:: CARD_FIN semantic presentation

REAL is non empty non trivial non finite set
NAT is non empty non trivial V4() V5() V6() non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty non trivial non finite set
{} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
RAT is non empty non trivial non finite set
the empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
{{},1} is non empty finite V60() set
COMPLEX is non empty non trivial non finite set
NAT is non empty non trivial V4() V5() V6() non finite cardinal limit_cardinal set
bool NAT is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
INT is non empty non trivial non finite set
[:COMPLEX,COMPLEX:] is non empty non trivial Relation-like non finite set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial Relation-like non finite set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:REAL,REAL:] is non empty non trivial Relation-like non finite set
bool [:REAL,REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is non empty non trivial Relation-like non finite set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:RAT,RAT:] is non empty non trivial Relation-like non finite set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is non empty non trivial Relation-like non finite set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is non empty non trivial Relation-like non finite set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is non empty non trivial Relation-like non finite set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is non empty non trivial Relation-like non finite set
[:[:NAT,NAT:],NAT:] is non empty non trivial Relation-like non finite set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
[:COMPLEX,REAL:] is non empty non trivial Relation-like non finite set
bool [:COMPLEX,REAL:] is non empty non trivial non finite set
2 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
3 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
0 is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of NAT
union {} is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
dom {} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
rng {} is empty trivial V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V36() V37() V38() V39() V42() V43() V44() V45() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V201() V205() set
card {} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
{{}} is non empty trivial functional finite V60() 1 -element set
addint is non empty Relation-like [:INT,INT:] -defined INT -valued Function-like V25([:INT,INT:]) quasi_total V32() V33() V34() V128( INT ) commutative associative Element of bool [:[:INT,INT:],INT:]
addnat is non empty Relation-like [:NAT,NAT:] -defined NAT -valued Function-like V25([:NAT,NAT:]) quasi_total V32() V33() V34() V35() V52() V128( NAT ) commutative associative Element of bool [:[:NAT,NAT:],NAT:]
the_unity_wrt addint is V11() ext-real V53() integer Element of INT
0 ! is V11() ext-real V53() Element of REAL
Funcs ({},{}) is non empty functional finite set
id {} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional V25( {} ) V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
{(id {})} is non empty trivial functional finite V60() 1 -element set
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 \ X is finite Element of bool S1
bool S1 is non empty finite V60() set
card (S1 \ X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) - (card X) is V11() ext-real V53() integer Element of INT
X is finite set
S1 is finite set
Funcs (X,S1) is functional finite set
card (Funcs (X,S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is set
{S2} is non empty trivial finite 1 -element set
X \/ {S2} is non empty finite set
x is set
{x} is non empty trivial finite 1 -element set
S1 \/ {x} is non empty finite set
[:(X \/ {S2}),(S1 \/ {x}):] is non empty Relation-like finite set
bool [:(X \/ {S2}),(S1 \/ {x}):] is non empty finite V60() set
{ b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is set
card { b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is V4() V5() V6() cardinal set
{ b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( S₁[b₁,X \/ {S2},S1 \/ {x}] & rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is set
y is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
y . S2 is set
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : S₁[b₁,X,S1] } is set
F1 is set
F1 is set
F1 is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
F1 | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng (F1 | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
F1 . S2 is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : S₁[b₁,X,S1] } is V4() V5() V6() cardinal set
card { b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( S₁[b₁,X \/ {S2},S1 \/ {x}] & rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is V4() V5() V6() cardinal set
F1 is set
F1 is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
