) is Element of the carrier of X
x * w1 is Element of the carrier of X
the multF of X . (x,w1) is Element of the carrier of X
Class ((X,Y),(x * w1)) is Element of bool the carrier of X
[x,y] is V26() set
{x,y} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,y},{x}} is non empty V41() V45() set
[w1,c6] is V26() set
{w1,c6} is non empty V41() set
{w1} is non empty trivial V41() V48(1) set
{{w1,c6},{w1}} is non empty V41() V45() set
[(x * w1),(y * c6)] is V26() set
{(x * w1),(y * c6)} is non empty V41() set
{(x * w1)} is non empty trivial V41() V48(1) set
{{(x * w1),(y * c6)},{(x * w1)}} is non empty V41() V45() set
w2 is set
v1 is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
Class (v1,x) is Element of bool the carrier of X
Class (v1,w1) is Element of bool the carrier of X
Class (v1,(x * w1)) is Element of bool the carrier of X
X is set
Y is set
[:X,Y:] is Relation-like set
bool [:X,Y:] is non empty set
[:X,X:] is Relation-like set
bool [:X,X:] is non empty set
f is Relation-like X -defined Y -valued Function-like quasi_total Element of bool [:X,Y:]
y is set
f . y is set
y is set
f . y is set
x is set
f . x is set
y is set
f . y is set
x is set
f . x is set
c6 is set
f . c6 is set
y is Relation-like X -defined X -valued total quasi_total V260() V265() Element of bool [:X,X:]
y is Relation-like X -defined X -valued total quasi_total V260() V265() Element of bool [:X,X:]
x is Relation-like X -defined X -valued total quasi_total V260() V265() Element of bool [:X,X:]
c6 is set
w1 is set
[c6,w1] is V26() set
{c6,w1} is non empty V41() set
{c6} is non empty trivial V41() V48(1) set
{{c6,w1},{c6}} is non empty V41() V45() set
f . c6 is set
f . w1 is set
w2 is set
v1 is set
f . w2 is set
f . v1 is set
[w2,v1] is V26() set
{w2,v1} is non empty V41() set
{w2} is non empty trivial V41() V48(1) set
{{w2,v1},{w2}} is non empty V41() V45() set
c6 is set
w1 is set
[c6,w1] is V26() set
{c6,w1} is non empty V41() set
{c6} is non empty trivial V41() V48(1) set
{{c6,w1},{c6}} is non empty V41() V45() set
f . c6 is set
f . w1 is set
w2 is set
v1 is set
f . w2 is set
f . v1 is set
[w2,v1] is V26() set
{w2,v1} is non empty V41() set
{w2} is non empty trivial V41() V48(1) set
{{w2,v1},{w2}} is non empty V41() V45() set
X is non empty multMagma
the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
bool [: the carrier of X, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
( the carrier of X, the carrier of Y,f) is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() Element of bool [: the carrier of X, the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
x is Element of the carrier of X
c6 is Element of the carrier of X
Class (( the carrier of X, the carrier of Y,f),c6) is Element of bool the carrier of X
bool the carrier of X is non empty set
w1 is Element of the carrier of X
w2 is Element of the carrier of X
Class (( the carrier of X, the carrier of Y,f),w2) is Element of bool the carrier of X
x * w1 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 total 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:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (x,w1) is Element of the carrier of X
c6 * w2 is Element of the carrier of X
the multF of X . (c6,w2) is Element of the carrier of X
Class (( the carrier of X, the carrier of Y,f),(c6 * w2)) is Element of bool the carrier of X
[c6,x] is V26() set
{c6,x} is non empty V41() set
{c6} is non empty trivial V41() V48(1) set
{{c6,x},{c6}} is non empty V41() V45() set
[w2,w1] is V26() set
{w2,w1} is non empty V41() set
{w2} is non empty trivial V41() V48(1) set
{{w2,w1},{w2}} is non empty V41() V45() set
f . w2 is Element of the carrier of Y
f . w1 is Element of the carrier of Y
f . (c6 * w2) is Element of the carrier of Y
f . c6 is Element of the carrier of Y
(f . c6) * (f . w2) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((f . c6),(f . w2)) is Element of the carrier of Y
f . x is Element of the carrier of Y
(f . x) * (f . w1) is Element of the carrier of Y
the multF of Y . ((f . x),(f . w1)) is Element of the carrier of Y
f . (x * w1) is Element of the carrier of Y
[(c6 * w2),(x * w1)] is V26() set
{(c6 * w2),(x * w1)} is non empty V41() set
{(c6 * w2)} is non empty trivial V41() V48(1) set
{{(c6 * w2),(x * w1)},{(c6 * w2)}} is non empty V41() V45() set
X is non empty multMagma
the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of Y, the carrier of X:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of X:] is non empty set
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
f is non empty Relation-like the carrier of Y -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of X:]
( the carrier of Y, the carrier of X,f) is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() Element of bool [: the carrier of Y, the carrier of Y:]
bool [: the carrier of Y, the carrier of Y:] is non empty set
{ [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } is set
id { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } is Relation-like { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } -defined { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } -valued Function-like one-to-one total quasi_total Element of bool [: { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } , { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } :]
[: { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } , { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } :] is Relation-like set
bool [: { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } , { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } :] is non empty set
dom (id { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } ) is Element of bool { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 }
bool { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } is non empty set
rng (id { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } ) is Element of bool { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 }
c6 is set
w1 is set
(id { [b1,b2] where b1, b2 is Element of the carrier of Y : f . b1 = f . b2 } ) . w1 is set
w2 is Element of the carrier of Y
v1 is Element of the carrier of Y
[w2,v1] is V26() set
{w2,v1} is non empty V41() set
{w2} is non empty trivial V41() V48(1) set
{{w2,v1},{w2}} is non empty V41() V45() set
f . w2 is Element of the carrier of X
f . v1 is Element of the carrier of X
c6 is Relation-like [: the carrier of Y, the carrier of Y:] -valued Function-like set
(Y,c6) is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:]
proj1 c6 is set
{ b1 where b1 is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:] : for b2 being set
for b3, b4 being Element of the carrier of Y holds
( not b2 in proj1 c6 or not c6 . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
meet { b1 where b1 is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:] : for b2 being set
for b3, b4 being Element of the carrier of Y holds
( not b2 in proj1 c6 or not c6 . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
w1 is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:]
w2 is set
c6 . w2 is set
v1 is Element of the carrier of Y
v2 is Element of the carrier of Y
[v1,v2] is V26() set
{v1,v2} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,v2},{v1}} is non empty V41() V45() set
Class (w1,v2) is Element of bool the carrier of Y
bool the carrier of Y is non empty set
c11 is Element of the carrier of Y
w9 is Element of the carrier of Y
[c11,w9] is V26() set
{c11,w9} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,w9},{c11}} is non empty V41() V45() set
f . c11 is Element of the carrier of X
f . w9 is Element of the carrier of X
w2 is set
v1 is set
v2 is set
[v1,v2] is V26() set
{v1,v2} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,v2},{v1}} is non empty V41() V45() set
f . v1 is set
f . v2 is set
z1 is set
z2 is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:]
c11 is Element of the carrier of Y
w9 is Element of the carrier of Y
[c11,w9] is V26() set
{c11,w9} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,w9},{c11}} is non empty V41() V45() set
c6 . [c11,w9] is set
Class (z2,w9) is Element of bool the carrier of Y
bool the carrier of Y is non empty set
X is multMagma
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the empty trivial V63() {} -element multMagma is empty trivial V63() {} -element multMagma
the carrier of the empty trivial V63() {} -element multMagma is empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() set
f is set
[:f,f:] is Relation-like set
the multF of the empty trivial V63() {} -element multMagma is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [: the carrier of the empty trivial V63() {} -element multMagma , the carrier of the empty trivial V63() {} -element multMagma :] -defined the carrier of the empty trivial V63() {} -element multMagma -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[: the carrier of the empty trivial V63() {} -element multMagma , the carrier of the empty trivial V63() {} -element multMagma :], the carrier of the empty trivial V63() {} -element multMagma :]
[: the carrier of the empty trivial V63() {} -element multMagma , the carrier of the empty trivial V63() {} -element multMagma :] is Relation-like V41() set
[:[: the carrier of the empty trivial V63() {} -element multMagma , the carrier of the empty trivial V63() {} -element multMagma :], the carrier of the empty trivial V63() {} -element multMagma :] is Relation-like V41() set
bool [:[: the carrier of the empty trivial V63() {} -element multMagma , the carrier of the empty trivial V63() {} -element multMagma :], the carrier of the empty trivial V63() {} -element multMagma :] is non empty V41() V45() set
the multF of X | {} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding {} -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
y is set
the multF of X | y is Relation-like [: the carrier of X, the carrier of X:] -defined y -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of the empty trivial V63() {} -element multMagma is Relation-like Function-like set
X is multMagma
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
multMagma(# the carrier of X, the multF of X #) is strict multMagma
the multF of multMagma(# the carrier of X, the multF of X #) is Relation-like [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined the carrier of multMagma(# the carrier of X, the multF of X #) -valued Function-like quasi_total Element of bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):]
the carrier of multMagma(# the carrier of X, the multF of X #) is set
[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like set
[:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like set
bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):] is non empty set
the multF of multMagma(# the carrier of X, the multF of X #) | [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined the carrier of multMagma(# the carrier of X, the multF of X #) -valued Function-like Element of bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):]
the multF of X || the carrier of multMagma(# the carrier of X, the multF of X #) is Relation-like Function-like set
f is (X)
X is non empty multMagma
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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
multMagma(# the carrier of X, the multF of X #) is strict multMagma
the multF of multMagma(# the carrier of X, the multF of X #) is Relation-like [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined the carrier of multMagma(# the carrier of X, the multF of X #) -valued Function-like quasi_total Element of bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):]
the carrier of multMagma(# the carrier of X, the multF of X #) is set
[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like set
[:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like set
bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):] is non empty set
the multF of multMagma(# the carrier of X, the multF of X #) | [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] is Relation-like [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined [: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):] -defined the carrier of multMagma(# the carrier of X, the multF of X #) -valued Function-like Element of bool [:[: the carrier of multMagma(# the carrier of X, the multF of X #), the carrier of multMagma(# the carrier of X, the multF of X #):], the carrier of multMagma(# the carrier of X, the multF of X #):]
the multF of X || the carrier of multMagma(# the carrier of X, the multF of X #) is Relation-like Function-like set
f is (X)
X is multMagma
Y is (X)
the carrier of Y is set
the multF of Y is Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
multMagma(# the carrier of Y, the multF of Y #) is strict multMagma
f is (X)
the carrier of f is set
the multF of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like quasi_total Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[: the carrier of f, the carrier of f:] is Relation-like set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is Relation-like set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
multMagma(# the carrier of f, the multF of f #) is strict multMagma
the multF of f || the carrier of Y is Relation-like Function-like set
the multF of Y || the carrier of f is Relation-like Function-like set
( the multF of Y || the carrier of f) || the carrier of Y is Relation-like Function-like set
the multF of Y | [: the carrier of f, the carrier of f:] is Relation-like [: the carrier of Y, the carrier of Y:] -defined [: the carrier of f, the carrier of f:] -defined [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
( the multF of Y | [: the carrier of f, the carrier of f:]) || the carrier of Y is Relation-like Function-like set
( the multF of Y | [: the carrier of f, the carrier of f:]) | [: the carrier of Y, the carrier of Y:] is Relation-like [: the carrier of Y, the carrier of Y:] -defined [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
X is multMagma
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
multMagma(# the carrier of X, the multF of X #) is strict multMagma
Y is (X)
the carrier of Y is set
the multF of Y is Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
multMagma(# the carrier of Y, the multF of Y #) is strict multMagma
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of X is Relation-like Function-like set
f is (X)
the multF of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like quasi_total Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
the carrier of f is set
[: the carrier of f, the carrier of f:] is Relation-like set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is Relation-like set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
the multF of Y || the carrier of f is Relation-like Function-like set
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is Element of bool the carrier of X
f is Element of the carrier of X
y is Element of the carrier of X
f * y is Element of the carrier of X
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is (X)
the carrier of Y is set
f is Element of the carrier of X
y is Element of the carrier of X
f * y is Element of the carrier of X
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
[f,y] is V26() set
{f,y} is non empty V41() set
{f} is non empty trivial V41() V48(1) set
{{f,y},{f}} is non empty V41() V45() set
[: the carrier of Y, the carrier of Y:] is Relation-like set
x is Element of the carrier of Y
c6 is Element of the carrier of Y
x * c6 is Element of the carrier of Y
the multF of Y is Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . (x,c6) is Element of the carrier of Y
[x,c6] is V26() set
{x,c6} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,c6},{x}} is non empty V41() V45() set
the multF of Y . [x,c6] is set
the multF of X || the carrier of Y is Relation-like Function-like set
( the multF of X || the carrier of Y) . [x,c6] is set
the multF of X | [: the carrier of Y, the carrier of Y:] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of Y, the carrier of Y:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
( the multF of X | [: the carrier of Y, the carrier of Y:]) . [f,y] is set
the multF of X . [f,y] is set
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is (X)
the carrier of Y is set
f is Element of bool the carrier of X
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is Relation-like Function-like set
proj1 Y is set
meet Y is set
f is set
proj2 Y is set
meet (proj2 Y) is set
y is set
Y . y is set
f is Element of the carrier of X
y is Element of the carrier of X
f * y is Element of the carrier of X
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
proj2 Y is set
meet (proj2 Y) is set
proj2 Y is set
meet (proj2 Y) is set
x is set
c6 is set
Y . c6 is set
X is non empty multMagma
the carrier of X is non empty set
bool the carrier of X is non empty set
Y is Element of bool the carrier of X
Y * Y is Element of bool the carrier of X
f is set
y is Element of the carrier of X
x is Element of the carrier of X
y * x 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (y,x) is Element of the carrier of X
f is Element of the carrier of X
y is Element of the carrier of X
f * y 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
X is non empty multMagma
the carrier of X is non empty set
bool the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
bool [: the carrier of X, the carrier of Y:] is non empty set
bool the carrier of Y is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
y is (X) Element of bool the carrier of X
f .: y is Element of bool the carrier of Y
x is Element of the carrier of Y
c6 is Element of the carrier of Y
x * c6 is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . (x,c6) is Element of the carrier of Y
dom f is non empty Element of bool the carrier of X
w1 is set
f . w1 is set
w2 is set
f . w2 is set
v1 is Element of the carrier of X
v2 is Element of the carrier of X
v1 * v2 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (v1,v2) is Element of the carrier of X
f . (v1 * v2) is Element of the carrier of Y
X is non empty multMagma
the carrier of X is non empty set
bool the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of Y, the carrier of X:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of X:] is non empty set
bool the carrier of Y is non empty set
f is non empty Relation-like the carrier of Y -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of X:]
y is (X) Element of bool the carrier of X
f " y is Element of bool the carrier of Y
x is Element of the carrier of Y
c6 is Element of the carrier of Y
x * c6 is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . (x,c6) is Element of the carrier of Y
dom f is non empty Element of bool the carrier of Y
f . x is Element of the carrier of X
f . c6 is Element of the carrier of X
(f . x) * (f . c6) 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . ((f . x),(f . c6)) is Element of the carrier of X
f . (x * c6) is Element of the carrier of X
X is non empty multMagma
the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of Y, the carrier of X:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of X:] is non empty set
bool the carrier of Y is non empty set
f is non empty Relation-like the carrier of Y -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of X:]
y is non empty Relation-like the carrier of Y -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of X:]
{ b1 where b1 is Element of the carrier of Y : f . b1 = y . b1 } is set
c6 is set
w1 is Element of the carrier of Y
f . w1 is Element of the carrier of X
y . w1 is Element of the carrier of X
c6 is Element of bool the carrier of Y
w1 is Element of the carrier of Y
w2 is Element of the carrier of Y
w1 * w2 is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . (w1,w2) is Element of the carrier of Y
v1 is Element of the carrier of Y
f . v1 is Element of the carrier of X
y . v1 is Element of the carrier of X
v2 is Element of the carrier of Y
f . v2 is Element of the carrier of X
y . v2 is Element of the carrier of X
f . (w1 * w2) is Element of the carrier of X
y . w1 is Element of the carrier of X
y . w2 is Element of the carrier of X
(y . w1) * (y . w2) 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . ((y . w1),(y . w2)) is Element of the carrier of X
y . (w1 * w2) is Element of the carrier of X
X is multMagma
the carrier of X is set
bool the carrier of X is non empty 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
Y is (X) Element of bool the carrier of X
the multF of X || Y is Relation-like Function-like set
[:Y,Y:] is Relation-like set
[:[:Y,Y:],Y:] is Relation-like set
bool [:[:Y,Y:],Y:] is non empty set
[:Y,Y:] 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
f is set
the multF of X . f is set
y is set
x is set
[y,x] is V26() set
{y,x} is non empty V41() set
{y} is non empty trivial V41() V48(1) set
{{y,x},{y}} is non empty V41() V45() set
c6 is Element of the carrier of X
w1 is Element of the carrier of X
c6 * w1 is Element of the carrier of X
the multF of X . (c6,w1) is Element of the carrier of X
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is Element of bool the carrier of X
bool the carrier of X is non empty Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty set
f is set
id f is Relation-like f -defined f -valued Function-like one-to-one total quasi_total Element of bool [:f,f:]
[:f,f:] is Relation-like set
bool [:f,f:] is non empty set
meet (id f) is set
dom (id f) is Element of bool f
bool f is non empty set
c6 is set
(id f) . c6 is set
w1 is strict (X)
the carrier of w1 is set
c6 is (X) Element of bool the carrier of X
(X,c6) is Relation-like [:c6,c6:] -defined c6 -valued Function-like quasi_total Element of bool [:[:c6,c6:],c6:]
[:c6,c6:] is Relation-like set
[:[:c6,c6:],c6:] is Relation-like set
bool [:[:c6,c6:],c6:] is non empty 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || c6 is Relation-like Function-like set
multMagma(# c6,(X,c6) #) is strict multMagma
the carrier of multMagma(# c6,(X,c6) #) is set
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of X is Relation-like Function-like set
multMagma(# the carrier of X, the multF of X #) is strict multMagma
w2 is strict (X)
the carrier of w2 is set
w2 is strict (X)
the carrier of w2 is set
v1 is strict (X)
the carrier of v1 is set
w2 is set
v1 is set
v2 is strict (X)
the carrier of v2 is set
meet f is set
rng (id f) is Element of bool f
meet (rng (id f)) is set
w2 is strict (X)
the carrier of w2 is set
v1 is set
rng (id f) is Element of bool f
meet (rng (id f)) is set
meet f is set
[: the carrier of w2, the carrier of w2:] is Relation-like set
the multF of X | [: the carrier of w2, the carrier of w2:] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of w2, the carrier of w2:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of w2 is Relation-like Function-like set
the multF of w2 is Relation-like [: the carrier of w2, the carrier of w2:] -defined the carrier of w2 -valued Function-like quasi_total Element of bool [:[: the carrier of w2, the carrier of w2:], the carrier of w2:]
[:[: the carrier of w2, the carrier of w2:], the carrier of w2:] is Relation-like set
bool [:[: the carrier of w2, the carrier of w2:], the carrier of w2:] is non empty set
the multF of multMagma(# c6,(X,c6) #) is Relation-like [: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] -defined the carrier of multMagma(# c6,(X,c6) #) -valued Function-like quasi_total Element of bool [:[: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):], the carrier of multMagma(# c6,(X,c6) #):]