F1 | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng (F1 | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
F1 . S2 is set
F1 is set
F1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
X is finite set
S1 is finite set
Funcs (X,S1) is functional finite set
card (Funcs (X,S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is set
{S2} is non empty trivial finite 1 -element set
X \/ {S2} is non empty finite set
[:(X \/ {S2}),S1:] is Relation-like finite set
bool [:(X \/ {S2}),S1:] is non empty finite V60() set
x is set
{ b₁ where b₁ is Relation-like X \/ {S2} -defined S1 -valued Function-like quasi_total finite Element of bool [:(X \/ {S2}),S1:] : b₁ . S2 = x } is set
card { b₁ where b₁ is Relation-like X \/ {S2} -defined S1 -valued Function-like quasi_total finite Element of bool [:(X \/ {S2}),S1:] : b₁ . S2 = x } is V4() V5() V6() cardinal set
{x} is non empty trivial finite 1 -element set
S1 \/ {x} is non empty finite set
[:(X \/ {S2}),(S1 \/ {x}):] is non empty Relation-like finite set
bool [:(X \/ {S2}),(S1 \/ {x}):] is non empty finite V60() set
{ b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is set
c is set
F1 is Relation-like X \/ {S2} -defined S1 -valued Function-like quasi_total finite Element of bool [:(X \/ {S2}),S1:]
F1 . S2 is set
F1 | X is Relation-like X -defined X \/ {S2} -defined S1 -valued Function-like finite Element of bool [:(X \/ {S2}),S1:]
rng (F1 | X) is finite Element of bool S1
bool S1 is non empty finite V60() set
c is set
F1 is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
F1 | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng (F1 | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
F1 . S2 is set
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is finite set
Funcs (S1,X) is functional finite set
card (Funcs (S1,X)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) |^ (card S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is finite set
Funcs (x,h) is functional finite set
card (Funcs (x,h)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card h) |^ (card x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card h) |^ y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is Relation-like Function-like set
dom F1 is set
rng F1 is set
[:(card h),h:] is Relation-like finite set
bool [:(card h),h:] is non empty finite V60() set
XF is set
F1 is Relation-like card h -defined h -valued Function-like quasi_total finite Element of bool [:(card h),h:]
[:x,h:] is Relation-like finite set
bool [:x,h:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : S₂[b₁] } is set
card { b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : S₂[b₁] } is V4() V5() V6() cardinal set
n is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{XF} is non empty trivial finite 1 -element set
x \ {XF} is finite Element of bool x
bool x is non empty finite V60() set
u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n . u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . u is set
rng F1 is finite Element of bool h
bool h is non empty finite V60() set
{ b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : ( S₂[b₁] & b₁ . XF = F1 . u ) } is set
(x \ {XF}) \/ {XF} is non empty finite set
[:((x \ {XF}) \/ {XF}),h:] is Relation-like finite set
bool [:((x \ {XF}) \/ {XF}),h:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like (x \ {XF}) \/ {XF} -defined h -valued Function-like quasi_total finite Element of bool [:((x \ {XF}) \/ {XF}),h:] : b₁ . XF = F1 . u } is set
Fyy is set
y is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:]
y . XF is set
Fyy is set
y is Relation-like (x \ {XF}) \/ {XF} -defined h -valued Function-like quasi_total finite Element of bool [:((x \ {XF}) \/ {XF}),h:]
y . XF is set
card (x \ {XF}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
Funcs ((x \ {XF}),h) is functional finite set
card (Funcs ((x \ {XF}),h)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card { b₁ where b₁ is Relation-like (x \ {XF}) \/ {XF} -defined h -valued Function-like quasi_total finite Element of bool [:((x \ {XF}) \/ {XF}),h:] : b₁ . XF = F1 . u } is V4() V5() V6() cardinal set
n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n . n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
len n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(card h) |^ S2 is V11() ext-real V53() integer set
(len n) * ((card h) |^ S2) is V11() ext-real V53() integer Element of INT
u is set
u is set
n is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:]
u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n . u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card h) * ((card h) |^ S2) is V11() ext-real V53() integer Element of INT
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
Funcs (S2,x) is functional finite set
card (Funcs (S2,x)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card x) |^ (card S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is finite set
S1 is finite set
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is one-to-one } is V4() V5() V6() cardinal set
S2 is set
{S2} is non empty trivial finite 1 -element set
X \/ {S2} is non empty finite set
x is set
{x} is non empty trivial finite 1 -element set
S1 \/ {x} is non empty finite set
[:(X \/ {S2}),(S1 \/ {x}):] is non empty Relation-like finite set
bool [:(X \/ {S2}),(S1 \/ {x}):] is non empty finite V60() set
{ b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( b₁ is one-to-one & b₁ . S2 = x ) } is set
card { b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( b₁ is one-to-one & b₁ . S2 = x ) } is V4() V5() V6() cardinal set
h is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
h . S2 is set
h | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite set
rng (h | X) is finite set
h | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
(h | X) | X is Relation-like X -defined X -defined S1 \/ {x} -valued Function-like finite set
rng ((h | X) | X) is finite set
(X \/ {S2}) /\ X is finite set
dom h is non empty finite Element of bool (X \/ {S2})
bool (X \/ {S2}) is non empty finite V60() set
dom (h | X) is finite Element of bool X
bool X is non empty finite V60() set
(h | X) | X is Relation-like X -defined X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng ((h | X) | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
rng (h | X) is finite Element of bool (S1 \/ {x})
{ b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( S₁[b₁,X \/ {S2},S1 \/ {x}] & rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is set
y is set
c is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
c . S2 is set
c | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng (c | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
dom c is non empty finite Element of bool (X \/ {S2})
bool (X \/ {S2}) is non empty finite V60() set
F1 is set
dom (c | X) is finite Element of bool X
bool X is non empty finite V60() set
F1 is set
(c | X) . F1 is set
c . F1 is set
y is set
c is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
c | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite set
rng (c | X) is finite set
c | X is Relation-like X -defined X \/ {S2} -defined S1 \/ {x} -valued Function-like finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):]
rng (c | X) is finite Element of bool (S1 \/ {x})
bool (S1 \/ {x}) is non empty finite V60() set
c . S2 is set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : ( b₁ is one-to-one & rng (b₁ | X) c= S1 ) } is set
F1 is set
F1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
F1 | X is Relation-like X -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
rng (F1 | X) is finite Element of bool S1
bool S1 is non empty finite V60() set
rng F1 is finite Element of bool S1
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : S₁[b₁,X,S1] } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : S₁[b₁,X,S1] } is V4() V5() V6() cardinal set
card { b₁ where b₁ is non empty Relation-like X \/ {S2} -defined S1 \/ {x} -valued Function-like V25(X \/ {S2}) quasi_total finite Element of bool [:(X \/ {S2}),(S1 \/ {x}):] : ( S₁[b₁,X \/ {S2},S1 \/ {x}] & rng (b₁ | X) c= S1 & b₁ . S2 = x ) } is V4() V5() V6() cardinal set
F1 is set
F1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
F1 | X is Relation-like X -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
rng (F1 | X) is finite Element of bool S1
bool S1 is non empty finite V60() set
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
X ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
X -' S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(X -' S1) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(X !) / ((X -' S1) !) is V11() ext-real non negative V53() Element of REAL
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card S1) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) -' (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card S1) -' (card X)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card S1) !) / (((card S1) -' (card X)) !) is V11() ext-real non negative V53() Element of REAL
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is finite set
card h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:x,h:] is Relation-like finite set
bool [:x,h:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card h) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card h) -' (card x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card h) -' (card x)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card h) !) / (((card h) -' (card x)) !) is V11() ext-real non negative V53() Element of REAL
c is set
F1 is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:]
rng {} is empty trivial proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V36() V37() V38() V39() V42() V43() V44() V45() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V201() V205() Element of bool RAT
bool RAT is non empty non trivial non finite set
(card h) - (card x) is V11() ext-real V53() integer Element of INT
dom {} is empty proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool NAT