[: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] is Relation-like set
[:[: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):], the carrier of multMagma(# c6,(X,c6) #):] is Relation-like set
bool [:[: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):], the carrier of multMagma(# c6,(X,c6) #):] is non empty set
the multF of X | [: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
( the multF of X | [: the carrier of w2, the carrier of w2:]) | [: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of multMagma(# c6,(X,c6) #), the carrier of multMagma(# c6,(X,c6) #):] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of w2 || the carrier of multMagma(# c6,(X,c6) #) is Relation-like Function-like set
w2 is strict (X)
the carrier of w2 is set
f is strict (X)
the carrier of f is set
y is strict (X)
the carrier of y is set
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is Element of bool the carrier of X
(X,Y) is strict (X)
f is Element of the carrier of X
y is Element of the carrier of X
f * y is Element of the carrier of X
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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . (f,y) is Element of the carrier of X
f is (X) Element of bool the carrier of X
(X,f) is Relation-like [:f,f:] -defined f -valued Function-like quasi_total Element of bool [:[:f,f:],f:]
[:f,f:] is Relation-like set
[:[:f,f:],f:] is Relation-like set
bool [:[:f,f:],f:] is non empty 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || f is Relation-like Function-like set
multMagma(# f,(X,f) #) is strict multMagma
y is strict (X)
the carrier of (X,Y) is set
the carrier of y is set
X is multMagma
the carrier of X is set
bool the carrier of X is non empty set
Y is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool the carrier of X
(X,Y) is strict (X)
X is non empty multMagma
the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
bool [: the carrier of X, the carrier of Y:] is non empty set
bool the carrier of X is non empty set
f is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
y is Element of bool the carrier of X
(X,y) is strict (X)
the carrier of (X,y) is set
f .: the carrier of (X,y) is Element of bool the carrier of Y
bool the carrier of Y is non empty set
f .: y is Element of bool the carrier of Y
(Y,(f .: y)) is strict (Y)
the carrier of (Y,(f .: y)) is set
f " the carrier of (Y,(f .: y)) is Element of bool the carrier of X
c11 is strict (Y)
the carrier of c11 is set
w1 is (Y) Element of bool the carrier of Y
(Y,w1) is Relation-like [:w1,w1:] -defined w1 -valued Function-like quasi_total Element of bool [:[:w1,w1:],w1:]
[:w1,w1:] is Relation-like set
[:[:w1,w1:],w1:] is Relation-like set
bool [:[:w1,w1:],w1:] is non empty set
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y || w1 is Relation-like Function-like set
multMagma(# w1,(Y,w1) #) is strict multMagma
c11 is strict (Y)
w9 is strict (X)
the carrier of w9 is set
v2 is (X) Element of bool the carrier of X
(X,v2) is Relation-like [:v2,v2:] -defined v2 -valued Function-like quasi_total Element of bool [:[:v2,v2:],v2:]
[:v2,v2:] is Relation-like set
[:[:v2,v2:],v2:] is Relation-like set
bool [:[:v2,v2:],v2:] 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X || v2 is Relation-like Function-like set
multMagma(# v2,(X,v2) #) is strict multMagma
f .: (f " the carrier of (Y,(f .: y))) is Element of bool the carrier of Y
f " (f .: y) is Element of bool the carrier of X
dom f is non empty Element of bool the carrier of X
w9 is strict (X)
X is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(bool (the_universe_of (X \/ NAT))) ^omega is set
Y is set
f is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
(dom f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
x is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 - 1 is complex ext-real V53() V54() set
x is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 - 1 is complex ext-real V53() V54() set
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
y - 1 is complex ext-real V53() V54() set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg x is Element of bool NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f . c6 is set
y - c6 is complex ext-real V53() V54() set
f . (y - c6) is set
[:(f . c6),(f . (y - c6)):] is Relation-like set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f . w2 is set
y - w2 is complex ext-real V53() V54() set
f . (y - w2) is set
[:(f . w2),(f . (y - w2)):] is Relation-like set
c6 is Relation-like Function-like FinSequence-like set
dom c6 is Element of bool NAT
disjoin c6 is Relation-like Function-like set
Union (disjoin c6) is set
len c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 . w2 is set
f . w2 is set
y - w2 is complex ext-real V53() V54() set
f . (y - w2) is set
[:(f . w2),(f . (y - w2)):] is Relation-like set
w2 is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 - 1 is complex ext-real V53() V54() set
Y is Relation-like Function-like set
proj1 Y is set
f is set
Y . f is set
y is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x - 1 is complex ext-real V53() V54() set
(dom y) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
the_transitive-closure_of (X \/ NAT) is set
Tarski-Class (the_transitive-closure_of (X \/ NAT)) is set
x is set
union X is set
union NAT is epsilon-transitive epsilon-connected ordinal set
(union X) \/ (union NAT) is set
union (X \/ NAT) is set
union (the_transitive-closure_of (X \/ NAT)) is set
(dom y) - 1 is complex ext-real V53() V54() set
c6 is Relation-like Function-like FinSequence-like set
len c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin c6 is Relation-like Function-like set
Union (disjoin c6) is set
(dom y) -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 . w2 is set
y . w2 is set
(dom y) - w2 is complex ext-real V53() V54() set
y . ((dom y) - w2) is set
[:(y . w2),(y . ((dom y) - w2)):] is Relation-like set
Seg w1 is Element of bool NAT
- w2 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
- ((dom y) - 1) is complex ext-real V53() V54() set
(- w2) + (dom y) is complex ext-real V53() V54() set
(- 1) + (dom y) is complex ext-real V53() V54() set
(- ((dom y) - 1)) + (dom y) is complex ext-real V53() V54() set
(dom y) -' w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
w1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
rng y is V41() Element of bool (bool (the_universe_of (X \/ NAT)))
bool (bool (the_universe_of (X \/ NAT))) is non empty set
proj2 (disjoin c6) is set
w2 is set
proj1 (disjoin c6) is set
v1 is set
(disjoin c6) . v1 is set
dom c6 is Element of bool NAT
c6 . v1 is set
{v1} is non empty trivial V41() V48(1) set
[:(c6 . v1),{v1}:] is Relation-like set
Seg w1 is Element of bool NAT
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 . v2 is set
{v2} is non empty trivial V41() V45() V48(1) set
c11 is set
the_transitive-closure_of (X \/ NAT) is set
Tarski-Class (the_transitive-closure_of (X \/ NAT)) is set
c11 is set
union X is set
union NAT is epsilon-transitive epsilon-connected ordinal set
(union X) \/ (union NAT) is set
union (X \/ NAT) is set
union (the_transitive-closure_of (X \/ NAT)) is set
union (proj2 (disjoin c6)) is set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
[:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):] is Relation-like set
bool [:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):] is non empty set
f is Relation-like (bool (the_universe_of (X \/ NAT))) ^omega -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):]
y is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
y . 0 is Element of bool (the_universe_of (X \/ NAT))
y . 1 is Element of bool (the_universe_of (X \/ NAT))
dom {} is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool NAT
y | 0 is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined 0 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
((bool (the_universe_of (X \/ NAT))),f,(y | 0)) is Element of bool (the_universe_of (X \/ NAT))
dom y is non empty Element of bool NAT
y | 1 is T-Sequence-like Relation-like NAT -defined 1 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
dom (y | 1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
((bool (the_universe_of (X \/ NAT))),f,(y | 1)) is Element of bool (the_universe_of (X \/ NAT))
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x - 1 is complex ext-real V53() V54() set
y . x is set
y | x is T-Sequence-like Relation-like NAT -defined x -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
dom (y | x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool x
bool x is non empty V41() V45() set
((bool (the_universe_of (X \/ NAT))),f,(y | x)) is Element of bool (the_universe_of (X \/ NAT))
c6 is Relation-like Function-like FinSequence-like set
len c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin c6 is Relation-like Function-like set
Union (disjoin c6) is set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 . w1 is set
y . w1 is set
x - w1 is complex ext-real V53() V54() set
y . (x - w1) is set
[:(y . w1),(y . (x - w1)):] is Relation-like set
x -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
Seg (x -' 1) is Element of bool NAT
(x -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(y | x) . w1 is set
- w1 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
- (x - 1) is complex ext-real V53() V54() set
(- w1) + x is complex ext-real V53() V54() set
(- 1) + x is complex ext-real V53() V54() set
(- (x - 1)) + x is complex ext-real V53() V54() set
x -' w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
(y | x) . (x - w1) is set
[:((y | x) . w1),((y | x) . (x - w1)):] is Relation-like set
f is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
f . 0 is Element of bool (the_universe_of (X \/ NAT))
f . 1 is Element of bool (the_universe_of (X \/ NAT))
y is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
y . 0 is Element of bool (the_universe_of (X \/ NAT))
y . 1 is Element of bool (the_universe_of (X \/ NAT))
Y . {} is set
x is T-Sequence-like Relation-like bool (the_universe_of (X \/ NAT)) -valued Function-like V41() set
dom x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 - 1 is complex ext-real V53() V54() set
dom {} is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool NAT
[:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):] is Relation-like set
bool [:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):] is non empty set
x is Relation-like (bool (the_universe_of (X \/ NAT))) ^omega -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:((bool (the_universe_of (X \/ NAT))) ^omega),(bool (the_universe_of (X \/ NAT))):]
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f . c6 is set
f | c6 is T-Sequence-like Relation-like NAT -defined c6 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
((bool (the_universe_of (X \/ NAT))),x,(f | c6)) is Element of bool (the_universe_of (X \/ NAT))
c6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
x . {} is set
dom f is non empty Element of bool NAT
f | 1 is T-Sequence-like Relation-like NAT -defined 1 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
dom (f | 1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
dom f is non empty Element of bool NAT
dom (f | c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool c6
bool c6 is non empty V41() V45() set
c6 - 1 is complex ext-real V53() V54() set
w1 is Relation-like Function-like FinSequence-like set
len w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin w1 is Relation-like Function-like set
Union (disjoin w1) is set
w2 is Relation-like Function-like FinSequence-like set
len w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin w2 is Relation-like Function-like set
Union (disjoin w2) is set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 . v1 is set
w2 . v1 is set
(f | c6) . v1 is set
c6 - v1 is complex ext-real V53() V54() set
(f | c6) . (c6 - v1) is set
[:((f | c6) . v1),((f | c6) . (c6 - v1)):] is Relation-like set
f . v1 is set
f . (c6 - v1) is set
[:(f . v1),(f . (c6 - v1)):] is Relation-like set
c6 -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
Seg (c6 -' 1) is Element of bool NAT
(c6 -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
- v1 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
- (c6 - 1) is complex ext-real V53() V54() set
(- v1) + c6 is complex ext-real V53() V54() set
(- 1) + c6 is complex ext-real V53() V54() set
(- (c6 - 1)) + c6 is complex ext-real V53() V54() set
c6 -' v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y . c6 is set
y | c6 is T-Sequence-like Relation-like NAT -defined c6 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
((bool (the_universe_of (X \/ NAT))),x,(y | c6)) is Element of bool (the_universe_of (X \/ NAT))
c6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
x . {} is set
dom y is non empty Element of bool NAT
y | 1 is T-Sequence-like Relation-like NAT -defined 1 -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like V41() Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
dom (y | 1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
dom y is non empty Element of bool NAT
dom (y | c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool c6
bool c6 is non empty V41() V45() set
c6 - 1 is complex ext-real V53() V54() set
w1 is Relation-like Function-like FinSequence-like set
len w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin w1 is Relation-like Function-like set
Union (disjoin w1) is set
w2 is Relation-like Function-like FinSequence-like set
len w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin w2 is Relation-like Function-like set
Union (disjoin w2) is set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 . v1 is set
w2 . v1 is set
(y | c6) . v1 is set
c6 - v1 is complex ext-real V53() V54() set
(y | c6) . (c6 - v1) is set
[:((y | c6) . v1),((y | c6) . (c6 - v1)):] is Relation-like set
y . v1 is set
y . (c6 - v1) is set
[:(y . v1),(y . (c6 - v1)):] is Relation-like set
c6 -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
Seg (c6 -' 1) is Element of bool NAT
(c6 -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
- v1 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
- (c6 - 1) is complex ext-real V53() V54() set
(- v1) + c6 is complex ext-real V53() V54() set
(- 1) + c6 is complex ext-real V53() V54() set
(- (c6 - 1)) + c6 is complex ext-real V53() V54() set
c6 -' v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
X is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X) . Y is set
X is non empty set
Y is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X) . f is set
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
f - 1 is complex ext-real V53() V54() set
y is Relation-like Function-like FinSequence-like set
len y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin y is Relation-like Function-like set
Union (disjoin y) is set
2 - 1 is complex ext-real V53() V54() set
Seg (len y) is Element of bool NAT
dom y is Element of bool NAT
proj1 (disjoin y) is set
(disjoin y) . 1 is set
y . 1 is set
[:(y . 1),{1}:] is Relation-like set
(X) . 1 is Element of bool (the_universe_of (X \/ NAT))
(X) . (f - 1) is set
[:((X) . 1),((X) . (f - 1)):] is Relation-like set
(f - 1) + 1 is complex ext-real V53() V54() set
0 + f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
(- 1) + f is complex ext-real V53() V54() set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X) . x is set
c6 is set
proj2 (disjoin y) is set
union (proj2 (disjoin y)) is set
X is set
(X,0) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . 0 is set
X is set
(X,1) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . 1 is set
X is set
(X,2) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . 2 is set
[:X,X:] is Relation-like set
[:[:X,X:],{1}:] is Relation-like set
2 - 1 is complex ext-real V53() V54() set
(X) . 2 is Element of bool (the_universe_of (X \/ NAT))
Y is Relation-like Function-like FinSequence-like set
len Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin Y is Relation-like Function-like set
Union (disjoin Y) is set
Y . 1 is set
(X) . 1 is Element of bool (the_universe_of (X \/ NAT))
(X) . (2 - 1) is set
[:((X) . 1),((X) . (2 - 1)):] is Relation-like set
(X,1) is set
(X) . 1 is set
[:(X,1),X:] is Relation-like set
<*[:X,X:]*> is Relation-like Function-like set
proj2 (disjoin Y) is set
union (proj2 (disjoin Y)) is set
f is set
y is set
proj1 (disjoin Y) is set
x is set
(disjoin Y) . x is set
dom Y is Element of bool NAT
(disjoin Y) . 1 is set
{2} is non empty trivial V41() V45() V48(1) set
X is set
(X,3) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . 3 is set
[:X,X:] is Relation-like set
[:[:X,X:],{1}:] is Relation-like set
[:X,[:[:X,X:],{1}:]:] is Relation-like set
[:[:X,[:[:X,X:],{1}:]:],{1}:] is Relation-like set
[:[:[:X,X:],{1}:],X:] is Relation-like set
[:[:[:[:X,X:],{1}:],X:],{2}:] is Relation-like set
[:[:X,[:[:X,X:],{1}:]:],{1}:] \/ [:[:[:[:X,X:],{1}:],X:],{2}:] is Relation-like set
3 - 1 is complex ext-real V53() V54() set
(X) . 3 is Element of bool (the_universe_of (X \/ NAT))
y is Relation-like Function-like FinSequence-like set
len y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin y is Relation-like Function-like set
Union (disjoin y) is set
y . 1 is set
(X,1) is set
(X) . 1 is set
(X,2) is set
(X) . 2 is set
[:(X,1),(X,2):] is Relation-like set
[:(X,1),[:[:X,X:],{1}:]:] is Relation-like set
y . 2 is set
(X) . 2 is Element of bool (the_universe_of (X \/ NAT))
3 - 2 is complex ext-real V53() V54() set
(X) . (3 - 2) is set
[:((X) . 2),((X) . (3 - 2)):] is Relation-like set
[:(X,2),X:] is Relation-like set
proj2 (disjoin y) is set
union (proj2 (disjoin y)) is set
x is set
c6 is set
proj1 (disjoin y) is set
w1 is set
(disjoin y) . w1 is set
dom y is Element of bool NAT
(disjoin y) . 1 is set
(disjoin y) . 2 is set
X is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Y - 1 is complex ext-real V53() V54() set
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
f is Relation-like Function-like FinSequence-like set
len f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin f is Relation-like Function-like set
Union (disjoin f) is set
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f . y is set
(X,y) is set
(X) . y is set
Y -' y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
(X,(Y -' y)) is set
(X) . (Y -' y) is set
[:(X,y),(X,(Y -' y)):] is Relation-like set
- y is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
- (Y - 1) is complex ext-real V53() V54() set
(- y) + Y is complex ext-real V53() V54() set
(- 1) + Y is complex ext-real V53() V54() set
(- (Y - 1)) + Y is complex ext-real V53() V54() set
Y - y is complex ext-real V53() V54() set
X is set
Y is set
Y `2 is set
Y `1 is set
(Y `1) `1 is set
(Y `1) `2 is set
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,f) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . f is set
f - 1 is complex ext-real V53() V54() set
y is Relation-like Function-like FinSequence-like set
len y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin y is Relation-like Function-like set
Union (disjoin y) is set
proj2 (disjoin y) is set
union (proj2 (disjoin y)) is set
x is set
proj1 (disjoin y) is set
c6 is set
(disjoin y) . c6 is set
dom y is Element of bool NAT
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg (len y) is Element of bool NAT
y . w1 is set
(X) . w1 is set
f - w1 is complex ext-real V53() V54() set
(X) . (f - w1) is set
[:((X) . w1),((X) . (f - w1)):] is Relation-like set
{w1} is non empty trivial V41() V45() V48(1) set
[:[:((X) . w1),((X) . (f - w1)):],{w1}:] is Relation-like set
- (f - 1) is complex ext-real V53() V54() set
- w1 is complex ext-real non positive V53() V54() set
(- (f - 1)) + f is complex ext-real V53() V54() set
(- w1) + f is complex ext-real V53() V54() set
(X,w1) is set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,w2) is set
(X) . w2 is set
w1 + w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[:(X,w1),(X,w2):] is Relation-like set
[:[:(X,w1),(X,w2):],{w1}:] is Relation-like set
X is set
Y is set
f is set
[Y,f] is V26() set
{Y,f} is non empty V41() set
{Y} is non empty trivial V41() V48(1) set
{{Y,f},{Y}} is non empty V41() V45() set
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . y is set
[[Y,f],y] is V26() set
{[Y,f],y} is non empty V41() set
{[Y,f]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[Y,f],y},{[Y,f]}} is non empty V41() V45() set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,x) is set
(X) . x is set
y + x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,(y + x)) is set
(X) . (y + x) is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(y + x) - 1 is complex ext-real V53() V54() set
c6 is Relation-like Function-like FinSequence-like set
len c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin c6 is Relation-like Function-like set
Union (disjoin c6) is set
1 - 1 is complex ext-real V53() V54() set
x - 1 is complex ext-real V53() V54() set
0 + y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(x - 1) + y is complex ext-real V53() V54() set
c6 . y is set
(y + x) - y is complex ext-real V53() V54() set
(X) . ((y + x) - y) is set
[:((X) . y),((X) . ((y + x) - y)):] is Relation-like set
[:((X) . y),((X) . x):] is Relation-like set
{y} is non empty trivial V41() V45() V48(1) set
[:(c6 . y),{y}:] is Relation-like set
Seg (len c6) is Element of bool NAT
dom c6 is Element of bool NAT
(disjoin c6) . y is set
proj1 (disjoin c6) is set
proj2 (disjoin c6) is set
union (proj2 (disjoin c6)) is set
X is set
Y is set
f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,f) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . f is set
(Y,f) is set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) . f is set
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,y) is set
(X) . y is set
(Y,y) is set
(Y) . y is set
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
x is set
x `2 is set
y - 1 is complex ext-real V53() V54() set
x `1 is set
(x `1) `1 is set
(x `1) `2 is set
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,c6) is set
(X) . c6 is set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,w1) is set
(X) . w1 is set
c6 + w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[:(X,c6),(X,w1):] is Relation-like set
{c6} is non empty trivial V41() V45() V48(1) set
[:[:(X,c6),(X,w1):],{c6}:] is Relation-like set
w2 is Relation-like Function-like FinSequence-like set
len w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin w2 is Relation-like Function-like set
Union (disjoin w2) is set
w2 . c6 is set
(Y,c6) is set
(Y) . c6 is set
y -' c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
(Y,(y -' c6)) is set
(Y) . (y -' c6) is set
[:(Y,c6),(Y,(y -' c6)):] is Relation-like set
[(x `1),(x `2)] is V26() set
{(x `1),(x `2)} is non empty V41() set
{(x `1)} is non empty trivial V41() V48(1) set
{{(x `1),(x `2)},{(x `1)}} is non empty V41() V45() set
[((x `1) `1),((x `1) `2)] is V26() set
{((x `1) `1),((x `1) `2)} is non empty V41() set
{((x `1) `1)} is non empty trivial V41() V48(1) set
{{((x `1) `1),((x `1) `2)},{((x `1) `1)}} is non empty V41() V45() set
c6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(y - 1) + 1 is complex ext-real V53() V54() set
y - c6 is complex ext-real V53() V54() set
c6 - c6 is complex ext-real V53() V54() set
0 + w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[:(w2 . c6),{c6}:] is Relation-like set
Seg (len w2) is Element of bool NAT
dom w2 is Element of bool NAT
(disjoin w2) . c6 is set
proj1 (disjoin w2) is set
proj2 (disjoin w2) is set
union (proj2 (disjoin w2)) is set
X is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
X is set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) is set