y is set
c is Relation-like Function-like set
dom c is set
rng c is set
[:(card h),h:] is Relation-like finite set
bool [:(card h),h:] is non empty finite V60() set
F1 is Relation-like card h -defined h -valued Function-like quasi_total finite Element of bool [:(card h),h:]
{ b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : S₂[b₁] } is set
card { b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : S₂[b₁] } is V4() V5() V6() cardinal set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
S2 ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 !) / (((card h) -' (card x)) !) is V11() ext-real non negative V53() Element of REAL
(card x) - 1 is V11() ext-real V53() integer Element of INT
{y} is non empty trivial finite 1 -element set
x \ {y} is finite Element of bool x
bool x is non empty finite V60() set
(x \ {y}) \/ {y} is non empty finite set
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(XF + 1) - 1 is V11() ext-real V53() integer Element of INT
(S2 + 1) - 1 is V11() ext-real V53() integer Element of INT
XF - XF is V11() ext-real V53() integer Element of INT
S2 - XF is V11() ext-real V53() integer Element of INT
n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 . n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . n1 is set
{(F1 . n1)} is non empty trivial finite 1 -element set
h \ {(F1 . n1)} is finite Element of bool h
bool h is non empty finite V60() set
(h \ {(F1 . n1)}) \/ {(F1 . n1)} is non empty finite set
card (x \ {y}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (h \ {(F1 . n1)}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(x \ {y}),(h \ {(F1 . n1)}):] is Relation-like finite set
bool [:(x \ {y}),(h \ {(F1 . n1)}):] is non empty finite V60() set
{ b₁ where b₁ is Relation-like x \ {y} -defined h \ {(F1 . n1)} -valued Function-like quasi_total finite Element of bool [:(x \ {y}),(h \ {(F1 . n1)}):] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like x \ {y} -defined h \ {(F1 . n1)} -valued Function-like quasi_total finite Element of bool [:(x \ {y}),(h \ {(F1 . n1)}):] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card (h \ {(F1 . n1)})) -' (card (x \ {y})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card (h \ {(F1 . n1)})) -' (card (x \ {y}))) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 !) / (((card (h \ {(F1 . n1)})) -' (card (x \ {y}))) !) is V11() ext-real non negative V53() Element of REAL
(card x) - (card x) is V11() ext-real V53() integer Element of INT
(card h) - (card x) is V11() ext-real V53() integer Element of INT
(card (h \ {(F1 . n1)})) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((card (h \ {(F1 . n1)})) + 1) - 1 is V11() ext-real V53() integer Element of INT
(card (x \ {y})) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((card (x \ {y})) + 1) - 1 is V11() ext-real V53() integer Element of INT
(((card (h \ {(F1 . n1)})) + 1) - 1) - (((card (x \ {y})) + 1) - 1) is V11() ext-real V53() integer Element of INT
{ b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : ( b₁ is one-to-one & b₁ . y = F1 . n1 ) } is set
card { b₁ where b₁ is Relation-like x -defined h -valued Function-like quasi_total finite Element of bool [:x,h:] : ( b₁ is one-to-one & b₁ . y = F1 . n1 ) } is V4() V5() V6() cardinal set
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 . XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
len F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(len F1) * ((S2 !) / (((card h) -' (card x)) !)) is V11() ext-real non negative V53() Element of REAL
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 . XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 + 1) * ((S2 !) / (((card h) -' (card x)) !)) is V11() ext-real non negative V53() Element of REAL
(S2 + 1) * (S2 !) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((S2 + 1) * (S2 !)) / (((card h) -' (card x)) !) is V11() ext-real non negative V53() Element of REAL
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:S2,x:] is Relation-like finite set
bool [:S2,x:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like S2 -defined x -valued Function-like quasi_total finite Element of bool [:S2,x:] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like S2 -defined x -valued Function-like quasi_total finite Element of bool [:S2,x:] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card x) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card x) -' (card S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card x) -' (card S2)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card x) !) / (((card x) -' (card S2)) !) is V11() ext-real non negative V53() Element of REAL
(card x) - (card S2) is V11() ext-real V53() integer Element of INT
y is set
c is Relation-like S2 -defined x -valued Function-like quasi_total finite Element of bool [:S2,x:]
dom {} is empty proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool NAT
rng {} is empty trivial proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V36() V37() V38() V39() V42() V43() V44() V45() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V201() V205() Element of bool RAT
bool RAT is non empty non trivial non finite set
X is finite set
[:X,X:] is Relation-like finite set
bool [:X,X:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : b₁ is Relation-like X -defined X -valued Function-like one-to-one V25(X) quasi_total onto bijective finite Element of bool [:X,X:] } is set
card { b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : b₁ is Relation-like X -defined X -valued Function-like one-to-one V25(X) quasi_total onto bijective finite Element of bool [:X,X:] } is V4() V5() V6() cardinal set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{ b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : b₁ is one-to-one } is set
x is set
h is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:]
(card X) -' (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card X) -' (card X)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card X) !) / (((card X) -' (card X)) !) is V11() ext-real non negative V53() Element of REAL
x is set
h is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:]
X is finite set
S2 is set
x is set
{S2,x} is non empty finite set
Funcs (X,{S2,x}) is non empty functional finite FUNCTION_DOMAIN of X,{S2,x}
bool (Funcs (X,{S2,x})) is non empty finite V60() set
[:X,{S2,x}:] is Relation-like finite set
bool [:X,{S2,x}:] is non empty finite V60() set
{S2} is non empty trivial finite 1 -element set
S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h is set
y is set
c is Relation-like X -defined {S2,x} -valued Function-like V25(X) quasi_total finite Element of bool [:X,{S2,x}:]
c " {S2} is finite Element of bool X
bool X is non empty finite V60() set
card (c " {S2}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is functional finite Element of bool (Funcs (X,{S2,x}))
c is set
F1 is Relation-like X -defined {S2,x} -valued Function-like V25(X) quasi_total finite Element of bool [:X,{S2,x}:]
F1 " {S2} is finite Element of bool X
bool X is non empty finite V60() set
card (F1 " {S2}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is functional finite Element of bool (Funcs (X,{S2,x}))
y is functional finite Element of bool (Funcs (X,{S2,x}))
c is set
F1 is Relation-like X -defined {S2,x} -valued Function-like V25(X) quasi_total finite Element of bool [:X,{S2,x}:]
F1 " {S2} is finite Element of bool X
bool X is non empty finite V60() set
card (F1 " {S2}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is Relation-like X -defined {S2,x} -valued Function-like V25(X) quasi_total finite Element of bool [:X,{S2,x}:]
F1 " {S2} is finite Element of bool X
bool X is non empty finite V60() set
card (F1 " {S2}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(S1,S2,X,X) is functional finite Element of bool (Funcs (S1,{X,X}))
{X,X} is non empty finite set
Funcs (S1,{X,X}) is non empty functional finite FUNCTION_DOMAIN of S1,{X,X}
bool (Funcs (S1,{X,X})) is non empty finite V60() set
x is set
[:S1,{X,X}:] is Relation-like finite set
bool [:S1,{X,X}:] is non empty finite V60() set
{X} is non empty trivial finite 1 -element set
h is Relation-like S1 -defined {X,X} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{X,X}:]
h " {X} is finite Element of bool S1
bool S1 is non empty finite V60() set
card (h " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng h is finite Element of bool {X,X}
bool {X,X} is non empty finite V60() set
dom h is finite Element of bool S1
rng h is finite Element of bool {X,X}
bool {X,X} is non empty finite V60() set
dom h is finite Element of bool S1
rng h is finite Element of bool {X,X}
bool {X,X} is non empty finite V60() set
X is set
S1 is set
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(S2,x,X,S1) is functional finite Element of bool (Funcs (S2,{X,S1}))
{X,S1} is non empty finite set
Funcs (S2,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of S2,{X,S1}
bool (Funcs (S2,{X,S1})) is non empty finite V60() set
h is set
[:S2,{X,S1}:] is Relation-like finite set
bool [:S2,{X,S1}:] is non empty finite V60() set
{X} is non empty trivial finite 1 -element set
y is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
y " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is set
S2 is finite set
(S2,0,X,S1) is functional finite Element of bool (Funcs (S2,{X,S1}))
{X,S1} is non empty finite set
Funcs (S2,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of S2,{X,S1}
bool (Funcs (S2,{X,S1})) is non empty finite V60() set
card (S2,0,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom {} is empty proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool NAT
[:S2,{X,S1}:] is Relation-like finite set
bool [:S2,{X,S1}:] is non empty finite V60() set
x is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
{x} is non empty trivial functional finite V60() 1 -element Element of bool (bool [:S2,{X,S1}:])
bool (bool [:S2,{X,S1}:]) is non empty finite V60() set
h is set
{X} is non empty trivial finite 1 -element set