(X) . Y is set
{Y} is non empty trivial V41() V45() V48(1) set
[:(X,Y),{Y}:] is Relation-like set
f is set
dom (X) is non empty Element of bool NAT
dom ((X) | NATPLUS) is Element of bool NAT
(disjoin ((X) | NATPLUS)) . Y is set
((X) | NATPLUS) . Y is set
[:(((X) | NATPLUS) . Y),{Y}:] is Relation-like set
[:((X) . Y),{Y}:] is Relation-like set
proj1 (disjoin ((X) | NATPLUS)) is set
proj2 (disjoin ((X) | NATPLUS)) is set
union (proj2 (disjoin ((X) | NATPLUS))) is set
X is set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X) . Y is set
Y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
Y - 1 is complex ext-real V53() V54() set
f is Relation-like Function-like FinSequence-like set
len f is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin f is Relation-like Function-like set
Union (disjoin f) is set
proj2 (disjoin f) is set
y is set
proj1 (disjoin f) is set
x is set
(disjoin f) . x is set
dom f is Element of bool NAT
Seg (len f) is Element of bool NAT
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(Y - 1) + 1 is complex ext-real V53() V54() set
(X) . c6 is set
f . c6 is set
Y - c6 is complex ext-real V53() V54() set
(X) . (Y - c6) is set
[:((X) . c6),((X) . (Y - c6)):] is Relation-like set
(disjoin f) . c6 is set
{c6} is non empty trivial V41() V45() V48(1) set
[:(f . c6),{c6}:] is Relation-like set
union (proj2 (disjoin f)) is set
proj2 (disjoin ((X) | NATPLUS)) is set
Y is set
proj1 (disjoin ((X) | NATPLUS)) is set
f is set
(disjoin ((X) | NATPLUS)) . f is set
dom ((X) | NATPLUS) is Element of bool NAT
y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(disjoin ((X) | NATPLUS)) . y is set
((X) | NATPLUS) . y is set
{y} is non empty trivial V41() V45() V48(1) set
[:(((X) | NATPLUS) . y),{y}:] is Relation-like set
(X) . y is set
[:((X) . y),{y}:] is Relation-like set
[:{},{y}:] is Relation-like V41() set
union (proj2 (disjoin ((X) | NATPLUS))) is set
(X,1) is set
(X) . 1 is set
[:(X,1),{1}:] is Relation-like set
X is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
X is non empty set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is Element of (X)
Y `2 is set
proj2 (disjoin ((X) | NATPLUS)) is set
union (proj2 (disjoin ((X) | NATPLUS))) is set
f is set
proj1 (disjoin ((X) | NATPLUS)) is set
y is set
(disjoin ((X) | NATPLUS)) . y is set
dom ((X) | NATPLUS) is Element of bool NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
((X) | NATPLUS) . x is set
{x} is non empty trivial V41() V45() V48(1) set
[:(((X) | NATPLUS) . x),{x}:] is Relation-like set
c6 is set
X is non empty set
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is Element of (X)
Y `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(Y `2)) is non empty set
(X) . (Y `2) is set
{(Y `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(Y `2)),{(Y `2)}:] is non empty Relation-like set
proj2 (disjoin ((X) | NATPLUS)) is set
union (proj2 (disjoin ((X) | NATPLUS))) is set
f is set
proj1 (disjoin ((X) | NATPLUS)) is set
y is set
(disjoin ((X) | NATPLUS)) . y is set
dom ((X) | NATPLUS) is Element of bool NAT
((X) | NATPLUS) . y is set
(X) . y is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
((X) | NATPLUS) . x is set
{x} is non empty trivial V41() V45() V48(1) set
[:(((X) | NATPLUS) . x),{x}:] is Relation-like set
X is non empty set
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is Element of (X)
Y `1 is set
f is Element of (X)
f `1 is set
[(Y `1),(f `1)] is V26() set
{(Y `1),(f `1)} is non empty V41() set
{(Y `1)} is non empty trivial V41() V48(1) set
{{(Y `1),(f `1)},{(Y `1)}} is non empty V41() V45() set
Y `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[(Y `1),(f `1)],(Y `2)] is V26() set
{[(Y `1),(f `1)],(Y `2)} is non empty V41() set
{[(Y `1),(f `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(Y `1),(f `1)],(Y `2)},{[(Y `1),(f `1)]}} is non empty V41() V45() set
f `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(Y `2) + (f `2) is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[[(Y `1),(f `1)],(Y `2)],((Y `2) + (f `2))] is V26() set
{[[(Y `1),(f `1)],(Y `2)],((Y `2) + (f `2))} is non empty V41() set
{[[(Y `1),(f `1)],(Y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(Y `1),(f `1)],(Y `2)],((Y `2) + (f `2))},{[[(Y `1),(f `1)],(Y `2)]}} is non empty V41() V45() set
(X,(Y `2)) is non empty set
(X) . (Y `2) is set
{(Y `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(Y `2)),{(Y `2)}:] is non empty Relation-like set
(X,(f `2)) is non empty set
(X) . (f `2) is set
{(f `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(f `2)),{(f `2)}:] is non empty Relation-like set
(X,((Y `2) + (f `2))) is non empty set
(X) . ((Y `2) + (f `2)) is set
{((Y `2) + (f `2))} is non empty trivial V41() V45() V48(1) set
[[[(Y `1),(f `1)],(Y `2)],((Y `2) + (f `2))] `1 is set
[[[(Y `1),(f `1)],(Y `2)],((Y `2) + (f `2))] `2 is set
[:(X,((Y `2) + (f `2))),{((Y `2) + (f `2))}:] is non empty Relation-like set
X is set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
Y is set
(Y) is set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
f is set
y is non empty set
(y) is non empty set
y \/ NAT is non empty set
the_universe_of (y \/ NAT) is set
bool (the_universe_of (y \/ NAT)) is non empty Element of bool (bool (the_universe_of (y \/ NAT)))
bool (the_universe_of (y \/ NAT)) is non empty set
bool (bool (the_universe_of (y \/ NAT))) is non empty set
(y) is non empty Relation-like NAT -defined bool (the_universe_of (y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (y \/ NAT))):]
[:NAT,(bool (the_universe_of (y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (y \/ NAT))):] is non empty non trivial V41() set
(y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (y \/ NAT))):]
disjoin ((y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((y) | NATPLUS)) is set
x is Element of (y)
x `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(y,(x `2)) is non empty set
(y) . (x `2) is set
{(x `2)} is non empty trivial V41() V45() V48(1) set
[:(y,(x `2)),{(x `2)}:] is non empty Relation-like set
x `1 is set
c6 is non empty set
(c6,(x `2)) is non empty set
(c6) is non empty Relation-like NAT -defined bool (the_universe_of (c6 \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (c6 \/ NAT))):]
c6 \/ NAT is non empty set
the_universe_of (c6 \/ NAT) is set
bool (the_universe_of (c6 \/ NAT)) is non empty Element of bool (bool (the_universe_of (c6 \/ NAT)))
bool (the_universe_of (c6 \/ NAT)) is non empty set
bool (bool (the_universe_of (c6 \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (c6 \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (c6 \/ NAT))):] is non empty non trivial V41() set
(c6) . (x `2) is set
[(x `1),(x `2)] is V26() set
{(x `1),(x `2)} is non empty V41() set
{(x `1)} is non empty trivial V41() V48(1) set
{{(x `1),(x `2)},{(x `1)}} is non empty V41() V45() set
[:(c6,(x `2)),{(x `2)}:] is non empty Relation-like set
(c6) is non empty set
(c6) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (c6 \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (c6 \/ NAT))):]
disjoin ((c6) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((c6) | NATPLUS)) is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X is set
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
{Y} is non empty trivial V41() V45() V48(1) set
[:(X,Y),{Y}:] is Relation-like set
(X) is set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
X is set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
Y is non empty set
f is Element of Y
f `2 is set
f `1 is set
y is Element of Y
y `2 is set
y `1 is set
[(f `1),(y `1)] is V26() set
{(f `1),(y `1)} is non empty V41() set
{(f `1)} is non empty trivial V41() V48(1) set
{{(f `1),(y `1)},{(f `1)}} is non empty V41() V45() set
[[(f `1),(y `1)],(f `2)] is V26() set
{[(f `1),(y `1)],(f `2)} is non empty V41() set
{[(f `1),(y `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(f `1),(y `1)],(f `2)},{[(f `1),(y `1)]}} is non empty V41() V45() set
x is non empty set
(x) is non empty set
x \/ NAT is non empty set
the_universe_of (x \/ NAT) is set
bool (the_universe_of (x \/ NAT)) is non empty Element of bool (bool (the_universe_of (x \/ NAT)))
bool (the_universe_of (x \/ NAT)) is non empty set
bool (bool (the_universe_of (x \/ NAT))) is non empty set
(x) is non empty Relation-like NAT -defined bool (the_universe_of (x \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (x \/ NAT))):]
[:NAT,(bool (the_universe_of (x \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (x \/ NAT))):] is non empty non trivial V41() set
(x) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (x \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (x \/ NAT))):]
disjoin ((x) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((x) | NATPLUS)) is set
c6 is Element of (x)
c6 `1 is set
w1 is Element of (x)
w1 `1 is set
[(c6 `1),(w1 `1)] is V26() set
{(c6 `1),(w1 `1)} is non empty V41() set
{(c6 `1)} is non empty trivial V41() V48(1) set
{{(c6 `1),(w1 `1)},{(c6 `1)}} is non empty V41() V45() set
c6 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[(c6 `1),(w1 `1)],(c6 `2)] is V26() set
{[(c6 `1),(w1 `1)],(c6 `2)} is non empty V41() set
{[(c6 `1),(w1 `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(c6 `1),(w1 `1)],(c6 `2)},{[(c6 `1),(w1 `1)]}} is non empty V41() V45() set
w1 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(c6 `2) + (w1 `2) is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[[(c6 `1),(w1 `1)],(c6 `2)],((c6 `2) + (w1 `2))] is V26() set
{[[(c6 `1),(w1 `1)],(c6 `2)],((c6 `2) + (w1 `2))} is non empty V41() set
{[[(c6 `1),(w1 `1)],(c6 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(c6 `1),(w1 `1)],(c6 `2)],((c6 `2) + (w1 `2))},{[[(c6 `1),(w1 `1)],(c6 `2)]}} is non empty V41() V45() set
w2 is Element of Y
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 + v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(f `1),(y `1)],(f `2)],(v1 + v2)] is V26() set
{[[(f `1),(y `1)],(f `2)],(v1 + v2)} is non empty V41() set
{[[(f `1),(y `1)],(f `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(f `1),(y `1)],(f `2)],(v1 + v2)},{[[(f `1),(y `1)],(f `2)]}} is non empty V41() V45() set
[:Y,Y:] is non empty Relation-like set
[:[:Y,Y:],Y:] is non empty Relation-like set
bool [:[:Y,Y:],Y:] is non empty set
f is non empty Relation-like [:Y,Y:] -defined Y -valued Function-like total quasi_total Element of bool [:[:Y,Y:],Y:]
y is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x is Element of (X)
x `2 is set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is Element of (X)
c6 `2 is set
y . (x,c6) is Element of (X)
x `1 is set
c6 `1 is set
[(x `1),(c6 `1)] is V26() set
{(x `1),(c6 `1)} is non empty V41() set
{(x `1)} is non empty trivial V41() V48(1) set
{{(x `1),(c6 `1)},{(x `1)}} is non empty V41() V45() set
[[(x `1),(c6 `1)],(x `2)] is V26() set
{[(x `1),(c6 `1)],(x `2)} is non empty V41() set
{[(x `1),(c6 `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(x `1),(c6 `1)],(x `2)},{[(x `1),(c6 `1)]}} is non empty V41() V45() set
w1 + w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(x `1),(c6 `1)],(x `2)],(w1 + w2)] is V26() set
{[[(x `1),(c6 `1)],(x `2)],(w1 + w2)} is non empty V41() set
{[[(x `1),(c6 `1)],(x `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(x `1),(c6 `1)],(x `2)],(w1 + w2)},{[[(x `1),(c6 `1)],(x `2)]}} is non empty V41() V45() set
[:[:{},{}:],{}:] is Relation-like V41() set
bool [:[:{},{}:],{}:] is non empty V41() V45() set
Y is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:{},{}:] -defined {} -valued Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:{},{}:],{}:]
dom Y is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:{},{}:]
f is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:{},{}:] -defined {} -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:{},{}:],{}:]
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y is Element of (X)
y `2 is set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x is Element of (X)
x `2 is set
f . (y,x) is set
y `1 is set
x `1 is set
[(y `1),(x `1)] is V26() set
{(y `1),(x `1)} is non empty V41() set
{(y `1)} is non empty trivial V41() V48(1) set
{{(y `1),(x `1)},{(y `1)}} is non empty V41() V45() set
[[(y `1),(x `1)],(y `2)] is V26() set
{[(y `1),(x `1)],(y `2)} is non empty V41() set
{[(y `1),(x `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(y `1),(x `1)],(y `2)},{[(y `1),(x `1)]}} is non empty V41() V45() set
c6 + w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(y `1),(x `1)],(y `2)],(c6 + w1)] is V26() set
{[[(y `1),(x `1)],(y `2)],(c6 + w1)} is non empty V41() set
{[[(y `1),(x `1)],(y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(y `1),(x `1)],(y `2)],(c6 + w1)},{[[(y `1),(x `1)],(y `2)]}} is non empty V41() V45() set
Y is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
f is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
y is Element of (X)
x is Element of (X)
Y . (y,x) is Element of (X)
f . (y,x) is Element of (X)
y `2 is set
x `2 is set
y `1 is set
x `1 is set
[(y `1),(x `1)] is V26() set
{(y `1),(x `1)} is non empty V41() set
{(y `1)} is non empty trivial V41() V48(1) set
{{(y `1),(x `1)},{(y `1)}} is non empty V41() V45() set
[[(y `1),(x `1)],(y `2)] is V26() set
{[(y `1),(x `1)],(y `2)} is non empty V41() set
{[(y `1),(x `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(y `1),(x `1)],(y `2)},{[(y `1),(x `1)]}} is non empty V41() V45() set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 + v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(y `1),(x `1)],(y `2)],(w2 + v1)] is V26() set
{[[(y `1),(x `1)],(y `2)],(w2 + v1)} is non empty V41() set
{[[(y `1),(x `1)],(y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(y `1),(x `1)],(y `2)],(w2 + v1)},{[[(y `1),(x `1)],(y `2)]}} is non empty V41() V45() set
x is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
c6 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
w1 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
w2 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
f is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
y is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
x is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
c6 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
w1 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
w2 is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
X is set
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
X is set
(X) is multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
X is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() set
(X) is strict multMagma
(X) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding [:(X),(X):] -defined (X) -valued Function-like one-to-one constant functional quasi_total V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like V41() set
[:[:(X),(X):],(X):] is Relation-like V41() set
bool [:[:(X),(X):],(X):] is non empty V41() V45() set
multMagma(# (X),(X) #) is strict multMagma
X is non empty set
(X) is strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
Y is Element of the carrier of (X)
Y `2 is set
f is set
X is set
bool X is non empty set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
Y is Element of bool X
(Y) is strict multMagma
(Y) is set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is Relation-like set
[:[:(Y),(Y):],(Y):] is Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is set
the carrier of (X) is set
[: the carrier of (X), the carrier of (X):] is Relation-like 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):], the carrier of (X):] is Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
f is set
[:f,f:] is Relation-like set
the multF of (X) | [:f,f:] is Relation-like [: the carrier of (X), the carrier of (X):] -defined [:f,f:] -defined [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
the multF of (X) || the carrier of (Y) is Relation-like Function-like set
the multF of (Y) is Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[: the carrier of (Y), the carrier of (Y):] is Relation-like set
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
the multF of (X) | {} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding {} -defined [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
[: the carrier of (Y), the carrier of (Y):] is Relation-like set
the multF of (Y) is Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
dom the multF of (Y) is Relation-like the carrier of (Y) -defined the carrier of (Y) -valued Element of bool [: the carrier of (Y), the carrier of (Y):]
bool [: the carrier of (Y), the carrier of (Y):] 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
proj1 ( the multF of (X) || the carrier of (Y)) is set
y is set
the multF of (Y) . y is set
( the multF of (X) || the carrier of (Y)) . y is set
x is set
c6 is set
[x,c6] is V26() set
{x,c6} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,c6},{x}} is non empty V41() V45() set
w1 is Element of (Y)
w1 `2 is set
w2 is Element of (Y)
w2 `2 is set
the multF of (Y) . (w1,w2) is set
w1 `1 is set
w2 `1 is set
[(w1 `1),(w2 `1)] is V26() set
{(w1 `1),(w2 `1)} is non empty V41() set
{(w1 `1)} is non empty trivial V41() V48(1) set
{{(w1 `1),(w2 `1)},{(w1 `1)}} is non empty V41() V45() set
[[(w1 `1),(w2 `1)],(w1 `2)] is V26() set
{[(w1 `1),(w2 `1)],(w1 `2)} is non empty V41() set
{[(w1 `1),(w2 `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(w1 `1),(w2 `1)],(w1 `2)},{[(w1 `1),(w2 `1)]}} is non empty V41() V45() set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 + v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(w1 `1),(w2 `1)],(w1 `2)],(v1 + v2)] is V26() set
{[[(w1 `1),(w2 `1)],(w1 `2)],(v1 + v2)} is non empty V41() set
{[[(w1 `1),(w2 `1)],(w1 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(w1 `1),(w2 `1)],(w1 `2)],(v1 + v2)},{[[(w1 `1),(w2 `1)],(w1 `2)]}} is non empty V41() V45() set
c11 is Element of (X)
w9 is Element of (X)
(X) . (c11,w9) is Element of (X)
the multF of (X) . y is set
( the multF of (X) | [:f,f:]) . y is set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
Y `2 is set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
{ (b1 `1) where b1 is Element of the carrier of (X) : (X,b1) = 1 } is set
Y is set
(X,1) is set
(X) . 1 is set
[Y,1] is V26() set
{Y,1} is non empty V41() set
{Y} is non empty trivial V41() V48(1) set
{{Y,1},{Y}} is non empty V41() V45() set
[:(X,1),{1}:] is Relation-like set
[Y,1] `2 is set
f is Element of the carrier of (X)
(X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[Y,1] `1 is set
f is Element of the carrier of (X)
f `1 is set
(X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f `2 is set
[:X,{1}:] is Relation-like set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
Y `1 is set
Y `2 is set
(X,Y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f is Element of the carrier of (X)
Y * f is Element of the carrier of (X)
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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (Y,f) is Element of the carrier of (X)
f `1 is set
[(Y `1),(f `1)] is V26() set
{(Y `1),(f `1)} is non empty V41() set
{(Y `1)} is non empty trivial V41() V48(1) set
{{(Y `1),(f `1)},{(Y `1)}} is non empty V41() V45() set
[[(Y `1),(f `1)],(Y `2)] is V26() set
{[(Y `1),(f `1)],(Y `2)} is non empty V41() set
{[(Y `1),(f `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(Y `1),(f `1)],(Y `2)},{[(Y `1),(f `1)]}} is non empty V41() V45() set
(X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) + (X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))] is V26() set
{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))} is non empty V41() set
{[[(Y `1),(f `1)],(Y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))},{[[(Y `1),(f `1)],(Y `2)]}} is non empty V41() V45() set
f `2 is set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
Y `1 is set
Y `2 is set
[(Y `1),(Y `2)] is V26() set
{(Y `1),(Y `2)} is non empty V41() set
{(Y `1)} is non empty trivial V41() V48(1) set
{{(Y `1),(Y `2)},{(Y `1)}} is non empty V41() V45() set
(X,Y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f is non empty set
(f) is non empty strict multMagma
(f) is non empty set
f \/ NAT is non empty set
the_universe_of (f \/ NAT) is set
bool (the_universe_of (f \/ NAT)) is non empty Element of bool (bool (the_universe_of (f \/ NAT)))
bool (the_universe_of (f \/ NAT)) is non empty set
bool (bool (the_universe_of (f \/ NAT))) is non empty set
(f) is non empty Relation-like NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
[:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial V41() set
(f) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
disjoin ((f) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((f) | NATPLUS)) is set
(f) is non empty Relation-like [:(f),(f):] -defined (f) -valued Function-like total quasi_total Element of bool [:[:(f),(f):],(f):]
[:(f),(f):] is non empty Relation-like set
[:[:(f),(f):],(f):] is non empty Relation-like set
bool [:[:(f),(f):],(f):] is non empty set
multMagma(# (f),(f) #) is strict multMagma
the carrier of (f) is non empty set
y is Element of the carrier of (f)
y `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(y `2)) is set
(X) . (y `2) is set
{(y `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(y `2)),{(y `2)}:] is Relation-like set
x is set
c6 is set
[x,c6] is V26() set
{x,c6} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,c6},{x}} is non empty V41() V45() set
x is Element of (f)
x `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(f,y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(f,y) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
(X,Y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f is Element of the carrier of (X)
Y * f is Element of the carrier of (X)
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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (Y,f) is Element of the carrier of (X)
(X,(Y * f)) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) + (X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Y `1 is set
f `1 is set
[(Y `1),(f `1)] is V26() set
{(Y `1),(f `1)} is non empty V41() set
{(Y `1)} is non empty trivial V41() V48(1) set
{{(Y `1),(f `1)},{(Y `1)}} is non empty V41() V45() set
Y `2 is set
[[(Y `1),(f `1)],(Y `2)] is V26() set
{[(Y `1),(f `1)],(Y `2)} is non empty V41() set
{[(Y `1),(f `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(Y `1),(f `1)],(Y `2)},{[(Y `1),(f `1)]}} is non empty V41() V45() set
[[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))] is V26() set
{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))} is non empty V41() set
{[[(Y `1),(f `1)],(Y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))},{[[(Y `1),(f `1)],(Y `2)]}} is non empty V41() V45() set
[[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))] `2 is set
(X,Y) + 0 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
(X,Y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
f is non empty set
(f) is non empty strict multMagma
(f) is non empty set
f \/ NAT is non empty set
the_universe_of (f \/ NAT) is set
bool (the_universe_of (f \/ NAT)) is non empty Element of bool (bool (the_universe_of (f \/ NAT)))
bool (the_universe_of (f \/ NAT)) is non empty set
bool (bool (the_universe_of (f \/ NAT))) is non empty set
(f) is non empty Relation-like NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
[:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial V41() set
(f) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
disjoin ((f) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((f) | NATPLUS)) is set
(f) is non empty Relation-like [:(f),(f):] -defined (f) -valued Function-like total quasi_total Element of bool [:[:(f),(f):],(f):]
[:(f),(f):] is non empty Relation-like set
[:[:(f),(f):],(f):] is non empty Relation-like set
bool [:[:(f),(f):],(f):] is non empty set
multMagma(# (f),(f) #) is strict multMagma
the carrier of (f) is non empty set
y is Element of the carrier of (f)
y `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(y `2)) is set
(X) . (y `2) is set
{(y `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(y `2)),{(y `2)}:] is Relation-like set
(X,Y) - 1 is complex ext-real V53() V54() set
(X) . (X,Y) is set
c6 is Relation-like Function-like FinSequence-like set
len c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
disjoin c6 is Relation-like Function-like set
Union (disjoin c6) is set
y `1 is set
proj2 (disjoin c6) is set
union (proj2 (disjoin c6)) is set
w1 is set
proj1 (disjoin c6) is set
w2 is set
(disjoin c6) . w2 is set
dom c6 is Element of bool NAT
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg (len c6) is Element of bool NAT
c6 . v1 is set
(X) . v1 is set
(X,Y) - v1 is complex ext-real V53() V54() set
(X) . ((X,Y) - v1) is set
[:((X) . v1),((X) . ((X,Y) - v1)):] is Relation-like set
{v1} is non empty trivial V41() V45() V48(1) set
[:[:((X) . v1),((X) . ((X,Y) - v1)):],{v1}:] is Relation-like set
(y `1) `1 is set
(y `1) `2 is set
((y `1) `1) `1 is set
((y `1) `1) `2 is set
- ((X,Y) - 1) is complex ext-real V53() V54() set
- v1 is complex ext-real non positive V53() V54() set
(- ((X,Y) - 1)) + (X,Y) is complex ext-real V53() V54() set
(- v1) + (X,Y) is complex ext-real V53() V54() set
[(((y `1) `1) `1),v1] is V26() set
{(((y `1) `1) `1),v1} is non empty V41() set
{(((y `1) `1) `1)} is non empty trivial V41() V48(1) set
{{(((y `1) `1) `1),v1},{(((y `1) `1) `1)}} is non empty V41() V45() set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[(((y `1) `1) `2),v2] is V26() set
{(((y `1) `1) `2),v2} is non empty V41() set
{(((y `1) `1) `2)} is non empty trivial V41() V48(1) set
{{(((y `1) `1) `2),v2},{(((y `1) `1) `2)}} is non empty V41() V45() set
(X,v1) is set
[:(X,v1),{v1}:] is Relation-like set
{v2} is non empty trivial V41() V45() V48(1) set
(X,v2) is set
(X) . v2 is set
[:(X,v2),{v2}:] is Relation-like set
z2 is Element of (X)
z1 is Element of (X)
[(((y `1) `1) `1),v1] `2 is set
z2 is Element of the carrier of (X)
(X,z2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
z is Element of the carrier of (X)
(X,z) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[(((y `1) `1) `2),v2] `2 is set
[(((y `1) `1) `1),v1] `1 is set
[(((y `1) `1) `2),v2] `1 is set
[((y `1) `1),((y `1) `2)] is V26() set
{((y `1) `1),((y `1) `2)} is non empty V41() set
{((y `1) `1)} is non empty trivial V41() V48(1) set
{{((y `1) `1),((y `1) `2)},{((y `1) `1)}} is non empty V41() V45() set
[((y `1) `1),v1] is V26() set
{((y `1) `1),v1} is non empty V41() set
{{((y `1) `1),v1},{((y `1) `1)}} is non empty V41() V45() set
f9 is set
fs is set
[f9,fs] is V26() set
{f9,fs} is non empty V41() set
{f9} is non empty trivial V41() V48(1) set
{{f9,fs},{f9}} is non empty V41() V45() set
z2 * z is Element of the carrier of (X)
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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (z2,z) is Element of the carrier of (X)
[(y `1),(y `2)] is V26() set
{(y `1),(y `2)} is non empty V41() set
{(y `1)} is non empty trivial V41() V48(1) set
{{(y `1),(y `2)},{(y `1)}} is non empty V41() V45() set
[[((y `1) `1),v1],(X,Y)] is V26() set
{[((y `1) `1),v1],(X,Y)} is non empty V41() set
{[((y `1) `1),v1]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[((y `1) `1),v1],(X,Y)},{[((y `1) `1),v1]}} is non empty V41() V45() set
z2 `2 is set
[((y `1) `1),(z2 `2)] is V26() set
{((y `1) `1),(z2 `2)} is non empty V41() set
{{((y `1) `1),(z2 `2)},{((y `1) `1)}} is non empty V41() V45() set
(X,z2) + (X,z) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[((y `1) `1),(z2 `2)],((X,z2) + (X,z))] is V26() set
{[((y `1) `1),(z2 `2)],((X,z2) + (X,z))} is non empty V41() set
{[((y `1) `1),(z2 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[((y `1) `1),(z2 `2)],((X,z2) + (X,z))},{[((y `1) `1),(z2 `2)]}} is non empty V41() V45() set
[(((y `1) `1) `1),(((y `1) `1) `2)] is V26() set
{(((y `1) `1) `1),(((y `1) `1) `2)} is non empty V41() set
{{(((y `1) `1) `1),(((y `1) `1) `2)},{(((y `1) `1) `1)}} is non empty V41() V45() set
[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)] is V26() set
{[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)} is non empty V41() set
{[(((y `1) `1) `1),(((y `1) `1) `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)},{[(((y `1) `1) `1),(((y `1) `1) `2)]}} is non empty V41() V45() set
[[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)],((X,z2) + (X,z))] is V26() set
{[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)],((X,z2) + (X,z))} is non empty V41() set
{[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)],((X,z2) + (X,z))},{[[(((y `1) `1) `1),(((y `1) `1) `2)],(z2 `2)]}} is non empty V41() V45() set
z `1 is set
[(((y `1) `1) `1),(z `1)] is V26() set
{(((y `1) `1) `1),(z `1)} is non empty V41() set
{{(((y `1) `1) `1),(z `1)},{(((y `1) `1) `1)}} is non empty V41() V45() set
[[(((y `1) `1) `1),(z `1)],(z2 `2)] is V26() set
{[(((y `1) `1) `1),(z `1)],(z2 `2)} is non empty V41() set
{[(((y `1) `1) `1),(z `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(((y `1) `1) `1),(z `1)],(z2 `2)},{[(((y `1) `1) `1),(z `1)]}} is non empty V41() V45() set
[[[(((y `1) `1) `1),(z `1)],(z2 `2)],((X,z2) + (X,z))] is V26() set
{[[(((y `1) `1) `1),(z `1)],(z2 `2)],((X,z2) + (X,z))} is non empty V41() set
{[[(((y `1) `1) `1),(z `1)],(z2 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(((y `1) `1) `1),(z `1)],(z2 `2)],((X,z2) + (X,z))},{[[(((y `1) `1) `1),(z `1)],(z2 `2)]}} is non empty V41() V45() set
z2 `1 is set
[(z2 `1),(z `1)] is V26() set
{(z2 `1),(z `1)} is non empty V41() set
{(z2 `1)} is non empty trivial V41() V48(1) set
{{(z2 `1),(z `1)},{(z2 `1)}} is non empty V41() V45() set
[[(z2 `1),(z `1)],(z2 `2)] is V26() set
{[(z2 `1),(z `1)],(z2 `2)} is non empty V41() set
{[(z2 `1),(z `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(z2 `1),(z `1)],(z2 `2)},{[(z2 `1),(z `1)]}} is non empty V41() V45() set
[[[(z2 `1),(z `1)],(z2 `2)],((X,z2) + (X,z))] is V26() set
{[[(z2 `1),(z `1)],(z2 `2)],((X,z2) + (X,z))} is non empty V41() set
{[[(z2 `1),(z `1)],(z2 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(z2 `1),(z `1)],(z2 `2)],((X,z2) + (X,z))},{[[(z2 `1),(z `1)],(z2 `2)]}} is non empty V41() V45() set
f9 is set
fs is set
[f9,fs] is V26() set
{f9,fs} is non empty V41() set
{f9} is non empty trivial V41() V48(1) set
{{f9,fs},{f9}} is non empty V41() V45() set
f9 is set
fs is set
[f9,fs] is V26() set
{f9,fs} is non empty V41() set
{f9} is non empty trivial V41() V48(1) set
{{f9,fs},{f9}} is non empty V41() V45() set
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
((X,Y) - 1) + 1 is complex ext-real V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
(X,Y) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(- v1) + ((X,Y) + 1) is complex ext-real V53() V54() set
(- 1) + ((X,Y) + 1) is complex ext-real V53() V54() set
v2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
X is set
(X) is strict multMagma
(X) is set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
Y is Element of the carrier of (X)
f is Element of the carrier of (X)
Y * f is Element of the carrier of (X)
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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (Y,f) is Element of the carrier of (X)
y is Element of the carrier of (X)
x is Element of the carrier of (X)
y * x is Element of the carrier of (X)
the multF of (X) . (y,x) is Element of the carrier of (X)
Y `1 is set
f `1 is set
[(Y `1),(f `1)] is V26() set
{(Y `1),(f `1)} is non empty V41() set
{(Y `1)} is non empty trivial V41() V48(1) set
{{(Y `1),(f `1)},{(Y `1)}} is non empty V41() V45() set
Y `2 is set
[[(Y `1),(f `1)],(Y `2)] is V26() set
{[(Y `1),(f `1)],(Y `2)} is non empty V41() set
{[(Y `1),(f `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(Y `1),(f `1)],(Y `2)},{[(Y `1),(f `1)]}} is non empty V41() V45() set
(X,Y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) + (X,f) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))] is V26() set
{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))} is non empty V41() set
{[[(Y `1),(f `1)],(Y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(Y `1),(f `1)],(Y `2)],((X,Y) + (X,f))},{[[(Y `1),(f `1)],(Y `2)]}} is non empty V41() V45() set
y `1 is set
x `1 is set
[(y `1),(x `1)] is V26() set
{(y `1),(x `1)} is non empty V41() set
{(y `1)} is non empty trivial V41() V48(1) set
{{(y `1),(x `1)},{(y `1)}} is non empty V41() V45() set
y `2 is set
[[(y `1),(x `1)],(y `2)] is V26() set
{[(y `1),(x `1)],(y `2)} is non empty V41() set
{[(y `1),(x `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(y `1),(x `1)],(y `2)},{[(y `1),(x `1)]}} is non empty V41() V45() set
(X,y) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,y) + (X,x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(y `1),(x `1)],(y `2)],((X,y) + (X,x))] is V26() set
{[[(y `1),(x `1)],(y `2)],((X,y) + (X,x))} is non empty V41() set
{[[(y `1),(x `1)],(y `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(y `1),(x `1)],(y `2)],((X,y) + (X,x))},{[[(y `1),(x `1)],(y `2)]}} is non empty V41() V45() set
f `2 is set
x `2 is set
[(Y `1),(Y `2)] is V26() set
{(Y `1),(Y `2)} is non empty V41() set
{{(Y `1),(Y `2)},{(Y `1)}} is non empty V41() V45() set
[(y `1),(y `2)] is V26() set
{(y `1),(y `2)} is non empty V41() set
{{(y `1),(y `2)},{(y `1)}} is non empty V41() V45() set
[(f `1),(f `2)] is V26() set
{(f `1),(f `2)} is non empty V41() set
{(f `1)} is non empty trivial V41() V48(1) set
{{(f `1),(f `2)},{(f `1)}} is non empty V41() V45() set
[(x `1),(x `2)] is V26() set
{(x `1),(x `2)} is non empty V41() set
{(x `1)} is non empty trivial V41() V48(1) set
{{(x `1),(x `2)},{(x `1)}} is non empty V41() V45() set
X is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
(X) is strict multMagma
(X) is set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
[:(X,Y), the carrier of (X):] is Relation-like set
bool [:(X,Y), the carrier of (X):] is non empty set
f is set
[f,Y] is V26() set
{f,Y} is non empty V41() set
{f} is non empty trivial V41() V48(1) set
{{f,Y},{f}} is non empty V41() V45() set
{Y} is non empty trivial V41() V45() V48(1) set
[:(X,Y),{Y}:] is Relation-like set
f is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom f is Element of bool (X,Y)
bool (X,Y) is non empty set
y is set
f . y is set
[y,Y] is V26() set
{y,Y} is non empty V41() set
{y} is non empty trivial V41() V48(1) set
{{y,Y},{y}} is non empty V41() V45() set
dom {} is empty proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional V41() V42() V45() V46() V48( {} ) V53() V54() Element of bool NAT
proj2 {} is empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex ext-real non positive non negative Relation-like non-empty empty-yielding Function-like one-to-one constant functional with_non-empty_elements V41() V42() V45() V46() V48( {} ) V53() V54() set
y is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
f is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom f is Element of bool (X,Y)
bool (X,Y) is non empty set
y is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom y is Element of bool (X,Y)
x is set
f . x is set
y . x is set
[x,Y] is V26() set
{x,Y} is non empty V41() set
{x} is non empty trivial V41() V48(1) set
{{x,Y},{x}} is non empty V41() V45() set
y is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom y is Element of bool (X,Y)
bool (X,Y) is non empty set
x is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom x is Element of bool (X,Y)
c6 is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
w1 is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
w2 is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
dom w2 is Element of bool (X,Y)
v1 is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
X is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
(X) is strict multMagma
(X) is set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
[:(X,Y), the carrier of (X):] is Relation-like set
bool [:(X,Y), the carrier of (X):] is non empty set
dom (X,Y) is Element of bool (X,Y)
bool (X,Y) is non empty set
f is set
y is set
(X,Y) . f is set
(X,Y) . y is set
[f,Y] is V26() set
{f,Y} is non empty V41() set
{f} is non empty trivial V41() V48(1) set
{{f,Y},{f}} is non empty V41() V45() set
[y,Y] is V26() set
{y,Y} is non empty V41() set
{y} is non empty trivial V41() V48(1) set
{{y,Y},{y}} is non empty V41() V45() set
X is set
Y is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,Y) is Relation-like (X,Y) -defined the carrier of (X) -valued Function-like quasi_total Element of bool [:(X,Y), the carrier of (X):]
(X,Y) is set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . Y is set
(X) is strict multMagma
(X) is set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is Relation-like [:(X),(X):] -defined (X) -valued Function-like quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is Relation-like set
[:[:(X),(X):],(X):] is Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is set
[:(X,Y), the carrier of (X):] is Relation-like set
bool [:(X,Y), the carrier of (X):] is non empty set
X is non empty set
(X,1) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . 1 is set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X) is non empty strict multMagma
(X) is non empty set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
Y is set
(X,1) . Y is set
[Y,1] is V26() set
{Y,1} is non empty V41() set
{Y} is non empty trivial V41() V48(1) set
{{Y,1},{Y}} is non empty V41() V45() set
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
bool the carrier of (X) is non empty set
(X,1) is non empty set
(X) . 1 is set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) .: X is Element of bool the carrier of (X)
Y is Element of bool the carrier of (X)
((X),Y) is strict ((X))
y is set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
(X,1) . y is set
the carrier of ((X),Y) is set
y is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
c6 is Element of the carrier of (X)
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is Element of the carrier of (X)
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 `1 is set
c6 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[(c6 `1),(c6 `2)] is V26() set
{(c6 `1),(c6 `2)} is non empty V41() set
{(c6 `1)} is non empty trivial V41() V48(1) set
{{(c6 `1),(c6 `2)},{(c6 `1)}} is non empty V41() V45() set
[(c6 `1),1] is V26() set
{(c6 `1),1} is non empty V41() set
{{(c6 `1),1},{(c6 `1)}} is non empty V41() V45() set
{ (b1 `1) where b1 is Element of the carrier of (X) : (X,b1) = 1 } is set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
(X,1) . (c6 `1) is set
c6 is Element of the carrier of (X)
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is Element of the carrier of (X)
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is Element of the carrier of (X)
w2 is Element of the carrier of (X)
w1 * w2 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (w1,w2) is Element of the carrier of (X)
(X,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,w2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[w1,w2] is V26() set
{w1,w2} is non empty V41() set
{w1} is non empty trivial V41() V48(1) set
{{w1,w2},{w1}} is non empty V41() V45() set
[: the carrier of ((X),Y), the carrier of ((X),Y):] is Relation-like set
v1 is Element of the carrier of ((X),Y)
v2 is Element of the carrier of ((X),Y)
v1 * v2 is Element of the carrier of ((X),Y)
the multF of ((X),Y) is Relation-like [: the carrier of ((X),Y), the carrier of ((X),Y):] -defined the carrier of ((X),Y) -valued Function-like quasi_total Element of bool [:[: the carrier of ((X),Y), the carrier of ((X),Y):], the carrier of ((X),Y):]
[:[: the carrier of ((X),Y), the carrier of ((X),Y):], the carrier of ((X),Y):] is Relation-like set
bool [:[: the carrier of ((X),Y), the carrier of ((X),Y):], the carrier of ((X),Y):] is non empty set
the multF of ((X),Y) . (v1,v2) is Element of the carrier of ((X),Y)
[v1,v2] is V26() set
{v1,v2} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,v2},{v1}} is non empty V41() V45() set
the multF of ((X),Y) . [v1,v2] is set
the multF of (X) || the carrier of ((X),Y) is Relation-like Function-like set
( the multF of (X) || the carrier of ((X),Y)) . [w1,w2] is set
the multF of (X) | [: the carrier of ((X),Y), the carrier of ((X),Y):] is Relation-like [: the carrier of (X), the carrier of (X):] -defined [: the carrier of ((X),Y), the carrier of ((X),Y):] -defined [: the carrier of (X), the carrier of (X):] -defined the carrier of (X) -valued Function-like Element of bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):]
( the multF of (X) | [: the carrier of ((X),Y), the carrier of ((X),Y):]) . [w1,w2] is set
the multF of (X) . [w1,w2] is set
c6 is Element of the carrier of (X)
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x is Element of the carrier of (X)
(X,x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is Element of the carrier of (X)
(X,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
[: the carrier of (X), the carrier of (X):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (X):] is non empty set
(X,1) is non empty set
(X) . 1 is set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) .: X is Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
Y is Relation-like the carrier of (X) -defined the carrier of (X) -valued total quasi_total V260() V265() ((X)) Element of bool [: the carrier of (X), the carrier of (X):]
((X),Y) is non empty strict multMagma
Class Y is non empty with_non-empty_elements a_partition of the carrier of (X)
((X),Y) is non empty Relation-like [:(Class Y),(Class Y):] -defined Class Y -valued Function-like total quasi_total Element of bool [:[:(Class Y),(Class Y):],(Class Y):]
[:(Class Y),(Class Y):] is non empty Relation-like set
[:[:(Class Y),(Class Y):],(Class Y):] is non empty Relation-like set
bool [:[:(Class Y),(Class Y):],(Class Y):] is non empty set
multMagma(# (Class Y),((X),Y) #) is strict multMagma
the carrier of ((X),Y) is non empty set
((X),Y) is non empty Relation-like the carrier of (X) -defined the carrier of ((X),Y) -valued Function-like total quasi_total onto multiplicative Element of bool [: the carrier of (X), the carrier of ((X),Y):]
[: the carrier of (X), the carrier of ((X),Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of ((X),Y):] is non empty set
((X),Y) .: ((X,1) .: X) is Element of bool the carrier of ((X),Y)
bool the carrier of ((X),Y) is non empty set
(((X),Y),(((X),Y) .: ((X,1) .: X))) is strict (((X),Y))
y is Element of bool the carrier of (X)
((X),y) is strict ((X))
the carrier of ((X),y) is set
rng ((X),Y) is non empty Element of bool the carrier of ((X),Y)
dom ((X),Y) is non empty Element of bool the carrier of (X)
((X),Y) .: (dom ((X),Y)) is Element of bool the carrier of ((X),Y)
((X),Y) .: the carrier of ((X),y) is Element of bool the carrier of ((X),Y)
((X),Y) .: y is Element of bool the carrier of ((X),Y)
(((X),Y),(((X),Y) .: y)) is strict (((X),Y))
the carrier of (((X),Y),(((X),Y) .: y)) is set
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(Y,1) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) . 1 is set
(Y) is non empty strict multMagma
(Y) is non empty set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
[:X, the carrier of (Y):] is non empty Relation-like set
bool [:X, the carrier of (Y):] is non empty set
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(Y,1) * f is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
dom f is non empty Element of bool X
bool X is non empty set
dom (Y,1) is non empty Element of bool (Y,1)
bool (Y,1) is non empty set
rng f is non empty Element of bool Y
bool Y is non empty set
dom ((Y,1) * f) is Element of bool X
rng ((Y,1) * f) is Element of bool the carrier of (Y)
bool the carrier of (Y) is non empty set
rng (Y,1) is non empty Element of bool the carrier of (Y)
X is non empty set
f is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,f) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) . f is set
Y is non empty multMagma
the carrier of Y is non empty set
[:(X,f), the carrier of Y:] is non empty Relation-like set
bool [:(X,f), the carrier of Y:] is non empty set
y is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,y) is non empty set
(X) . y is set
[:(X,y), the carrier of Y:] is non empty Relation-like set
bool [:(X,y), the carrier of Y:] is non empty set
[:(X,f),(X,y):] is non empty Relation-like set
{f} is non empty trivial V41() V45() V48(1) set
[:[:(X,f),(X,y):],{f}:] is non empty Relation-like set
[:[:[:(X,f),(X,y):],{f}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,f),(X,y):],{f}:], the carrier of Y:] is non empty set
x is non empty Relation-like (X,f) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,f), the carrier of Y:]
c6 is non empty Relation-like (X,y) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,y), the carrier of Y:]
w2 is set
w2 `1 is set
(w2 `1) `1 is set
(w2 `1) `2 is set
v1 is Element of (X,f)
x . v1 is Element of the carrier of Y
v2 is Element of (X,y)
c6 . v2 is Element of the carrier of Y
(x . v1) * (c6 . v2) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((x . v1),(c6 . v2)) is Element of the carrier of Y
w9 is Element of [:[:(X,f),(X,y):],{f}:]
z2 is Element of (X,f)
w9 `1 is set
(w9 `1) `1 is set
z1 is Element of (X,y)
(w9 `1) `2 is set
x . z2 is Element of the carrier of Y
c6 . z1 is Element of the carrier of Y
(x . z2) * (c6 . z1) is Element of the carrier of Y
the multF of Y . ((x . z2),(c6 . z1)) is Element of the carrier of Y
w2 is non empty Relation-like [:[:(X,f),(X,y):],{f}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,f),(X,y):],{f}:], the carrier of Y:]
v2 is Element of (X,f)
v1 is Element of [:[:(X,f),(X,y):],{f}:]
v1 `1 is set
(v1 `1) `1 is set
c11 is Element of (X,y)
(v1 `1) `2 is set
w2 . v1 is Element of the carrier of Y
x . v2 is Element of the carrier of Y
c6 . c11 is Element of the carrier of Y
(x . v2) * (c6 . c11) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((x . v2),(c6 . c11)) is Element of the carrier of Y
w1 is non empty Relation-like [:[:(X,f),(X,y):],{f}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,f),(X,y):],{f}:], the carrier of Y:]
w2 is non empty Relation-like [:[:(X,f),(X,y):],{f}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,f),(X,y):],{f}:], the carrier of Y:]
v1 is set
w1 . v1 is set
w2 . v1 is set
v2 is Element of [:[:(X,f),(X,y):],{f}:]
v2 `1 is set
(v2 `1) `1 is set
(v2 `1) `2 is set
c11 is Element of (X,f)
x . c11 is Element of the carrier of Y
w9 is Element of (X,y)
c6 . w9 is Element of the carrier of Y
(x . c11) * (c6 . w9) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((x . c11),(c6 . w9)) is Element of the carrier of Y
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X,1) is non empty set
(X) . 1 is set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) " is Relation-like Function-like one-to-one set
Y is non empty multMagma
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
[: the carrier of (X), the carrier of Y:] is non empty Relation-like set
bool [: the carrier of (X), the carrier of Y:] is non empty set
f is non empty Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:X, the carrier of Y:]
((X,1) ") * f is Relation-like the carrier of Y -valued Function-like set
y is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,x) is set