y is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
y " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is finite Element of bool S2
{X} is non empty trivial finite 1 -element set
x " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card {} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of NAT
{S1} is non empty trivial finite 1 -element set
x is Relation-like Function-like set
dom x is set
rng x is set
[:S2,{X,S1}:] is Relation-like finite set
bool [:S2,{X,S1}:] is non empty finite V60() set
{x} is non empty trivial functional finite 1 -element set
h is set
{X} is non empty trivial finite 1 -element set
y is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
y " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng y is finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
dom y is finite Element of bool S2
{X} is non empty trivial finite 1 -element set
x " {X} is set
card {} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of NAT
X is set
S1 is set
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2,(card S2),X,S1) is functional finite Element of bool (Funcs (S2,{X,S1}))
{X,S1} is non empty finite set
Funcs (S2,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of S2,{X,S1}
bool (Funcs (S2,{X,S1})) is non empty finite V60() set
card (S2,(card S2),X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom {} is empty proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool NAT
[:S2,{X,S1}:] is Relation-like finite set
bool [:S2,{X,S1}:] is non empty finite V60() set
x is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
{x} is non empty trivial functional finite V60() 1 -element Element of bool (bool [:S2,{X,S1}:])
bool (bool [:S2,{X,S1}:]) is non empty finite V60() set
h is set
{X} is non empty trivial finite 1 -element set
y is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
y " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is finite Element of bool S2
{X} is non empty trivial finite 1 -element set
x " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
{X} is non empty trivial finite 1 -element set
x is Relation-like Function-like set
dom x is set
rng x is set
[:S2,{X,S1}:] is Relation-like finite set
bool [:S2,{X,S1}:] is non empty finite V60() set
{x} is non empty trivial functional finite 1 -element set
h is set
y is Relation-like S2 -defined {X,S1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{X,S1}:]
y " {X} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng y is finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
{S1} is non empty trivial finite 1 -element set
y " {S1} is finite Element of bool S2
c is set
y . c is set
rng y is finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
rng y is finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
rng y is finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
dom y is finite Element of bool S2
x " {X} is set
card (x " {X}) is V4() V5() V6() cardinal set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is set
{S1,S2} is non empty finite set
x is finite set
x \/ {X} is non empty finite set
[:(x \/ {X}),{S1,S2}:] is non empty Relation-like finite set
bool [:(x \/ {X}),{S1,S2}:] is non empty finite V60() set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(x,h,S1,S2) is functional finite Element of bool (Funcs (x,{S1,S2}))
Funcs (x,{S1,S2}) is non empty functional finite FUNCTION_DOMAIN of x,{S1,S2}
bool (Funcs (x,{S1,S2})) is non empty finite V60() set
card (x,h,S1,S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
{ b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:] : ( card (b₁ " {S1}) = h + 1 & b₁ . X = S1 ) } is set
card { b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:] : ( card (b₁ " {S1}) = h + 1 & b₁ . X = S1 ) } is V4() V5() V6() cardinal set
{S1,S2} \/ {S1} is non empty finite set
[:(x \/ {X}),({S1,S2} \/ {S1}):] is non empty Relation-like finite set
bool [:(x \/ {X}),({S1,S2} \/ {S1}):] is non empty finite V60() set
c is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):]
c . X is set
c | x is Relation-like x -defined x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):]
(c | x) | x is Relation-like x -defined x -defined x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):]
F1 is Relation-like Function-like set
F1 | x is Relation-like Function-like finite set
(F1 | x) " {S1} is finite set
card ((F1 | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is Relation-like Function-like set
F1 | x is Relation-like Function-like finite set
(F1 | x) " {S1} is finite set
card ((F1 | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{ b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):] : ( S₁[b₁,x \/ {X},{S1,S2} \/ {S1}] & rng (b₁ | x) c= {S1,S2} & b₁ . X = S1 ) } is set
[:x,{S1,S2}:] is Relation-like finite set
bool [:x,{S1,S2}:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:] : S₁[b₁,x,{S1,S2}] } is set
{S1} \/ {S1,S2} is non empty finite set
{S1,S1,S2} is non empty finite set
F1 is set