(X) . x is set
Funcs ((X,x), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,x), the carrier of Y
y is Relation-like Function-like set
proj1 y is set
Funcs (X, the carrier of Y) is non empty functional FUNCTION_DOMAIN of X, the carrier of Y
y . 1 is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,x) is set
(X) . x is set
Funcs ((X,x), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,x), the carrier of Y
proj2 y is set
union (proj2 y) is set
Union y is set
x is non empty set
x ^omega is set
c6 is set
w1 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
(dom w1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
v1 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 - 1 is complex ext-real V53() V54() set
c11 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
v1 . c11 is set
(X,c11) is non empty set
(X) . c11 is set
[:(X,c11), the carrier of Y:] is non empty Relation-like set
bool [:(X,c11), the carrier of Y:] is non empty set
v1 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 - 1 is complex ext-real V53() V54() set
c11 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
v1 . c11 is set
(X,c11) is non empty set
(X) . c11 is set
[:(X,c11), the carrier of Y:] is non empty Relation-like set
bool [:(X,c11), the carrier of Y:] is non empty set
w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 -' 1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
2 - 1 is complex ext-real V53() V54() set
w2 - 1 is complex ext-real V53() V54() set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg v1 is Element of bool NAT
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(w2 - 1) + 1 is complex ext-real V53() V54() set
(v2 + 1) - v2 is complex ext-real V53() V54() set
w2 - v2 is complex ext-real V53() V54() set
w2 -' v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
c11 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,c11) is non empty set
(X) . c11 is set
[:(X,c11), the carrier of Y:] is non empty Relation-like set
bool [:(X,c11), the carrier of Y:] is non empty set
w1 . c11 is set
- v2 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
(- v2) + w2 is complex ext-real V53() V54() set
(- 1) + w2 is complex ext-real V53() V54() set
w9 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,w9) is non empty set
(X) . w9 is set
[:(X,w9), the carrier of Y:] is non empty Relation-like set
bool [:(X,w9), the carrier of Y:] is non empty set
w1 . w9 is set
z2 is non empty Relation-like (X,c11) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,c11), the carrier of Y:]
z1 is non empty Relation-like (X,w9) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,w9), the carrier of Y:]
(X,Y,c11,w9,z2,z1) is non empty Relation-like [:[:(X,c11),(X,w9):],{c11}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,c11),(X,w9):],{c11}:], the carrier of Y:]
[:(X,c11),(X,w9):] is non empty Relation-like set
{c11} is non empty trivial V41() V45() V48(1) set
[:[:(X,c11),(X,w9):],{c11}:] is non empty Relation-like set
[:[:[:(X,c11),(X,w9):],{c11}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,c11),(X,w9):],{c11}:], the carrier of Y:] is non empty set
z is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 - z is complex ext-real V53() V54() set
v2 is Relation-like Function-like FinSequence-like set
dom v2 is Element of bool NAT
Union v2 is set
w9 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w9 - 1 is complex ext-real V53() V54() set
len v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w9 - z2 is complex ext-real V53() V54() set
v2 . z2 is set
z2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(w2 - 1) + 1 is complex ext-real V53() V54() set
(z2 + 1) - z2 is complex ext-real V53() V54() set
w2 - z2 is complex ext-real V53() V54() set
w2 -' z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
z1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z1) is non empty set
(X) . z1 is set
[:(X,z1), the carrier of Y:] is non empty Relation-like set
bool [:(X,z1), the carrier of Y:] is non empty set
w1 . z1 is set
- z2 is complex ext-real non positive V53() V54() set
- 1 is complex ext-real non positive V53() V54() set
(- z2) + w2 is complex ext-real V53() V54() set
(- 1) + w2 is complex ext-real V53() V54() set
z2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z2) is non empty set
(X) . z2 is set
[:(X,z2), the carrier of Y:] is non empty Relation-like set
bool [:(X,z2), the carrier of Y:] is non empty set
w1 . z2 is set
z is non empty Relation-like (X,z1) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z1), the carrier of Y:]
f9 is non empty Relation-like (X,z2) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z2), the carrier of Y:]
(X,Y,z1,z2,z,f9) is non empty Relation-like [:[:(X,z1),(X,z2):],{z1}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:]
[:(X,z1),(X,z2):] is non empty Relation-like set
{z1} is non empty trivial V41() V45() V48(1) set
[:[:(X,z1),(X,z2):],{z1}:] is non empty Relation-like set
[:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty set
fs is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,fs) is non empty set
(X) . fs is set
[:(X,fs), the carrier of Y:] is non empty Relation-like set
bool [:(X,fs), the carrier of Y:] is non empty set
p is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,p) is non empty set
(X) . p is set
[:(X,p), the carrier of Y:] is non empty Relation-like set
bool [:(X,p), the carrier of Y:] is non empty set
m1 is non empty Relation-like (X,fs) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,fs), the carrier of Y:]
w1 . fs is set
m2 is non empty Relation-like (X,p) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,p), the carrier of Y:]
w1 . p is set
(X,Y,fs,p,m1,m2) is non empty Relation-like [:[:(X,fs),(X,p):],{fs}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,fs),(X,p):],{fs}:], the carrier of Y:]
[:(X,fs),(X,p):] is non empty Relation-like set
{fs} is non empty trivial V41() V45() V48(1) set
[:[:(X,fs),(X,p):],{fs}:] is non empty Relation-like set
[:[:[:(X,fs),(X,p):],{fs}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,fs),(X,p):],{fs}:], the carrier of Y:] is non empty set
w9 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w9 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
z2 - 1 is complex ext-real V53() V54() set
z1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w9 . z1 is set
(X,z1) is non empty set
(X) . z1 is set
[:(X,z1), the carrier of Y:] is non empty Relation-like set
bool [:(X,z1), the carrier of Y:] is non empty set
z2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w1 . z2 is set
(X,z2) is non empty set
(X) . z2 is set
[:(X,z2), the carrier of Y:] is non empty Relation-like set
bool [:(X,z2), the carrier of Y:] is non empty set
w1 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
w2 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 - 1 is complex ext-real V53() V54() set
v2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w2 . v2 is set
(X,v2) is non empty set
(X) . v2 is set
[:(X,v2), the carrier of Y:] is non empty Relation-like set
bool [:(X,v2), the carrier of Y:] is non empty set
c11 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w1 . c11 is set
(X,c11) is non empty set
(X) . c11 is set
[:(X,c11), the carrier of Y:] is non empty Relation-like set
bool [:(X,c11), the carrier of Y:] is non empty set
w1 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
w2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w1 . w2 is set
(X,w2) is non empty set
(X) . w2 is set
[:(X,w2), the carrier of Y:] is non empty Relation-like set
bool [:(X,w2), the carrier of Y:] is non empty set
c6 is Relation-like Function-like set
proj1 c6 is set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 - 1 is complex ext-real V53() V54() set
(X,w1) is set
(X) . w1 is set
Funcs ((X,w1), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,w1), the carrier of Y
w2 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
c6 . w2 is set
v1 is Relation-like Function-like FinSequence-like set
len v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
Union v1 is set
v1 is Relation-like Function-like FinSequence-like set
len v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
Union v1 is set
v2 is set
proj2 v1 is set
union (proj2 v1) is set
c11 is set
dom v1 is Element of bool NAT
w9 is set
v1 . w9 is set
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg (len v1) is Element of bool NAT
w1 - z2 is complex ext-real V53() V54() set
v1 . z2 is set
z1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z1) is non empty set
(X) . z1 is set
[:(X,z1), the carrier of Y:] is non empty Relation-like set
bool [:(X,z1), the carrier of Y:] is non empty set
z2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z2) is non empty set
(X) . z2 is set
[:(X,z2), the carrier of Y:] is non empty Relation-like set
bool [:(X,z2), the carrier of Y:] is non empty set
z is non empty Relation-like (X,z1) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z1), the carrier of Y:]
w2 . z1 is set
f9 is non empty Relation-like (X,z2) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z2), the carrier of Y:]
w2 . z2 is set
(X,Y,z1,z2,z,f9) is non empty Relation-like [:[:(X,z1),(X,z2):],{z1}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:]
[:(X,z1),(X,z2):] is non empty Relation-like set
{z1} is non empty trivial V41() V45() V48(1) set
[:[:(X,z1),(X,z2):],{z1}:] is non empty Relation-like set
[:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty set
fs is set
p is set
[fs,p] is V26() set
{fs,p} is non empty V41() set
{fs} is non empty trivial V41() V48(1) set
{{fs,p},{fs}} is non empty V41() V45() set
v2 is set
c11 is set
[v2,c11] is V26() set
{v2,c11} is non empty V41() set
{v2} is non empty trivial V41() V48(1) set
{{v2,c11},{v2}} is non empty V41() V45() set
w9 is set
[v2,w9] is V26() set
{v2,w9} is non empty V41() set
{{v2,w9},{v2}} is non empty V41() V45() set
proj2 v1 is set
union (proj2 v1) is set
z2 is set
dom v1 is Element of bool NAT
z1 is set
v1 . z1 is set
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
Seg (len v1) is Element of bool NAT
w1 - z2 is complex ext-real V53() V54() set
v1 . z2 is set
z is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z) is non empty set
(X) . z is set
[:(X,z), the carrier of Y:] is non empty Relation-like set
bool [:(X,z), the carrier of Y:] is non empty set
f9 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,f9) is non empty set
(X) . f9 is set
[:(X,f9), the carrier of Y:] is non empty Relation-like set
bool [:(X,f9), the carrier of Y:] is non empty set
fs is non empty Relation-like (X,z) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z), the carrier of Y:]
w2 . z is set
p is non empty Relation-like (X,f9) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,f9), the carrier of Y:]
w2 . f9 is set
(X,Y,z,f9,fs,p) is non empty Relation-like [:[:(X,z),(X,f9):],{z}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:]
[:(X,z),(X,f9):] is non empty Relation-like set
{z} is non empty trivial V41() V45() V48(1) set
[:[:(X,z),(X,f9):],{z}:] is non empty Relation-like set
[:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty set
dom (X,Y,z,f9,fs,p) is non empty Relation-like [:(X,z),(X,f9):] -defined {z} -valued Element of bool [:[:(X,z),(X,f9):],{z}:]
bool [:[:(X,z),(X,f9):],{z}:] is non empty set
v2 `2 is set
m1 is set
m2 is set
v1 . m2 is set
f1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 - f1 is complex ext-real V53() V54() set
v1 . f1 is set
f2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,f2) is non empty set
(X) . f2 is set
[:(X,f2), the carrier of Y:] is non empty Relation-like set
bool [:(X,f2), the carrier of Y:] is non empty set
x is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,x) is non empty set
(X) . x is set
[:(X,x), the carrier of Y:] is non empty Relation-like set
bool [:(X,x), the carrier of Y:] is non empty set
y is non empty Relation-like (X,f2) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,f2), the carrier of Y:]
w2 . f2 is set
z is non empty Relation-like (X,x) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,x), the carrier of Y:]
w2 . x is set
(X,Y,f2,x,y,z) is non empty Relation-like [:[:(X,f2),(X,x):],{f2}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,f2),(X,x):],{f2}:], the carrier of Y:]
[:(X,f2),(X,x):] is non empty Relation-like set
{f2} is non empty trivial V41() V45() V48(1) set
[:[:(X,f2),(X,x):],{f2}:] is non empty Relation-like set
[:[:[:(X,f2),(X,x):],{f2}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,f2),(X,x):],{f2}:], the carrier of Y:] is non empty set
dom (X,Y,f2,x,y,z) is non empty Relation-like [:(X,f2),(X,x):] -defined {f2} -valued Element of bool [:[:(X,f2),(X,x):],{f2}:]
bool [:[:(X,f2),(X,x):],{f2}:] is non empty set
v2 `1 is set
(v2 `1) `1 is set
(v2 `1) `2 is set
fa is Element of [:[:(X,z),(X,f9):],{z}:]
(X,Y,z,f9,fs,p) . fa is Element of the carrier of Y
fb is Element of (X,z)
fs . fb is Element of the carrier of Y
z1 is Element of (X,f9)
p . z1 is Element of the carrier of Y
(fs . fb) * (p . z1) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((fs . fb),(p . z1)) is Element of the carrier of Y
y29 is Element of (X,f2)
y . y29 is Element of the carrier of Y
z2 is Element of (X,x)
z . z2 is Element of the carrier of Y
(y . y29) * (z . z2) is Element of the carrier of Y
the multF of Y . ((y . y29),(z . z2)) is Element of the carrier of Y
x2 is Element of [:[:(X,f2),(X,x):],{f2}:]
(X,Y,f2,x,y,z) . x2 is Element of the carrier of Y
v2 is Relation-like Function-like set
c11 is set
c11 `2 is set
c11 `1 is set
(c11 `1) `1 is set
(c11 `1) `2 is set
w9 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,w9) is set
(X) . w9 is set
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,z2) is set
(X) . z2 is set
w9 + z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[:(X,w9),(X,z2):] is Relation-like set
{w9} is non empty trivial V41() V45() V48(1) set
[:[:(X,w9),(X,z2):],{w9}:] is Relation-like set
w1 - w9 is complex ext-real V53() V54() set
v1 . w9 is set
z1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z1) is non empty set
(X) . z1 is set
[:(X,z1), the carrier of Y:] is non empty Relation-like set
bool [:(X,z1), the carrier of Y:] is non empty set
z2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z2) is non empty set
(X) . z2 is set
[:(X,z2), the carrier of Y:] is non empty Relation-like set
bool [:(X,z2), the carrier of Y:] is non empty set
z is non empty Relation-like (X,z1) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z1), the carrier of Y:]
w2 . z1 is set
f9 is non empty Relation-like (X,z2) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z2), the carrier of Y:]
w2 . z2 is set
(X,Y,z1,z2,z,f9) is non empty Relation-like [:[:(X,z1),(X,z2):],{z1}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:]
[:(X,z1),(X,z2):] is non empty Relation-like set
{z1} is non empty trivial V41() V45() V48(1) set
[:[:(X,z1),(X,z2):],{z1}:] is non empty Relation-like set
[:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z1),(X,z2):],{z1}:], the carrier of Y:] is non empty set
p is Element of (X,z1)
z . p is Element of the carrier of Y
m1 is Element of (X,z2)
f9 . m1 is Element of the carrier of Y
(z . p) * (f9 . m1) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((z . p),(f9 . m1)) is Element of the carrier of Y
dom (X,Y,z1,z2,z,f9) is non empty Relation-like [:(X,z1),(X,z2):] -defined {z1} -valued Element of bool [:[:(X,z1),(X,z2):],{z1}:]
bool [:[:(X,z1),(X,z2):],{z1}:] is non empty set
fs is Element of [:[:(X,z1),(X,z2):],{z1}:]
(X,Y,z1,z2,z,f9) . fs is Element of the carrier of Y
m2 is set
f1 is set
[c11,f1] is V26() set
{c11,f1} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,f1},{c11}} is non empty V41() V45() set
Seg (len v1) is Element of bool NAT
dom v1 is Element of bool NAT
proj2 v1 is set
union (proj2 v1) is set
w9 is set
[c11,w9] is V26() set
{c11,w9} is non empty V41() set
{{c11,w9},{c11}} is non empty V41() V45() set
z2 is set
z1 is set
v1 . z1 is set
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 - z2 is complex ext-real V53() V54() set
v1 . z2 is set
z is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z) is non empty set
(X) . z is set
[:(X,z), the carrier of Y:] is non empty Relation-like set
bool [:(X,z), the carrier of Y:] is non empty set
f9 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,f9) is non empty set
(X) . f9 is set
[:(X,f9), the carrier of Y:] is non empty Relation-like set
bool [:(X,f9), the carrier of Y:] is non empty set
fs is non empty Relation-like (X,z) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z), the carrier of Y:]
w2 . z is set
p is non empty Relation-like (X,f9) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,f9), the carrier of Y:]
w2 . f9 is set
(X,Y,z,f9,fs,p) is non empty Relation-like [:[:(X,z),(X,f9):],{z}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:]
[:(X,z),(X,f9):] is non empty Relation-like set
{z} is non empty trivial V41() V45() V48(1) set
[:[:(X,z),(X,f9):],{z}:] is non empty Relation-like set
[:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty set
dom (X,Y,z,f9,fs,p) is non empty Relation-like [:(X,z),(X,f9):] -defined {z} -valued Element of bool [:[:(X,z),(X,f9):],{z}:]
bool [:[:(X,z),(X,f9):],{z}:] is non empty set
[(c11 `1),(c11 `2)] is V26() set
{(c11 `1),(c11 `2)} is non empty V41() set
{(c11 `1)} is non empty trivial V41() V48(1) set
{{(c11 `1),(c11 `2)},{(c11 `1)}} is non empty V41() V45() set
[((c11 `1) `1),((c11 `1) `2)] is V26() set
{((c11 `1) `1),((c11 `1) `2)} is non empty V41() set
{((c11 `1) `1)} is non empty trivial V41() V48(1) set
{{((c11 `1) `1),((c11 `1) `2)},{((c11 `1) `1)}} is non empty V41() V45() set
[[((c11 `1) `1),((c11 `1) `2)],(c11 `2)] is V26() set
{[((c11 `1) `1),((c11 `1) `2)],(c11 `2)} is non empty V41() set
{[((c11 `1) `1),((c11 `1) `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[((c11 `1) `1),((c11 `1) `2)],(c11 `2)},{[((c11 `1) `1),((c11 `1) `2)]}} is non empty V41() V45() set
z + f9 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(z + f9)) is non empty set
(X) . (z + f9) is set
proj1 v2 is set
proj2 v2 is set
c11 is set
w9 is set
v2 . w9 is set
[w9,c11] is V26() set
{w9,c11} is non empty V41() set
{w9} is non empty trivial V41() V48(1) set
{{w9,c11},{w9}} is non empty V41() V45() set
proj2 v1 is set
union (proj2 v1) is set
z2 is set
dom v1 is Element of bool NAT
z1 is set
v1 . z1 is set
Seg (len v1) is Element of bool NAT
z2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 - z2 is complex ext-real V53() V54() set
v1 . z2 is set
z is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z) is non empty set
(X) . z is set
[:(X,z), the carrier of Y:] is non empty Relation-like set
bool [:(X,z), the carrier of Y:] is non empty set
f9 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,f9) is non empty set
(X) . f9 is set
[:(X,f9), the carrier of Y:] is non empty Relation-like set
bool [:(X,f9), the carrier of Y:] is non empty set
fs is non empty Relation-like (X,z) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,z), the carrier of Y:]
w2 . z is set
p is non empty Relation-like (X,f9) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,f9), the carrier of Y:]
w2 . f9 is set
(X,Y,z,f9,fs,p) is non empty Relation-like [:[:(X,z),(X,f9):],{z}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:]
[:(X,z),(X,f9):] is non empty Relation-like set
{z} is non empty trivial V41() V45() V48(1) set
[:[:(X,z),(X,f9):],{z}:] is non empty Relation-like set
[:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,z),(X,f9):],{z}:], the carrier of Y:] is non empty set
rng (X,Y,z,f9,fs,p) is non empty Element of bool the carrier of Y
bool the carrier of Y is non empty set
w1 is set
c6 . w1 is set
w2 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 - 1 is complex ext-real V53() V54() set
(dom w2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
Funcs ({}, the carrier of Y) is non empty functional FUNCTION_DOMAIN of {} , the carrier of Y
y . 0 is set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,v1) is set
(X) . v1 is set
Funcs ((X,v1), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,v1), the carrier of Y
(dom w2) - 1 is complex ext-real V53() V54() set
(X,(dom w2)) is set
(X) . (dom w2) is set
Funcs ((X,(dom w2)), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,(dom w2)), the carrier of Y
v2 is Relation-like Function-like FinSequence-like set
len v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
Union v2 is set
y . (dom w2) is set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,v2) is set
(X) . v2 is set
Funcs ((X,v2), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,v2), the carrier of Y
w2 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w2 . v1 is set
(X,v1) is non empty set
(X) . v1 is set
[:(X,v1), the carrier of Y:] is non empty Relation-like set
bool [:(X,v1), the carrier of Y:] is non empty set
w2 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom w2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
v1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w2 . v1 is set
(X,v1) is non empty set
(X) . v1 is set
[:(X,v1), the carrier of Y:] is non empty Relation-like set
bool [:(X,v1), the carrier of Y:] is non empty set
[:(x ^omega),x:] is Relation-like set
bool [:(x ^omega),x:] is non empty set
w1 is Relation-like x ^omega -defined x -valued Function-like total quasi_total Element of bool [:(x ^omega),x:]
[:NAT,x:] is non empty non trivial Relation-like V41() set
bool [:NAT,x:] is non empty non trivial V41() set
w2 is non empty Relation-like NAT -defined x -valued Function-like total quasi_total Element of bool [:NAT,x:]
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v2 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 . v2 is set
(X,v2) is set
(X) . v2 is set
[:(X,v2), the carrier of Y:] is Relation-like set
bool [:(X,v2), the carrier of Y:] is non empty set
w2 | v2 is T-Sequence-like Relation-like NAT -defined v2 -defined NAT -defined x -valued Function-like V41() Element of bool [:NAT,x:]
dom (w2 | v2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool v2
bool v2 is non empty V41() V45() set
c11 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(w2 | v2) . c11 is set
(X,c11) is non empty set
(X) . c11 is set
[:(X,c11), the carrier of Y:] is non empty Relation-like set
bool [:(X,c11), the carrier of Y:] is non empty set
w2 . c11 is set
c11 is T-Sequence-like Relation-like x -valued Function-like V41() set
dom c11 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
(x,w1,c11) is Element of x
w9 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w9 - 1 is complex ext-real V53() V54() set
dom w2 is non empty Element of bool NAT
(dom w2) /\ v2 is V41() Element of bool NAT
v2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
Funcs ({}, the carrier of Y) is non empty functional FUNCTION_DOMAIN of {} , the carrier of Y
Funcs ((X,v2), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,v2), the carrier of Y
w9 is Relation-like Function-like set
proj1 w9 is set
proj2 w9 is set
v2 - 1 is complex ext-real V53() V54() set
Funcs ((X,v2), the carrier of Y) is non empty functional FUNCTION_DOMAIN of (X,v2), the carrier of Y
w9 is Relation-like Function-like FinSequence-like set
len w9 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
Union w9 is set
w9 is Relation-like Function-like set
proj1 w9 is set
proj2 w9 is set
v1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 . v1 is set
(X,v1) is set
(X) . v1 is set
[:(X,v1), the carrier of Y:] is Relation-like set
bool [:(X,v1), the carrier of Y:] is non empty set
v1 is set
v2 is set
c11 is Element of the carrier of (X)
c11 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(c11 `2)) is non empty set
(X) . (c11 `2) is set
[:(X,(c11 `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,(c11 `2)), the carrier of Y:] is non empty set
w2 . (c11 `2) is set
w9 is non empty Relation-like (X,(c11 `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,(c11 `2)), the carrier of Y:]
c11 `1 is set
w9 . (c11 `1) is set
{(c11 `2)} is non empty trivial V41() V45() V48(1) set
[:(X,(c11 `2)),{(c11 `2)}:] is non empty Relation-like set
z1 is Element of the carrier of (X)
z1 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(z1 `2)) is non empty set
(X) . (z1 `2) is set
[:(X,(z1 `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,(z1 `2)), the carrier of Y:] is non empty set
z2 is non empty Relation-like (X,(z1 `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,(z1 `2)), the carrier of Y:]
w2 . (z1 `2) is set
z1 `1 is set
z2 . (z1 `1) is set
[:v1, the carrier of Y:] is Relation-like set
bool [:v1, the carrier of Y:] is non empty set
v2 is Relation-like v1 -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:v1, the carrier of Y:]
c11 is non empty Relation-like the carrier of (X) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of Y:]
w9 is Element of the carrier of (X)
z2 is Element of the carrier of (X)
w9 * z2 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (w9,z2) is Element of the carrier of (X)
c11 . (w9 * z2) is Element of the carrier of Y
c11 . w9 is Element of the carrier of Y
c11 . z2 is Element of the carrier of Y
(c11 . w9) * (c11 . z2) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((c11 . w9),(c11 . z2)) is Element of the carrier of Y
(w9 * z2) `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,((w9 * z2) `2)) is non empty set
(X) . ((w9 * z2) `2) is set
[:(X,((w9 * z2) `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,((w9 * z2) `2)), the carrier of Y:] is non empty set
w2 . ((w9 * z2) `2) is set
w9 `1 is set
z2 `1 is set
[(w9 `1),(z2 `1)] is V26() set
{(w9 `1),(z2 `1)} is non empty V41() set
{(w9 `1)} is non empty trivial V41() V48(1) set
{{(w9 `1),(z2 `1)},{(w9 `1)}} is non empty V41() V45() set
w9 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[(w9 `1),(z2 `1)],(w9 `2)] is V26() set
{[(w9 `1),(z2 `1)],(w9 `2)} is non empty V41() set
{[(w9 `1),(z2 `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(w9 `1),(z2 `1)],(w9 `2)},{[(w9 `1),(z2 `1)]}} is non empty V41() V45() set
(X,w9) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,z2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,w9) + (X,z2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(w9 `1),(z2 `1)],(w9 `2)],((X,w9) + (X,z2))] is V26() set
{[[(w9 `1),(z2 `1)],(w9 `2)],((X,w9) + (X,z2))} is non empty V41() set
{[[(w9 `1),(z2 `1)],(w9 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(w9 `1),(z2 `1)],(w9 `2)],((X,w9) + (X,z2))},{[[(w9 `1),(z2 `1)],(w9 `2)]}} is non empty V41() V45() set
(w9 * z2) `1 is set
z1 is non empty Relation-like (X,((w9 * z2) `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,((w9 * z2) `2)), the carrier of Y:]
w2 | ((X,w9) + (X,z2)) is T-Sequence-like Relation-like NAT -defined (X,w9) + (X,z2) -defined NAT -defined x -valued Function-like V41() Element of bool [:NAT,x:]
(x,w1,(w2 | ((X,w9) + (X,z2)))) is Element of x
dom (w2 | ((X,w9) + (X,z2))) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
z2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(w2 | ((X,w9) + (X,z2))) . z2 is set
(X,z2) is non empty set
(X) . z2 is set
[:(X,z2), the carrier of Y:] is non empty Relation-like set
bool [:(X,z2), the carrier of Y:] is non empty set
w2 . z2 is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
dom w2 is non empty Element of bool NAT
(dom w2) /\ ((X,w9) + (X,z2)) is V41() Element of bool NAT
((X,w9) + (X,z2)) - 1 is complex ext-real V53() V54() set
z is Relation-like Function-like FinSequence-like set
len z is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of NAT
Union z is set
{((w9 * z2) `2)} is non empty trivial V41() V45() V48(1) set
[:(X,((w9 * z2) `2)),{((w9 * z2) `2)}:] is non empty Relation-like set
dom z1 is non empty Element of bool (X,((w9 * z2) `2))
bool (X,((w9 * z2) `2)) is non empty set
z1 . ((w9 * z2) `1) is set
[((w9 * z2) `1),(z1 . ((w9 * z2) `1))] is V26() set
{((w9 * z2) `1),(z1 . ((w9 * z2) `1))} is non empty V41() set
{((w9 * z2) `1)} is non empty trivial V41() V48(1) set
{{((w9 * z2) `1),(z1 . ((w9 * z2) `1))},{((w9 * z2) `1)}} is non empty V41() V45() set
proj2 z is set
union (proj2 z) is set
f9 is set
dom z is Element of bool NAT
fs is set
z . fs is set
Seg (len z) is Element of bool NAT
p is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
((X,w9) + (X,z2)) - p is complex ext-real V53() V54() set
z . p is set
m1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,m1) is non empty set
(X) . m1 is set
[:(X,m1), the carrier of Y:] is non empty Relation-like set
bool [:(X,m1), the carrier of Y:] is non empty set
m2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,m2) is non empty set
(X) . m2 is set
[:(X,m2), the carrier of Y:] is non empty Relation-like set
bool [:(X,m2), the carrier of Y:] is non empty set
f1 is non empty Relation-like (X,m1) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,m1), the carrier of Y:]
(w2 | ((X,w9) + (X,z2))) . m1 is set
f2 is non empty Relation-like (X,m2) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,m2), the carrier of Y:]
(w2 | ((X,w9) + (X,z2))) . m2 is set
(X,Y,m1,m2,f1,f2) is non empty Relation-like [:[:(X,m1),(X,m2):],{m1}:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[:[:(X,m1),(X,m2):],{m1}:], the carrier of Y:]
[:(X,m1),(X,m2):] is non empty Relation-like set
{m1} is non empty trivial V41() V45() V48(1) set
[:[:(X,m1),(X,m2):],{m1}:] is non empty Relation-like set
[:[:[:(X,m1),(X,m2):],{m1}:], the carrier of Y:] is non empty Relation-like set
bool [:[:[:(X,m1),(X,m2):],{m1}:], the carrier of Y:] is non empty set
dom (X,Y,m1,m2,f1,f2) is non empty Relation-like [:(X,m1),(X,m2):] -defined {m1} -valued Element of bool [:[:(X,m1),(X,m2):],{m1}:]
bool [:[:(X,m1),(X,m2):],{m1}:] is non empty set
((w9 * z2) `1) `1 is set
((w9 * z2) `1) `2 is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,z2) + (X,w9) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
0 + (X,w9) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
0 + (X,z2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
x is Element of [:[:(X,m1),(X,m2):],{m1}:]
x `1 is set
(x `1) `1 is set
(x `1) `2 is set
(X,Y,m1,m2,f1,f2) . x is Element of the carrier of Y
y is Element of (X,m1)
f1 . y is Element of the carrier of Y
z is Element of (X,m2)
f2 . z is Element of the carrier of Y
(f1 . y) * (f2 . z) is Element of the carrier of Y
the multF of Y . ((f1 . y),(f2 . z)) is Element of the carrier of Y
(X,(w9 `2)) is non empty set
(X) . (w9 `2) is set
[:(X,(w9 `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,(w9 `2)), the carrier of Y:] is non empty set
w2 . (w9 `2) is set
z2 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(X,(z2 `2)) is non empty set
(X) . (z2 `2) is set
[:(X,(z2 `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,(z2 `2)), the carrier of Y:] is non empty set
w2 . (z2 `2) is set
w2 . m1 is set
fa is non empty Relation-like (X,(w9 `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,(w9 `2)), the carrier of Y:]
fa . (w9 `1) is set
w2 . m2 is set
fb is non empty Relation-like (X,(z2 `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,(z2 `2)), the carrier of Y:]
fb . (z2 `1) is set
proj1 (((X,1) ") * f) is set
dom c11 is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
z2 is set
proj1 ((X,1) ") is set
rng (X,1) is non empty Element of bool the carrier of (X)
(dom c11) /\ (proj1 (((X,1) ") * f)) is Element of bool the carrier of (X)
z2 is set
c11 . z2 is set
(((X,1) ") * f) . z2 is set
proj1 ((X,1) ") is set
rng (X,1) is non empty Element of bool the carrier of (X)
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
z1 is set
(X,1) . z1 is set
[:(X,1),{1}:] is non empty Relation-like set
[:X,{1}:] is non empty Relation-like set
[z1,1] is V26() set
{z1,1} is non empty V41() set
{z1} is non empty trivial V41() V48(1) set
{{z1,1},{z1}} is non empty V41() V45() set
((X,1) ") . z2 is set
z2 is Element of the carrier of (X)
z2 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w2 . (z2 `2) is set
(X,(z2 `2)) is non empty set
(X) . (z2 `2) is set
[:(X,(z2 `2)), the carrier of Y:] is non empty Relation-like set
bool [:(X,(z2 `2)), the carrier of Y:] is non empty set
f9 is non empty Relation-like (X,(z2 `2)) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:(X,(z2 `2)), the carrier of Y:]
w2 . 1 is Element of x
w2 | 1 is T-Sequence-like Relation-like NAT -defined 1 -defined NAT -defined x -valued Function-like V41() Element of bool [:NAT,x:]
(x,w1,(w2 | 1)) is Element of x
dom (w2 | 1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
fs is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
(w2 | 1) . fs is set
(X,fs) is non empty set
(X) . fs is set
[:(X,fs), the carrier of Y:] is non empty Relation-like set
bool [:(X,fs), the carrier of Y:] is non empty set
w2 . fs is set
dom w2 is non empty Element of bool NAT
fs is T-Sequence-like Relation-like x -valued Function-like V41() set
dom fs is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() Element of bool NAT
(dom w2) /\ 1 is V41() Element of bool NAT
z2 `1 is set
f9 . (z2 `1) is set
f9 . z1 is set
f . (((X,1) ") . z2) is set
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X,1) is non empty set
(X) . 1 is set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) " is Relation-like Function-like one-to-one set
Y is non empty multMagma
the carrier of Y is non empty set
[:X, the carrier of Y:] is non empty Relation-like set
bool [:X, the carrier of Y:] is non empty set
[: the carrier of (X), the carrier of Y:] is non empty Relation-like set
bool [: the carrier of (X), the carrier of Y:] is non empty set
f is non empty Relation-like X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:X, the carrier of Y:]
((X,1) ") * f is Relation-like the carrier of Y -valued Function-like set
y is non empty Relation-like the carrier of (X) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of Y:]
x is non empty Relation-like the carrier of (X) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of Y:]
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is Element of the carrier of (X)
(X,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y . w1 is Element of the carrier of Y
x . w1 is Element of the carrier of Y
w1 `1 is set
w1 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[(w1 `1),(w1 `2)] is V26() set
{(w1 `1),(w1 `2)} is non empty V41() set
{(w1 `1)} is non empty trivial V41() V48(1) set
{{(w1 `1),(w1 `2)},{(w1 `1)}} is non empty V41() V45() set
(X,w1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
{ (b1 `1) where b1 is Element of the carrier of (X) : (X,b1) = 1 } is set
proj1 (((X,1) ") * f) is set
dom y is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
dom x is non empty Element of bool the carrier of (X)
(X,1) . (w1 `1) is set
[(w1 `1),1] is V26() set
{(w1 `1),1} is non empty V41() set
{{(w1 `1),1},{(w1 `1)}} is non empty V41() V45() set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
rng (X,1) is non empty Element of bool the carrier of (X)
proj1 ((X,1) ") is set
dom f is non empty Element of bool X
bool X is non empty set
proj2 ((X,1) ") is set
(dom y) /\ (proj1 (((X,1) ") * f)) is Element of bool the carrier of (X)
(dom x) /\ (proj1 (((X,1) ") * f)) is Element of bool the carrier of (X)
(((X,1) ") * f) . w1 is set
w2 is Element of the carrier of (X)
v1 is Element of the carrier of (X)
w2 * v1 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (w2,v1) is Element of the carrier of (X)
(X,w2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(X,v1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y . w2 is Element of the carrier of Y
x . w2 is Element of the carrier of Y
y . v1 is Element of the carrier of Y
x . v1 is Element of the carrier of Y
y . (w2 * v1) is Element of the carrier of Y
(x . w2) * (x . v1) is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . ((x . w2),(x . v1)) is Element of the carrier of Y
c6 is Element of the carrier of (X)
y . c6 is Element of the carrier of Y
x . c6 is Element of the carrier of Y
(X,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w2 is Element of the carrier of (X)
(X,w2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
y . w2 is Element of the carrier of Y
x . w2 is Element of the carrier of Y
c6 is set
y . c6 is set
x . c6 is set
X is non empty multMagma
the carrier of X is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
[: the carrier of Y, the carrier of X:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of X:] is non empty set
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of Y:] is non empty set
f is non empty Relation-like the carrier of Y -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of X:]
rng f is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
( the carrier of Y, the carrier of X,f) is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() Element of bool [: the carrier of Y, the carrier of Y:]
y is non empty (X)
the carrier of y is non empty set
x is Relation-like the carrier of Y -defined the carrier of Y -valued total quasi_total V260() V265() (Y) Element of bool [: the carrier of Y, the carrier of Y:]
(Y,x) is non empty strict multMagma
Class x is non empty with_non-empty_elements a_partition of the carrier of Y
(Y,x) is non empty Relation-like [:(Class x),(Class x):] -defined Class x -valued Function-like total quasi_total Element of bool [:[:(Class x),(Class x):],(Class x):]
[:(Class x),(Class x):] is non empty Relation-like set
[:[:(Class x),(Class x):],(Class x):] is non empty Relation-like set
bool [:[:(Class x),(Class x):],(Class x):] is non empty set
multMagma(# (Class x),(Y,x) #) is strict multMagma
the carrier of (Y,x) is non empty set
[: the carrier of (Y,x), the carrier of y:] is non empty Relation-like set
bool [: the carrier of (Y,x), the carrier of y:] is non empty set
(Y,x) is non empty Relation-like the carrier of Y -defined the carrier of (Y,x) -valued Function-like total quasi_total onto multiplicative Element of bool [: the carrier of Y, the carrier of (Y,x):]
[: the carrier of Y, the carrier of (Y,x):] is non empty Relation-like set
bool [: the carrier of Y, the carrier of (Y,x):] is non empty set
(Y,x) ~ is Relation-like the carrier of (Y,x) -defined the carrier of Y -valued Element of bool [: the carrier of (Y,x), the carrier of Y:]
[: the carrier of (Y,x), the carrier of Y:] is non empty Relation-like set
bool [: the carrier of (Y,x), the carrier of Y:] is non empty set
((Y,x) ~) * f is Relation-like the carrier of (Y,x) -defined the carrier of X -valued Element of bool [: the carrier of (Y,x), the carrier of X:]
[: the carrier of (Y,x), the carrier of X:] is non empty Relation-like set
bool [: the carrier of (Y,x), the carrier of X:] is non empty set
w1 is set
w2 is set
[w1,w2] is V26() set
{w1,w2} is non empty V41() set
{w1} is non empty trivial V41() V48(1) set
{{w1,w2},{w1}} is non empty V41() V45() set
v1 is set
[w1,v1] is V26() set
{w1,v1} is non empty V41() set
{{w1,v1},{w1}} is non empty V41() V45() set
v2 is set
[w1,v2] is V26() set
{w1,v2} is non empty V41() set
{{w1,v2},{w1}} is non empty V41() V45() set
[v2,w2] is V26() set
{v2,w2} is non empty V41() set
{v2} is non empty trivial V41() V48(1) set
{{v2,w2},{v2}} is non empty V41() V45() set
c11 is set
[w1,c11] is V26() set
{w1,c11} is non empty V41() set
{{w1,c11},{w1}} is non empty V41() V45() set
[c11,v1] is V26() set
{c11,v1} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,v1},{c11}} is non empty V41() V45() set
[v2,w1] is V26() set
{v2,w1} is non empty V41() set
{{v2,w1},{v2}} is non empty V41() V45() set
[c11,w1] is V26() set
{c11,w1} is non empty V41() set
{{c11,w1},{c11}} is non empty V41() V45() set
dom (Y,x) is non empty Element of bool the carrier of Y
bool the carrier of Y is non empty set
w9 is Element of the carrier of Y
(Y,x) . w9 is Element of the carrier of (Y,x)
z2 is Element of the carrier of Y
(Y,x) . z2 is Element of the carrier of (Y,x)
f . w9 is Element of the carrier of X
f . z2 is Element of the carrier of X
Class (x,w9) is Element of bool the carrier of Y
Class (x,z2) is Element of bool the carrier of Y
[w9,z2] is V26() set
{w9,z2} is non empty V41() set
{w9} is non empty trivial V41() V48(1) set
{{w9,z2},{w9}} is non empty V41() V45() set
rng (Y,x) is non empty Element of bool the carrier of (Y,x)
bool the carrier of (Y,x) is non empty set
dom ((Y,x) ~) is Element of bool the carrier of (Y,x)
dom f is non empty Element of bool the carrier of Y
bool the carrier of Y is non empty set
dom (Y,x) is non empty Element of bool the carrier of Y
rng ((Y,x) ~) is Element of bool the carrier of Y
w1 is Relation-like Function-like set
proj1 w1 is set
proj2 w1 is set
w2 is non empty Relation-like the carrier of (Y,x) -defined the carrier of y -valued Function-like total quasi_total Element of bool [: the carrier of (Y,x), the carrier of y:]
w2 * (Y,x) is non empty Relation-like the carrier of Y -defined the carrier of y -valued Function-like total quasi_total Element of bool [: the carrier of Y, the carrier of y:]
[: the carrier of Y, the carrier of y:] is non empty Relation-like set
bool [: the carrier of Y, the carrier of y:] is non empty set
dom w2 is non empty Element of bool the carrier of (Y,x)
v1 is set
v2 is set
w2 . v1 is set
w2 . v2 is set
[v1,(w2 . v1)] is V26() set
{v1,(w2 . v1)} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,(w2 . v1)},{v1}} is non empty V41() V45() set
w9 is set
[v1,w9] is V26() set
{v1,w9} is non empty V41() set
{{v1,w9},{v1}} is non empty V41() V45() set
[w9,(w2 . v1)] is V26() set
{w9,(w2 . v1)} is non empty V41() set
{w9} is non empty trivial V41() V48(1) set
{{w9,(w2 . v1)},{w9}} is non empty V41() V45() set
[v2,(w2 . v1)] is V26() set
{v2,(w2 . v1)} is non empty V41() set
{v2} is non empty trivial V41() V48(1) set
{{v2,(w2 . v1)},{v2}} is non empty V41() V45() set
z2 is set
[v2,z2] is V26() set
{v2,z2} is non empty V41() set
{{v2,z2},{v2}} is non empty V41() V45() set
[z2,(w2 . v1)] is V26() set
{z2,(w2 . v1)} is non empty V41() set
{z2} is non empty trivial V41() V48(1) set
{{z2,(w2 . v1)},{z2}} is non empty V41() V45() set
[w9,v1] is V26() set
{w9,v1} is non empty V41() set
{{w9,v1},{w9}} is non empty V41() V45() set
[z2,v2] is V26() set
{z2,v2} is non empty V41() set
{{z2,v2},{z2}} is non empty V41() V45() set
z1 is Element of the carrier of Y
z2 is Element of the carrier of Y
f . z1 is Element of the carrier of X
f . z2 is Element of the carrier of X
[z1,z2] is V26() set
{z1,z2} is non empty V41() set
{z1} is non empty trivial V41() V48(1) set
{{z1,z2},{z1}} is non empty V41() V45() set
Class (x,z1) is Element of bool the carrier of Y
(Y,x) . z1 is Element of the carrier of (Y,x)
(Y,x) . z2 is Element of the carrier of (Y,x)
Class (x,z2) is Element of bool the carrier of Y
v1 is set
(Y,x) . v1 is set
v1 is set
f . v1 is set
(Y,x) . v1 is set
w2 . ((Y,x) . v1) is set
[((Y,x) . v1),(w2 . ((Y,x) . v1))] is V26() set
{((Y,x) . v1),(w2 . ((Y,x) . v1))} is non empty V41() set
{((Y,x) . v1)} is non empty trivial V41() V48(1) set
{{((Y,x) . v1),(w2 . ((Y,x) . v1))},{((Y,x) . v1)}} is non empty V41() V45() set
c11 is set
[((Y,x) . v1),c11] is V26() set
{((Y,x) . v1),c11} is non empty V41() set
{{((Y,x) . v1),c11},{((Y,x) . v1)}} is non empty V41() V45() set
[c11,(w2 . ((Y,x) . v1))] is V26() set
{c11,(w2 . ((Y,x) . v1))} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,(w2 . ((Y,x) . v1))},{c11}} is non empty V41() V45() set
[c11,((Y,x) . v1)] is V26() set
{c11,((Y,x) . v1)} is non empty V41() set
{{c11,((Y,x) . v1)},{c11}} is non empty V41() V45() set
(Y,x) . c11 is set
f . c11 is set
w9 is Element of the carrier of Y
(Y,x) . w9 is Element of the carrier of (Y,x)
Class (x,w9) is Element of bool the carrier of Y
z2 is Element of the carrier of Y
(Y,x) . z2 is Element of the carrier of (Y,x)
Class (x,z2) is Element of bool the carrier of Y
[v1,c11] is V26() set
{v1,c11} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,c11},{v1}} is non empty V41() V45() set
v1 is Element of the carrier of (Y,x)
v2 is Element of the carrier of (Y,x)
v1 * v2 is Element of the carrier of (Y,x)
the multF of (Y,x) is non empty Relation-like [: the carrier of (Y,x), the carrier of (Y,x):] -defined the carrier of (Y,x) -valued Function-like total quasi_total Element of bool [:[: the carrier of (Y,x), the carrier of (Y,x):], the carrier of (Y,x):]
[: the carrier of (Y,x), the carrier of (Y,x):] is non empty Relation-like set
[:[: the carrier of (Y,x), the carrier of (Y,x):], the carrier of (Y,x):] is non empty Relation-like set
bool [:[: the carrier of (Y,x), the carrier of (Y,x):], the carrier of (Y,x):] is non empty set
the multF of (Y,x) . (v1,v2) is Element of the carrier of (Y,x)
w2 . (v1 * v2) is Element of the carrier of y
w2 . v1 is Element of the carrier of y
w2 . v2 is Element of the carrier of y
(w2 . v1) * (w2 . v2) is Element of the carrier of y
the multF of y is non empty Relation-like [: the carrier of y, the carrier of y:] -defined the carrier of y -valued Function-like total quasi_total Element of bool [:[: the carrier of y, the carrier of y:], the carrier of y:]
[: the carrier of y, the carrier of y:] is non empty Relation-like set
[:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty Relation-like set
bool [:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty set
the multF of y . ((w2 . v1),(w2 . v2)) is Element of the carrier of y
[(v1 * v2),(w2 . (v1 * v2))] is V26() set
{(v1 * v2),(w2 . (v1 * v2))} is non empty V41() set
{(v1 * v2)} is non empty trivial V41() V48(1) set
{{(v1 * v2),(w2 . (v1 * v2))},{(v1 * v2)}} is non empty V41() V45() set
c11 is set
[(v1 * v2),c11] is V26() set
{(v1 * v2),c11} is non empty V41() set
{{(v1 * v2),c11},{(v1 * v2)}} is non empty V41() V45() set
[c11,(w2 . (v1 * v2))] is V26() set
{c11,(w2 . (v1 * v2))} is non empty V41() set
{c11} is non empty trivial V41() V48(1) set
{{c11,(w2 . (v1 * v2))},{c11}} is non empty V41() V45() set
[c11,(v1 * v2)] is V26() set
{c11,(v1 * v2)} is non empty V41() set
{{c11,(v1 * v2)},{c11}} is non empty V41() V45() set
(Y,x) . c11 is set
f . c11 is set
[v1,(w2 . v1)] is V26() set
{v1,(w2 . v1)} is non empty V41() set
{v1} is non empty trivial V41() V48(1) set
{{v1,(w2 . v1)},{v1}} is non empty V41() V45() set
w9 is set
[v1,w9] is V26() set
{v1,w9} is non empty V41() set
{{v1,w9},{v1}} is non empty V41() V45() set
[w9,(w2 . v1)] is V26() set
{w9,(w2 . v1)} is non empty V41() set
{w9} is non empty trivial V41() V48(1) set
{{w9,(w2 . v1)},{w9}} is non empty V41() V45() set
[w9,v1] is V26() set
{w9,v1} is non empty V41() set
{{w9,v1},{w9}} is non empty V41() V45() set
(Y,x) . w9 is set
f . w9 is set
[v2,(w2 . v2)] is V26() set
{v2,(w2 . v2)} is non empty V41() set
{v2} is non empty trivial V41() V48(1) set
{{v2,(w2 . v2)},{v2}} is non empty V41() V45() set
z2 is set
[v2,z2] is V26() set
{v2,z2} is non empty V41() set
{{v2,z2},{v2}} is non empty V41() V45() set
[z2,(w2 . v2)] is V26() set
{z2,(w2 . v2)} is non empty V41() set
{z2} is non empty trivial V41() V48(1) set
{{z2,(w2 . v2)},{z2}} is non empty V41() V45() set
[z2,v2] is V26() set
{z2,v2} is non empty V41() set
{{z2,v2},{z2}} is non empty V41() V45() set
(Y,x) . z2 is set
f . z2 is set
z is Element of the carrier of Y
(Y,x) . z is Element of the carrier of (Y,x)
z1 is Element of the carrier of Y
z2 is Element of the carrier of Y
z1 * z2 is Element of the carrier of Y
the multF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like total quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the multF of Y . (z1,z2) is Element of the carrier of Y
(Y,x) . (z1 * z2) is Element of the carrier of (Y,x)
Class (x,(z1 * z2)) is Element of bool the carrier of Y
Class (x,z) is Element of bool the carrier of Y
[z,(z1 * z2)] is V26() set
{z,(z1 * z2)} is non empty V41() set
{z} is non empty trivial V41() V48(1) set
{{z,(z1 * z2)},{z}} is non empty V41() V45() set
f . z is Element of the carrier of X
f . (z1 * z2) is Element of the carrier of X
f . z1 is Element of the carrier of X
f . z2 is Element of the carrier of X
(f . z1) * (f . z2) 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 total 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 Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X . ((f . z1),(f . z2)) is Element of the carrier of X
[(w2 . v1),(w2 . v2)] is V26() set
{(w2 . v1),(w2 . v2)} is non empty V41() set
{(w2 . v1)} is non empty trivial V41() V48(1) set
{{(w2 . v1),(w2 . v2)},{(w2 . v1)}} is non empty V41() V45() set
the multF of X . [(w2 . v1),(w2 . v2)] is set
the multF of X | [: the carrier of y, the carrier of y:] is Relation-like [: the carrier of y, the carrier of y:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
( the multF of X | [: the carrier of y, the carrier of y:]) . [(w2 . v1),(w2 . v2)] is set
( the multF of X | [: the carrier of y, the carrier of y:]) . ((w2 . v1),(w2 . v2)) is set
the multF of X || the carrier of y is Relation-like Function-like set
( the multF of X || the carrier of y) . ((w2 . v1),(w2 . v2)) is set
X is non empty multMagma
the carrier of X is non empty set
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
Y is non empty multMagma
the carrier of Y is non empty set
f is Relation-like the carrier of X -defined the carrier of X -valued total quasi_total V260() V265() (X) Element of bool [: the carrier of X, the carrier of X:]
(X,f) is non empty strict multMagma
Class f is non empty with_non-empty_elements a_partition of the carrier of X
(X,f) is non empty Relation-like [:(Class f),(Class f):] -defined Class f -valued Function-like total quasi_total Element of bool [:[:(Class f),(Class f):],(Class f):]
[:(Class f),(Class f):] is non empty Relation-like set
[:[:(Class f),(Class f):],(Class f):] is non empty Relation-like set
bool [:[:(Class f),(Class f):],(Class f):] is non empty set
multMagma(# (Class f),(X,f) #) is strict multMagma
the carrier of (X,f) is non empty set
[: the carrier of (X,f), the carrier of Y:] is non empty Relation-like set
bool [: the carrier of (X,f), the carrier of Y:] is non empty set
(X,f) is non empty Relation-like the carrier of X -defined the carrier of (X,f) -valued Function-like total quasi_total onto multiplicative Element of bool [: the carrier of X, the carrier of (X,f):]
[: the carrier of X, the carrier of (X,f):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (X,f):] is non empty set
y is non empty Relation-like the carrier of (X,f) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of (X,f), the carrier of Y:]
y * (X,f) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
bool [: the carrier of X, the carrier of Y:] is non empty set
x is non empty Relation-like the carrier of (X,f) -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of (X,f), the carrier of Y:]
x * (X,f) is non empty Relation-like the carrier of X -defined the carrier of Y -valued Function-like total quasi_total Element of bool [: the carrier of X, the carrier of Y:]
rng (X,f) is non empty Element of bool the carrier of (X,f)
bool the carrier of (X,f) is non empty set
dom y is non empty Element of bool the carrier of (X,f)
dom x is non empty Element of bool the carrier of (X,f)
X is non empty multMagma
the carrier of X is non empty set
( the carrier of X) is non empty strict multMagma
( the carrier of X) is non empty set
the carrier of X \/ NAT is non empty set
the_universe_of ( the carrier of X \/ NAT) is set
bool (the_universe_of ( the carrier of X \/ NAT)) is non empty Element of bool (bool (the_universe_of ( the carrier of X \/ NAT)))
bool (the_universe_of ( the carrier of X \/ NAT)) is non empty set
bool (bool (the_universe_of ( the carrier of X \/ NAT))) is non empty set
( the carrier of X) is non empty Relation-like NAT -defined bool (the_universe_of ( the carrier of X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of ( the carrier of X \/ NAT))):]
[:NAT,(bool (the_universe_of ( the carrier of X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of ( the carrier of X \/ NAT))):] is non empty non trivial V41() set
( the carrier of X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of ( the carrier of X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of ( the carrier of X \/ NAT))):]
disjoin (( the carrier of X) | NATPLUS) is Relation-like Function-like set
Union (disjoin (( the carrier of X) | NATPLUS)) is set
( the carrier of X) is non empty Relation-like [:( the carrier of X),( the carrier of X):] -defined ( the carrier of X) -valued Function-like total 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 Relation-like set
[:[:( the carrier of X),( the carrier of X):],( the carrier of X):] is non empty Relation-like set
bool [:[:( the carrier of X),( the carrier of X):],( the carrier of X):] is non empty set
multMagma(# ( the carrier of X),( the carrier of X) #) is strict multMagma
the carrier of ( the carrier of X) is non empty set
[: the carrier of ( the carrier of X), the carrier of X:] is non empty Relation-like set
bool [: the carrier of ( the carrier of X), the carrier of X:] is non empty set
( the carrier of X,1) is non empty Relation-like ( the carrier of X,1) -defined the carrier of ( the carrier of X) -valued Function-like one-to-one total quasi_total Element of bool [:( the carrier of X,1), the carrier of ( the carrier of X):]
( the carrier of X,1) is non empty set
( the carrier of X) . 1 is set
[:( the carrier of X,1), the carrier of ( the carrier of X):] is non empty Relation-like set
bool [:( the carrier of X,1), the carrier of ( the carrier of X):] is non empty set
( the carrier of X,1) " is Relation-like Function-like one-to-one set
id the carrier of X is non empty Relation-like the carrier of X -defined the carrier of X -valued Function-like one-to-one total quasi_total Element of bool [: the carrier of X, the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
bool [: the carrier of X, the carrier of X:] is non empty set
(( the carrier of X,1) ") * (id the carrier of X) is Relation-like the carrier of X -valued Function-like one-to-one set
f is non empty Relation-like the carrier of ( the carrier of X) -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of ( the carrier of X), the carrier of X:]
[: the carrier of ( the carrier of X), the carrier of ( the carrier of X):] is non empty Relation-like set
( the carrier of ( the carrier of X), the carrier of X,f) is Relation-like the carrier of ( the carrier of X) -defined the carrier of ( the carrier of X) -valued total quasi_total V260() V265() Element of bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):]
bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):] is non empty set
y is Relation-like [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):] -valued Function-like set
(( the carrier of X),y) is Relation-like the carrier of ( the carrier of X) -defined the carrier of ( the carrier of X) -valued total quasi_total V260() V265() (( the carrier of X)) Element of bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):]
proj1 y is set
{ b1 where b1 is Relation-like the carrier of ( the carrier of X) -defined the carrier of ( the carrier of X) -valued total quasi_total V260() V265() (( the carrier of X)) Element of bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):] : for b2 being set
for b3, b4 being Element of the carrier of ( the carrier of X) holds
( not b2 in proj1 y or not y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } is set
meet { b1 where b1 is Relation-like the carrier of ( the carrier of X) -defined the carrier of ( the carrier of X) -valued total quasi_total V260() V265() (( the carrier of X)) Element of bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):] : for b2 being set
for b3, b4 being Element of the carrier of ( the carrier of X) holds
( not b2 in proj1 y or not y . b2 = [b3,b4] or b3 in Class (b1,b4) ) } 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 total 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:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the multF of X | [: the carrier of X, the carrier of X:] is Relation-like [: the carrier of X, the carrier of X:] -defined [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
the multF of X || the carrier of X is Relation-like Function-like set
dom f is non empty Element of bool the carrier of ( the carrier of X)
bool the carrier of ( the carrier of X) is non empty set
w1 is set
[w1,1] is V26() set
{w1,1} is non empty V41() set
{w1} is non empty trivial V41() V48(1) set
{{w1,1},{w1}} is non empty V41() V45() set
f . [w1,1] is set
[:( the carrier of X,1),{1}:] is non empty Relation-like set
[: the carrier of X,{1}:] is non empty Relation-like set
proj1 ((( the carrier of X,1) ") * (id the carrier of X)) is set
( the carrier of X,1) . w1 is set
dom ( the carrier of X,1) is non empty Element of bool ( the carrier of X,1)
bool ( the carrier of X,1) is non empty set
rng ( the carrier of X,1) is non empty Element of bool the carrier of ( the carrier of X)
proj1 (( the carrier of X,1) ") is set
dom (id the carrier of X) is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
proj2 (( the carrier of X,1) ") is set
(id the carrier of X) . w1 is set
(( the carrier of X,1) ") . [w1,1] is set
(id the carrier of X) . ((( the carrier of X,1) ") . [w1,1]) is set
((( the carrier of X,1) ") * (id the carrier of X)) . [w1,1] is set
rng f is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
c6 is non empty (X)
the carrier of c6 is non empty set
x is Relation-like the carrier of ( the carrier of X) -defined the carrier of ( the carrier of X) -valued total quasi_total V260() V265() (( the carrier of X)) Element of bool [: the carrier of ( the carrier of X), the carrier of ( the carrier of X):]
(( the carrier of X),x) is non empty strict multMagma
Class x is non empty with_non-empty_elements a_partition of the carrier of ( the carrier of X)
(( the carrier of X),x) is non empty Relation-like [:(Class x),(Class x):] -defined Class x -valued Function-like total quasi_total Element of bool [:[:(Class x),(Class x):],(Class x):]
[:(Class x),(Class x):] is non empty Relation-like set
[:[:(Class x),(Class x):],(Class x):] is non empty Relation-like set
bool [:[:(Class x),(Class x):],(Class x):] is non empty set
multMagma(# (Class x),(( the carrier of X),x) #) is strict multMagma
the carrier of (( the carrier of X),x) is non empty set
[: the carrier of (( the carrier of X),x), the carrier of c6:] is non empty Relation-like set
bool [: the carrier of (( the carrier of X),x), the carrier of c6:] is non empty set
(( the carrier of X),x) is non empty Relation-like the carrier of ( the carrier of X) -defined the carrier of (( the carrier of X),x) -valued Function-like total quasi_total onto multiplicative Element of bool [: the carrier of ( the carrier of X), the carrier of (( the carrier of X),x):]
[: the carrier of ( the carrier of X), the carrier of (( the carrier of X),x):] is non empty Relation-like set
bool [: the carrier of ( the carrier of X), the carrier of (( the carrier of X),x):] is non empty set
w1 is non empty Relation-like the carrier of (( the carrier of X),x) -defined the carrier of c6 -valued Function-like total quasi_total Element of bool [: the carrier of (( the carrier of X),x), the carrier of c6:]
w1 * (( the carrier of X),x) is non empty Relation-like the carrier of ( the carrier of X) -defined the carrier of c6 -valued Function-like total quasi_total Element of bool [: the carrier of ( the carrier of X), the carrier of c6:]
[: the carrier of ( the carrier of X), the carrier of c6:] is non empty Relation-like set
bool [: the carrier of ( the carrier of X), the carrier of c6:] is non empty set
(( the carrier of X),(( the carrier of X),y)) is non empty strict multMagma
Class (( the carrier of X),y) is non empty with_non-empty_elements a_partition of the carrier of ( the carrier of X)
(( the carrier of X),(( the carrier of X),y)) is non empty Relation-like [:(Class (( the carrier of X),y)),(Class (( the carrier of X),y)):] -defined Class (( the carrier of X),y) -valued Function-like total quasi_total Element of bool [:[:(Class (( the carrier of X),y)),(Class (( the carrier of X),y)):],(Class (( the carrier of X),y)):]
[:(Class (( the carrier of X),y)),(Class (( the carrier of X),y)):] is non empty Relation-like set
[:[:(Class (( the carrier of X),y)),(Class (( the carrier of X),y)):],(Class (( the carrier of X),y)):] is non empty Relation-like set
bool [:[:(Class (( the carrier of X),y)),(Class (( the carrier of X),y)):],(Class (( the carrier of X),y)):] is non empty set
multMagma(# (Class (( the carrier of X),y)),(( the carrier of X),(( the carrier of X),y)) #) is strict multMagma
the carrier of (( the carrier of X),(( the carrier of X),y)) is non empty set
[: the carrier of (( the carrier of X),(( the carrier of X),y)), the carrier of X:] is non empty Relation-like set
bool [: the carrier of (( the carrier of X),(( the carrier of X),y)), the carrier of X:] is non empty set
w2 is non empty Relation-like the carrier of (( the carrier of X),(( the carrier of X),y)) -defined the carrier of X -valued Function-like total quasi_total Element of bool [: the carrier of (( the carrier of X),(( the carrier of X),y)), the carrier of X:]
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X,1) is non empty set
(X) . 1 is set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) " is Relation-like Function-like one-to-one set
(Y,1) is non empty set
(Y) . 1 is set
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
(Y,1) * f is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
[:X, the carrier of (Y):] is non empty Relation-like set
bool [:X, the carrier of (Y):] is non empty set
((X,1) ") * ((Y,1) * f) is Relation-like the carrier of (Y) -valued Function-like set
y is non empty Relation-like X -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:X, the carrier of (Y):]
((X,1) ") * y is Relation-like the carrier of (Y) -valued Function-like set
x is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (Y):]
y is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (Y):]
x is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (Y):]
c6 is non empty Relation-like X -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:X, the carrier of (Y):]
((X,1) ") * c6 is Relation-like the carrier of (Y) -valued Function-like set
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(X,Y,f) is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (Y):]
the carrier of (X) is non empty set
the carrier of (Y) is non empty set
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
f is non empty set
[:Y,f:] is non empty Relation-like set
bool [:Y,f:] is non empty set
(f) is non empty strict multMagma
(f) is non empty set
f \/ NAT is non empty set
the_universe_of (f \/ NAT) is set
bool (the_universe_of (f \/ NAT)) is non empty Element of bool (bool (the_universe_of (f \/ NAT)))
bool (the_universe_of (f \/ NAT)) is non empty set
bool (bool (the_universe_of (f \/ NAT))) is non empty set
(f) is non empty Relation-like NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
[:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial V41() set
(f) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
disjoin ((f) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((f) | NATPLUS)) is set
(f) is non empty Relation-like [:(f),(f):] -defined (f) -valued Function-like total quasi_total Element of bool [:[:(f),(f):],(f):]
[:(f),(f):] is non empty Relation-like set
[:[:(f),(f):],(f):] is non empty Relation-like set
bool [:[:(f),(f):],(f):] is non empty set
multMagma(# (f),(f) #) is strict multMagma
the carrier of (f) is non empty set
y is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(X,Y,y) is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (Y):]
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
x is non empty Relation-like Y -defined f -valued Function-like total quasi_total Element of bool [:Y,f:]
x * y is non empty Relation-like X -defined f -valued Function-like total quasi_total Element of bool [:X,f:]
[:X,f:] is non empty Relation-like set
bool [:X,f:] is non empty set
(X,f,(x * y)) is non empty Relation-like the carrier of (X) -defined the carrier of (f) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (f):]
[: the carrier of (X), the carrier of (f):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (f):] is non empty set
(Y,f,x) is non empty Relation-like the carrier of (Y) -defined the carrier of (f) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (Y), the carrier of (f):]
[: the carrier of (Y), the carrier of (f):] is non empty Relation-like set
bool [: the carrier of (Y), the carrier of (f):] is non empty set
(Y,f,x) * (X,Y,y) is non empty Relation-like the carrier of (X) -defined the carrier of (f) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (f):]
[:X, the carrier of (f):] is non empty Relation-like set
bool [:X, the carrier of (f):] is non empty set
(f,1) is non empty set
(f) . 1 is set
(f,1) is non empty Relation-like (f,1) -defined the carrier of (f) -valued Function-like one-to-one total quasi_total Element of bool [:(f,1), the carrier of (f):]
[:(f,1), the carrier of (f):] is non empty Relation-like set
bool [:(f,1), the carrier of (f):] is non empty set
(f,1) * (x * y) is Relation-like X -defined the carrier of (f) -valued Function-like Element of bool [:X, the carrier of (f):]
w2 is Element of the carrier of (X)
v1 is Element of the carrier of (X)
w2 * v1 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (w2,v1) is Element of the carrier of (X)
((Y,f,x) * (X,Y,y)) . (w2 * v1) is Element of the carrier of (f)
((Y,f,x) * (X,Y,y)) . w2 is Element of the carrier of (f)
((Y,f,x) * (X,Y,y)) . v1 is Element of the carrier of (f)
(((Y,f,x) * (X,Y,y)) . w2) * (((Y,f,x) * (X,Y,y)) . v1) is Element of the carrier of (f)
the multF of (f) is non empty Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like total quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is non empty Relation-like set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is non empty Relation-like set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is non empty set
the multF of (f) . ((((Y,f,x) * (X,Y,y)) . w2),(((Y,f,x) * (X,Y,y)) . v1)) is Element of the carrier of (f)
dom ((Y,f,x) * (X,Y,y)) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
dom (X,Y,y) is non empty Element of bool the carrier of (X)
(X,Y,y) . (w2 * v1) is Element of the carrier of (Y)
(Y,f,x) . ((X,Y,y) . (w2 * v1)) is Element of the carrier of (f)
(X,Y,y) . w2 is Element of the carrier of (Y)
(X,Y,y) . v1 is Element of the carrier of (Y)
((X,Y,y) . w2) * ((X,Y,y) . v1) is Element of the carrier of (Y)
the multF of (Y) is non empty Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[: the carrier of (Y), the carrier of (Y):] is non empty Relation-like set
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
the multF of (Y) . (((X,Y,y) . w2),((X,Y,y) . v1)) is Element of the carrier of (Y)
(Y,f,x) . (((X,Y,y) . w2) * ((X,Y,y) . v1)) is Element of the carrier of (f)
(Y,f,x) . ((X,Y,y) . w2) is Element of the carrier of (f)
(Y,f,x) . ((X,Y,y) . v1) is Element of the carrier of (f)
((Y,f,x) . ((X,Y,y) . w2)) * ((Y,f,x) . ((X,Y,y) . v1)) is Element of the carrier of (f)
the multF of (f) . (((Y,f,x) . ((X,Y,y) . w2)),((Y,f,x) . ((X,Y,y) . v1))) is Element of the carrier of (f)
(((Y,f,x) * (X,Y,y)) . w2) * ((Y,f,x) . ((X,Y,y) . v1)) is Element of the carrier of (f)
the multF of (f) . ((((Y,f,x) * (X,Y,y)) . w2),((Y,f,x) . ((X,Y,y) . v1))) is Element of the carrier of (f)
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X,1) is non empty set
(X) . 1 is set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) " is Relation-like Function-like one-to-one set
w1 is non empty Relation-like X -defined the carrier of (f) -valued Function-like total quasi_total Element of bool [:X, the carrier of (f):]
((X,1) ") * w1 is Relation-like the carrier of (f) -valued Function-like set
proj1 (((X,1) ") * w1) is set
proj1 ((X,1) ") is set
rng (X,1) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
dom ((Y,f,x) * (X,Y,y)) is non empty Element of bool the carrier of (X)
w2 is set
((Y,f,x) * (X,Y,y)) . w2 is set
(((X,1) ") * w1) . w2 is set
(Y,1) is non empty set
(Y) . 1 is set
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
(Y,1) * y is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
[:X, the carrier of (Y):] is non empty Relation-like set
bool [:X, the carrier of (Y):] is non empty set
((X,1) ") * ((Y,1) * y) is Relation-like the carrier of (Y) -valued Function-like set
proj1 (((X,1) ") * ((Y,1) * y)) is set
dom (X,Y,y) is non empty Element of bool the carrier of (X)
dom y is non empty Element of bool X
bool X is non empty set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
proj2 ((X,1) ") is set
((X,1) ") * y is Relation-like Y -valued Function-like set
proj1 (((X,1) ") * y) is set
rng (((X,1) ") * y) is Element of bool Y
bool Y is non empty set
dom (Y,1) is non empty Element of bool (Y,1)
bool (Y,1) is non empty set
(((X,1) ") * y) * (Y,1) is Relation-like the carrier of (Y) -valued Function-like set
proj1 ((((X,1) ") * y) * (Y,1)) is set
(((X,1) ") * y) . w2 is set
(Y,1) " is Relation-like Function-like one-to-one set
(f,1) * x is Relation-like Y -defined the carrier of (f) -valued Function-like Element of bool [:Y, the carrier of (f):]
[:Y, the carrier of (f):] is non empty Relation-like set
bool [:Y, the carrier of (f):] is non empty set
((Y,1) ") * ((f,1) * x) is Relation-like the carrier of (f) -valued Function-like set
proj1 (((Y,1) ") * ((f,1) * x)) is set
dom (Y,f,x) is non empty Element of bool the carrier of (Y)
bool the carrier of (Y) is non empty set
(Y,1) . ((((X,1) ") * y) . w2) is set
rng (Y,1) is non empty Element of bool the carrier of (Y)
dom x is non empty Element of bool Y
rng x is non empty Element of bool f
bool f is non empty set
dom (f,1) is non empty Element of bool (f,1)
bool (f,1) is non empty set
dom ((f,1) * x) is Element of bool Y
proj2 ((Y,1) ") is set
proj1 ((Y,1) ") is set
(Y,1) * ((Y,1) ") is Relation-like (Y,1) -defined Function-like one-to-one set
dom ((Y,1) * ((Y,1) ")) is Element of bool (Y,1)
(((X,1) ") * y) * ((f,1) * x) is Relation-like the carrier of (f) -valued Function-like set
(((X,1) ") * y) * x is Relation-like f -valued Function-like set
((((X,1) ") * y) * x) * (f,1) is Relation-like the carrier of (f) -valued Function-like set
((X,1) ") * (x * y) is Relation-like f -valued Function-like set
(((X,1) ") * (x * y)) * (f,1) is Relation-like the carrier of (f) -valued Function-like set
((X,1) ") * ((f,1) * (x * y)) is Relation-like the carrier of (f) -valued Function-like set
(X,Y,y) . w2 is set
(Y,f,x) . ((X,Y,y) . w2) is set
(((X,1) ") * ((Y,1) * y)) . w2 is set
(Y,f,x) . ((((X,1) ") * ((Y,1) * y)) . w2) is set
((((X,1) ") * y) * (Y,1)) . w2 is set
(Y,f,x) . (((((X,1) ") * y) * (Y,1)) . w2) is set
(Y,f,x) . ((Y,1) . ((((X,1) ") * y) . w2)) is set
(((Y,1) ") * ((f,1) * x)) . ((Y,1) . ((((X,1) ") * y) . w2)) is set
(Y,1) * (((Y,1) ") * ((f,1) * x)) is Relation-like (Y,1) -defined the carrier of (f) -valued Function-like set
((Y,1) * (((Y,1) ") * ((f,1) * x))) . ((((X,1) ") * y) . w2) is set
((Y,1) * ((Y,1) ")) * ((f,1) * x) is Relation-like (Y,1) -defined the carrier of (f) -valued Function-like set
(((Y,1) * ((Y,1) ")) * ((f,1) * x)) . ((((X,1) ") * y) . w2) is set
((Y,1) * ((Y,1) ")) . ((((X,1) ") * y) . w2) is set
((f,1) * x) . (((Y,1) * ((Y,1) ")) . ((((X,1) ") * y) . w2)) is set
((f,1) * x) . ((((X,1) ") * y) . w2) is set
X is non empty set
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
f is non empty set
[:X,f:] is non empty Relation-like set
bool [:X,f:] is non empty set
(f) is non empty strict multMagma
(f) is non empty set
f \/ NAT is non empty set
the_universe_of (f \/ NAT) is set
bool (the_universe_of (f \/ NAT)) is non empty Element of bool (bool (the_universe_of (f \/ NAT)))
bool (the_universe_of (f \/ NAT)) is non empty set
bool (bool (the_universe_of (f \/ NAT))) is non empty set
(f) is non empty Relation-like NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
[:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (f \/ NAT))):] is non empty non trivial V41() set
(f) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (f \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (f \/ NAT))):]
disjoin ((f) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((f) | NATPLUS)) is set
(f) is non empty Relation-like [:(f),(f):] -defined (f) -valued Function-like total quasi_total Element of bool [:[:(f),(f):],(f):]
[:(f),(f):] is non empty Relation-like set
[:[:(f),(f):],(f):] is non empty Relation-like set
bool [:[:(f),(f):],(f):] is non empty set
multMagma(# (f),(f) #) is strict multMagma
the carrier of (f) is non empty set
y is non empty Relation-like X -defined f -valued Function-like total quasi_total Element of bool [:X,f:]
(X,f,y) is non empty Relation-like the carrier of (X) -defined the carrier of (f) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (f):]
[: the carrier of (X), the carrier of (f):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (f):] is non empty set
x is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(X,Y,x) is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (Y):]
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
c6 is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (Y):]
w1 is Element of the carrier of (X)
w2 is Element of the carrier of (X)
w1 * w2 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (w1,w2) is Element of the carrier of (X)
c6 . (w1 * w2) is Element of the carrier of (Y)
c6 . w1 is Element of the carrier of (Y)
c6 . w2 is Element of the carrier of (Y)
(c6 . w1) * (c6 . w2) is Element of the carrier of (Y)
the multF of (Y) is non empty Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[: the carrier of (Y), the carrier of (Y):] is non empty Relation-like set
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
the multF of (Y) . ((c6 . w1),(c6 . w2)) is Element of the carrier of (Y)
(X,f,y) . w1 is Element of the carrier of (f)
(X,f,y) . w2 is Element of the carrier of (f)
(f,((X,f,y) . w1)) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
((X,f,y) . w1) `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
c11 is Element of the carrier of (Y)
(Y,c11) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(f,((X,f,y) . w2)) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
((X,f,y) . w2) `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
w9 is Element of the carrier of (Y)
(Y,w9) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
((X,f,y) . w1) * ((X,f,y) . w2) is Element of the carrier of (f)
the multF of (f) is non empty Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like total quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is non empty Relation-like set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is non empty Relation-like set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is non empty set
the multF of (f) . (((X,f,y) . w1),((X,f,y) . w2)) is Element of the carrier of (f)
c11 `1 is set
w9 `1 is set
[(c11 `1),(w9 `1)] is V26() set
{(c11 `1),(w9 `1)} is non empty V41() set
{(c11 `1)} is non empty trivial V41() V48(1) set
{{(c11 `1),(w9 `1)},{(c11 `1)}} is non empty V41() V45() set
c11 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[[(c11 `1),(w9 `1)],(c11 `2)] is V26() set
{[(c11 `1),(w9 `1)],(c11 `2)} is non empty V41() set
{[(c11 `1),(w9 `1)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[(c11 `1),(w9 `1)],(c11 `2)},{[(c11 `1),(w9 `1)]}} is non empty V41() V45() set
(Y,c11) + (Y,w9) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
[[[(c11 `1),(w9 `1)],(c11 `2)],((Y,c11) + (Y,w9))] is V26() set
{[[(c11 `1),(w9 `1)],(c11 `2)],((Y,c11) + (Y,w9))} is non empty V41() set
{[[(c11 `1),(w9 `1)],(c11 `2)]} is non empty trivial Relation-like Function-like constant V41() V48(1) set
{{[[(c11 `1),(w9 `1)],(c11 `2)],((Y,c11) + (Y,w9))},{[[(c11 `1),(w9 `1)],(c11 `2)]}} is non empty V41() V45() set
rng x is non empty Element of bool Y
bool Y is non empty set
(Y,1) is non empty set
(Y) . 1 is set
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
dom (Y,1) is non empty Element of bool (Y,1)
bool (Y,1) is non empty set
(Y,1) * x is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
[:X, the carrier of (Y):] is non empty Relation-like set
bool [:X, the carrier of (Y):] is non empty set
dom ((Y,1) * x) is Element of bool X
bool X is non empty set
dom x is non empty Element of bool X
(X,1) is non empty set
(X) . 1 is set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
(X,1) " is Relation-like Function-like one-to-one set
proj2 ((X,1) ") is set
((X,1) ") * ((Y,1) * x) is Relation-like the carrier of (Y) -valued Function-like set
proj1 (((X,1) ") * ((Y,1) * x)) is set
proj1 ((X,1) ") is set
rng (X,1) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
dom c6 is non empty Element of bool the carrier of (X)
w1 is set
c6 . w1 is set
(((X,1) ") * ((Y,1) * x)) . w1 is set
(f,1) is non empty set
(f) . 1 is set
(f,1) is non empty Relation-like (f,1) -defined the carrier of (f) -valued Function-like one-to-one total quasi_total Element of bool [:(f,1), the carrier of (f):]
[:(f,1), the carrier of (f):] is non empty Relation-like set
bool [:(f,1), the carrier of (f):] is non empty set
(f,1) * y is Relation-like X -defined the carrier of (f) -valued Function-like Element of bool [:X, the carrier of (f):]
[:X, the carrier of (f):] is non empty Relation-like set
bool [:X, the carrier of (f):] is non empty set
((X,1) ") * ((f,1) * y) is Relation-like the carrier of (f) -valued Function-like set
proj1 (((X,1) ") * ((f,1) * y)) is set
dom (X,f,y) is non empty Element of bool the carrier of (X)
rng y is non empty Element of bool f
bool f is non empty set
dom (f,1) is non empty Element of bool (f,1)
bool (f,1) is non empty set
dom ((f,1) * y) is Element of bool X
dom y is non empty Element of bool X
(Y,1) * y is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
dom ((Y,1) * y) is Element of bool X
w2 is set
((f,1) * y) . w2 is set
((Y,1) * y) . w2 is set
y . w2 is set
(f,1) . (y . w2) is set
[(y . w2),1] is V26() set
{(y . w2),1} is non empty V41() set
{(y . w2)} is non empty trivial V41() V48(1) set
{{(y . w2),1},{(y . w2)}} is non empty V41() V45() set
(Y,1) . (y . w2) is set
X is non empty set
Y is non empty set
[:Y,X:] is non empty Relation-like set
bool [:Y,X:] is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
id Y is non empty Relation-like Y -defined Y -valued Function-like one-to-one total quasi_total Element of bool [:Y,Y:]
[:Y,Y:] is non empty Relation-like set
bool [:Y,Y:] is non empty set
(Y,Y,(id Y)) is non empty Relation-like the carrier of (Y) -defined the carrier of (Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (Y), the carrier of (Y):]
[: the carrier of (Y), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (Y), the carrier of (Y):] is non empty set
f is non empty Relation-like Y -defined X -valued Function-like total quasi_total Element of bool [:Y,X:]
(Y,X,f) is non empty Relation-like the carrier of (Y) -defined the carrier of (X) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (Y), the carrier of (X):]
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
[: the carrier of (Y), the carrier of (X):] is non empty Relation-like set
bool [: the carrier of (Y), the carrier of (X):] is non empty set
dom (Y,X,f) is non empty Element of bool the carrier of (Y)
bool the carrier of (Y) is non empty set
id (dom (Y,X,f)) is non empty Relation-like dom (Y,X,f) -defined dom (Y,X,f) -valued Function-like one-to-one total quasi_total Element of bool [:(dom (Y,X,f)),(dom (Y,X,f)):]
[:(dom (Y,X,f)),(dom (Y,X,f)):] is non empty Relation-like set
bool [:(dom (Y,X,f)),(dom (Y,X,f)):] is non empty set
dom (Y,Y,(id Y)) is non empty Element of bool the carrier of (Y)
y is set
(Y,Y,(id Y)) . y is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is Element of the carrier of (Y)
(Y,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(Y,Y,(id Y)) . c6 is Element of the carrier of (Y)
c6 `1 is set
c6 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[(c6 `1),(c6 `2)] is V26() set
{(c6 `1),(c6 `2)} is non empty V41() set
{(c6 `1)} is non empty trivial V41() V48(1) set
{{(c6 `1),(c6 `2)},{(c6 `1)}} is non empty V41() V45() set
(Y,c6) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
{ (b1 `1) where b1 is Element of the carrier of (Y) : (Y,b1) = 1 } is set
dom (id Y) is non empty Element of bool Y
bool Y is non empty set
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
(Y,1) is non empty set
(Y) . 1 is set
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
(Y,1) " is Relation-like Function-like one-to-one set
(Y,1) * (id Y) is Relation-like Y -defined the carrier of (Y) -valued Function-like one-to-one Element of bool [:Y, the carrier of (Y):]
[:Y, the carrier of (Y):] is non empty Relation-like set
bool [:Y, the carrier of (Y):] is non empty set
((Y,1) ") * ((Y,1) * (id Y)) is Relation-like the carrier of (Y) -valued Function-like one-to-one set
proj1 (((Y,1) ") * ((Y,1) * (id Y))) is set
(Y,1) . (c6 `1) is set
[(c6 `1),1] is V26() set
{(c6 `1),1} is non empty V41() set
{{(c6 `1),1},{(c6 `1)}} is non empty V41() V45() set
dom (Y,1) is non empty Element of bool (Y,1)
bool (Y,1) is non empty set
rng (Y,1) is non empty Element of bool the carrier of (Y)
proj1 ((Y,1) ") is set
rng (id Y) is non empty Element of bool Y
dom ((Y,1) * (id Y)) is Element of bool Y
proj2 ((Y,1) ") is set
((Y,1) ") . c6 is set
(((Y,1) ") * ((Y,1) * (id Y))) . c6 is set
((Y,1) * (id Y)) . (c6 `1) is set
(id Y) . (c6 `1) is set
(Y,1) . ((id Y) . (c6 `1)) is set
w1 is Element of the carrier of (Y)
w2 is Element of the carrier of (Y)
w1 * w2 is Element of the carrier of (Y)
the multF of (Y) is non empty Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
the multF of (Y) . (w1,w2) is Element of the carrier of (Y)
(Y,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(Y,w2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(Y,Y,(id Y)) . w1 is Element of the carrier of (Y)
(Y,Y,(id Y)) . w2 is Element of the carrier of (Y)
((Y,Y,(id Y)) . w1) * ((Y,Y,(id Y)) . w2) is Element of the carrier of (Y)
the multF of (Y) . (((Y,Y,(id Y)) . w1),((Y,Y,(id Y)) . w2)) is Element of the carrier of (Y)
w1 * ((Y,Y,(id Y)) . w2) is Element of the carrier of (Y)
the multF of (Y) . (w1,((Y,Y,(id Y)) . w2)) is Element of the carrier of (Y)
x is Element of the carrier of (Y)
(Y,Y,(id Y)) . x is Element of the carrier of (Y)
(Y,x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is Element of the carrier of (Y)
(Y,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(Y,Y,(id Y)) . w1 is Element of the carrier of (Y)
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(X,Y,f) is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (Y):]
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
f " is Relation-like Function-like set
f * (f ") is Relation-like X -defined Function-like set
dom f is non empty Element of bool X
bool X is non empty set
id (dom f) is non empty Relation-like dom f -defined dom f -valued Function-like one-to-one total quasi_total Element of bool [:(dom f),(dom f):]
[:(dom f),(dom f):] is non empty Relation-like set
bool [:(dom f),(dom f):] is non empty set
rng f is non empty Element of bool Y
bool Y is non empty set
[:X,(rng f):] is non empty Relation-like set
bool [:X,(rng f):] is non empty set
[:(rng f),X:] is non empty Relation-like set
bool [:(rng f),X:] is non empty set
x is non empty Relation-like X -defined rng f -valued Function-like total quasi_total Element of bool [:X,(rng f):]
x " is Relation-like Function-like set
c6 is non empty Relation-like rng f -defined X -valued Function-like total quasi_total Element of bool [:(rng f),X:]
c6 * x is non empty Relation-like X -defined X -valued Function-like total quasi_total Element of bool [:X,X:]
[:X,X:] is non empty Relation-like set
bool [:X,X:] is non empty set
id X is non empty Relation-like X -defined X -valued Function-like one-to-one total quasi_total Element of bool [:X,X:]
((rng f)) is non empty strict multMagma
((rng f)) is non empty set
(rng f) \/ NAT is non empty set
the_universe_of ((rng f) \/ NAT) is set
bool (the_universe_of ((rng f) \/ NAT)) is non empty Element of bool (bool (the_universe_of ((rng f) \/ NAT)))
bool (the_universe_of ((rng f) \/ NAT)) is non empty set
bool (bool (the_universe_of ((rng f) \/ NAT))) is non empty set
((rng f)) is non empty Relation-like NAT -defined bool (the_universe_of ((rng f) \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of ((rng f) \/ NAT))):]
[:NAT,(bool (the_universe_of ((rng f) \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of ((rng f) \/ NAT))):] is non empty non trivial V41() set
((rng f)) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of ((rng f) \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of ((rng f) \/ NAT))):]
disjoin (((rng f)) | NATPLUS) is Relation-like Function-like set
Union (disjoin (((rng f)) | NATPLUS)) is set
((rng f)) is non empty Relation-like [:((rng f)),((rng f)):] -defined ((rng f)) -valued Function-like total quasi_total Element of bool [:[:((rng f)),((rng f)):],((rng f)):]
[:((rng f)),((rng f)):] is non empty Relation-like set
[:[:((rng f)),((rng f)):],((rng f)):] is non empty Relation-like set
bool [:[:((rng f)),((rng f)):],((rng f)):] is non empty set
multMagma(# ((rng f)),((rng f)) #) is strict multMagma
the carrier of ((rng f)) is non empty set
(X,(rng f),x) is non empty Relation-like the carrier of (X) -defined the carrier of ((rng f)) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of ((rng f)):]
[: the carrier of (X), the carrier of ((rng f)):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of ((rng f)):] is non empty set
((rng f),X,c6) is non empty Relation-like the carrier of ((rng f)) -defined the carrier of (X) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of ((rng f)), the carrier of (X):]
[: the carrier of ((rng f)), the carrier of (X):] is non empty Relation-like set
bool [: the carrier of ((rng f)), the carrier of (X):] is non empty set
((rng f),X,c6) * (X,(rng f),x) is non empty Relation-like the carrier of (X) -defined the carrier of (X) -valued Function-like total quasi_total Element of bool [: the carrier of (X), the carrier of (X):]
[: the carrier of (X), the carrier of (X):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (X):] is non empty set
(X,X,(id X)) is non empty Relation-like the carrier of (X) -defined the carrier of (X) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (X):]
((rng f),X,c6) * (X,Y,f) is Relation-like the carrier of (X) -defined the carrier of (X) -valued Function-like Element of bool [: the carrier of (X), the carrier of (X):]
dom (X,Y,f) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
id (dom (X,Y,f)) is non empty Relation-like dom (X,Y,f) -defined dom (X,Y,f) -valued Function-like one-to-one total quasi_total Element of bool [:(dom (X,Y,f)),(dom (X,Y,f)):]
[:(dom (X,Y,f)),(dom (X,Y,f)):] is non empty Relation-like set
bool [:(dom (X,Y,f)),(dom (X,Y,f)):] is non empty set
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
bool [:X,Y:] is non empty set
(Y) is non empty strict multMagma
(Y) is non empty set
Y \/ NAT is non empty set
the_universe_of (Y \/ NAT) is set
bool (the_universe_of (Y \/ NAT)) is non empty Element of bool (bool (the_universe_of (Y \/ NAT)))
bool (the_universe_of (Y \/ NAT)) is non empty set
bool (bool (the_universe_of (Y \/ NAT))) is non empty set
(Y) is non empty Relation-like NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
[:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (Y \/ NAT))):] is non empty non trivial V41() set
(Y) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (Y \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (Y \/ NAT))):]
disjoin ((Y) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((Y) | NATPLUS)) is set
(Y) is non empty Relation-like [:(Y),(Y):] -defined (Y) -valued Function-like total quasi_total Element of bool [:[:(Y),(Y):],(Y):]
[:(Y),(Y):] is non empty Relation-like set
[:[:(Y),(Y):],(Y):] is non empty Relation-like set
bool [:[:(Y),(Y):],(Y):] is non empty set
multMagma(# (Y),(Y) #) is strict multMagma
the carrier of (Y) is non empty set
f is non empty Relation-like X -defined Y -valued Function-like total quasi_total Element of bool [:X,Y:]
(X,Y,f) is non empty Relation-like the carrier of (X) -defined the carrier of (Y) -valued Function-like total quasi_total multiplicative Element of bool [: the carrier of (X), the carrier of (Y):]
(X) is non empty strict multMagma
(X) is non empty set
X \/ NAT is non empty set
the_universe_of (X \/ NAT) is set
bool (the_universe_of (X \/ NAT)) is non empty Element of bool (bool (the_universe_of (X \/ NAT)))
bool (the_universe_of (X \/ NAT)) is non empty set
bool (bool (the_universe_of (X \/ NAT))) is non empty set
(X) is non empty Relation-like NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like total quasi_total Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
[:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial Relation-like V41() set
bool [:NAT,(bool (the_universe_of (X \/ NAT))):] is non empty non trivial V41() set
(X) | NATPLUS is Relation-like NAT -defined NATPLUS -defined NAT -defined bool (the_universe_of (X \/ NAT)) -valued Function-like Element of bool [:NAT,(bool (the_universe_of (X \/ NAT))):]
disjoin ((X) | NATPLUS) is Relation-like Function-like set
Union (disjoin ((X) | NATPLUS)) is set
(X) is non empty Relation-like [:(X),(X):] -defined (X) -valued Function-like total quasi_total Element of bool [:[:(X),(X):],(X):]
[:(X),(X):] is non empty Relation-like set
[:[:(X),(X):],(X):] is non empty Relation-like set
bool [:[:(X),(X):],(X):] is non empty set
multMagma(# (X),(X) #) is strict multMagma
the carrier of (X) is non empty set
[: the carrier of (X), the carrier of (Y):] is non empty Relation-like set
bool [: the carrier of (X), the carrier of (Y):] is non empty set
rng (X,Y,f) is non empty Element of bool the carrier of (Y)
bool the carrier of (Y) is non empty set
y is set
x is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is Element of the carrier of (Y)
(Y,c6) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 `1 is set
c6 `2 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
[(c6 `1),(c6 `2)] is V26() set
{(c6 `1),(c6 `2)} is non empty V41() set
{(c6 `1)} is non empty trivial V41() V48(1) set
{{(c6 `1),(c6 `2)},{(c6 `1)}} is non empty V41() V45() set
(Y,c6) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural complex ext-real positive non negative V41() V46() V53() V54() set
{ (b1 `1) where b1 is Element of the carrier of (Y) : (Y,b1) = 1 } is set
(X,1) is non empty Relation-like (X,1) -defined the carrier of (X) -valued Function-like one-to-one total quasi_total Element of bool [:(X,1), the carrier of (X):]
(X,1) is non empty set
(X) . 1 is set
[:(X,1), the carrier of (X):] is non empty Relation-like set
bool [:(X,1), the carrier of (X):] is non empty set
(X,1) " is Relation-like Function-like one-to-one set
(Y,1) is non empty set
(Y) . 1 is set
(Y,1) is non empty Relation-like (Y,1) -defined the carrier of (Y) -valued Function-like one-to-one total quasi_total Element of bool [:(Y,1), the carrier of (Y):]
[:(Y,1), the carrier of (Y):] is non empty Relation-like set
bool [:(Y,1), the carrier of (Y):] is non empty set
(Y,1) * f is Relation-like X -defined the carrier of (Y) -valued Function-like Element of bool [:X, the carrier of (Y):]
[:X, the carrier of (Y):] is non empty Relation-like set
bool [:X, the carrier of (Y):] is non empty set
((X,1) ") * ((Y,1) * f) is Relation-like the carrier of (Y) -valued Function-like set
proj1 (((X,1) ") * ((Y,1) * f)) is set
dom (X,Y,f) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
(Y,1) . (c6 `1) is set
[(c6 `1),1] is V26() set
{(c6 `1),1} is non empty V41() set
{{(c6 `1),1},{(c6 `1)}} is non empty V41() V45() set
rng f is non empty Element of bool Y
bool Y is non empty set
dom f is non empty Element of bool X
bool X is non empty set
w2 is set
f . w2 is set
[w2,1] is V26() set
{w2,1} is non empty V41() set
{w2} is non empty trivial V41() V48(1) set
{{w2,1},{w2}} is non empty V41() V45() set
[:(X,1),{1}:] is non empty Relation-like set
v1 is Element of the carrier of (X)
(X,Y,f) . v1 is Element of the carrier of (Y)
dom (X,1) is non empty Element of bool (X,1)
bool (X,1) is non empty set
(X,1) . w2 is set
rng (X,1) is non empty Element of bool the carrier of (X)
proj1 ((X,1) ") is set
dom (Y,1) is non empty Element of bool (Y,1)
bool (Y,1) is non empty set
dom ((Y,1) * f) is Element of bool X
proj2 ((X,1) ") is set
(((X,1) ") * ((Y,1) * f)) . v1 is set
((X,1) ") . v1 is set
((Y,1) * f) . (((X,1) ") . v1) is set
w1 is Element of the carrier of (Y)
w2 is Element of the carrier of (Y)
w1 * w2 is Element of the carrier of (Y)
the multF of (Y) is non empty Relation-like [: the carrier of (Y), the carrier of (Y):] -defined the carrier of (Y) -valued Function-like total quasi_total Element of bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):]
[: the carrier of (Y), the carrier of (Y):] is non empty Relation-like set
[:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty Relation-like set
bool [:[: the carrier of (Y), the carrier of (Y):], the carrier of (Y):] is non empty set
the multF of (Y) . (w1,w2) is Element of the carrier of (Y)
(Y,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
(Y,w2) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
v1 is Element of the carrier of (X)
(X,Y,f) . v1 is Element of the carrier of (Y)
v2 is Element of the carrier of (X)
(X,Y,f) . v2 is Element of the carrier of (Y)
v1 * v2 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 total 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 Relation-like set
[:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty Relation-like set
bool [:[: the carrier of (X), the carrier of (X):], the carrier of (X):] is non empty set
the multF of (X) . (v1,v2) is Element of the carrier of (X)
(X,Y,f) . (v1 * v2) is Element of the carrier of (Y)
x is Element of the carrier of (Y)
(Y,x) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
c6 is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
w1 is Element of the carrier of (Y)
(Y,w1) is epsilon-transitive epsilon-connected ordinal natural complex ext-real non negative V41() V46() V53() V54() set
dom (X,Y,f) is non empty Element of bool the carrier of (X)
bool the carrier of (X) is non empty set
x is Element of the carrier of (X)
(X,Y,f) . x is Element of the carrier of (Y)
(X,Y,f) . x is set
x is set
(X,Y,f) . x is set