XF is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):]
XF | x is Relation-like x -defined x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):]
rng (XF | x) is finite Element of bool ({S1,S2} \/ {S1})
bool ({S1,S2} \/ {S1}) is non empty finite V60() set
XF . X is set
dom XF is non empty finite Element of bool (x \/ {X})
bool (x \/ {X}) is non empty finite V60() set
(x \/ {X}) \ {X} is finite Element of bool (x \/ {X})
(XF | x) " {S1} is finite Element of bool (x \/ {X})
{X} \/ ((XF | x) " {S1}) is non empty finite set
XF " {S1} is finite Element of bool (x \/ {X})
(dom XF) /\ x is finite Element of bool (x \/ {X})
dom (XF | x) is finite Element of bool x
bool x is non empty finite V60() set
card ((XF | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
XF is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:]
XF | x is Relation-like x -defined x -defined {S1,S2} -valued Function-like finite Element of bool [:x,{S1,S2}:]
XF " {S1} is finite Element of bool x
bool x is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
XF is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:]
XF " {S1} is finite Element of bool x
bool x is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom XF is finite Element of bool x
n is Relation-like Function-like set
n | x is Relation-like Function-like finite set
(n | x) " {S1} is finite set
card ((n | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card { b₁ where b₁ is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:] : S₁[b₁,x,{S1,S2}] } is V4() V5() V6() cardinal set
card { b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S1} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S1}):] : ( S₁[b₁,x \/ {X},{S1,S2} \/ {S1}] & rng (b₁ | x) c= {S1,S2} & b₁ . X = S1 ) } is V4() V5() V6() cardinal set
F1 is set
XF is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:]
XF " {S1} is finite Element of bool (x \/ {X})
bool (x \/ {X}) is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF . X is set
dom XF is non empty finite Element of bool (x \/ {X})
(dom XF) /\ x is finite Element of bool (x \/ {X})
XF | x is Relation-like x -defined x \/ {X} -defined {S1,S2} -valued Function-like finite Element of bool [:(x \/ {X}),{S1,S2}:]
dom (XF | x) is finite Element of bool x
bool x is non empty finite V60() set
(XF | x) " {S1} is finite Element of bool (x \/ {X})
(x \/ {X}) \ {X} is finite Element of bool (x \/ {X})
((XF | x) " {S1}) \/ {X} is non empty finite set
card ((XF | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card ((XF | x) " {S1})) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
n is Relation-like Function-like set
n | x is Relation-like Function-like finite set
(n | x) " {S1} is finite set
card ((n | x) " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{S1} \/ {S1,S2} is non empty finite set
{S1,S1,S2} is non empty finite set
rng (XF | x) is finite Element of bool {S1,S2}
bool {S1,S2} is non empty finite V60() set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is set
{S1,S2} is non empty finite set
x is finite set
x \/ {X} is non empty finite set
[:(x \/ {X}),{S1,S2}:] is non empty Relation-like finite set
bool [:(x \/ {X}),{S1,S2}:] is non empty finite V60() set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(x,h,S1,S2) is functional finite Element of bool (Funcs (x,{S1,S2}))
Funcs (x,{S1,S2}) is non empty functional finite FUNCTION_DOMAIN of x,{S1,S2}
bool (Funcs (x,{S1,S2})) is non empty finite V60() set
card (x,h,S1,S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{ b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:] : ( card (b₁ " {S1}) = h & b₁ . X = S2 ) } is set
card { b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:] : ( card (b₁ " {S1}) = h & b₁ . X = S2 ) } is V4() V5() V6() cardinal set
{S2} is non empty trivial finite 1 -element set
{S1,S2} \/ {S2} is non empty finite set
[:(x \/ {X}),({S1,S2} \/ {S2}):] is non empty Relation-like finite set
bool [:(x \/ {X}),({S1,S2} \/ {S2}):] is non empty finite V60() set
y is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):]
y . X is set
y | x is Relation-like x -defined x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):]
(x \/ {X}) \ {X} is finite Element of bool (x \/ {X})
bool (x \/ {X}) is non empty finite V60() set
dom y is non empty finite Element of bool (x \/ {X})
(y | x) " {S1} is finite Element of bool (x \/ {X})
y " {S1} is finite Element of bool (x \/ {X})
c is Relation-like Function-like set
c " {S1} is set
card (c " {S1}) is V4() V5() V6() cardinal set
F1 is Relation-like Function-like set
F1 " {S1} is set
card (F1 " {S1}) is V4() V5() V6() cardinal set
{ b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):] : ( S₁[b₁,x \/ {X},{S1,S2} \/ {S2}] & rng (b₁ | x) c= {S1,S2} & b₁ . X = S2 ) } is set
[:x,{S1,S2}:] is Relation-like finite set
bool [:x,{S1,S2}:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:] : S₁[b₁,x,{S1,S2}] } is set
card { b₁ where b₁ is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:] : S₁[b₁,x,{S1,S2}] } is V4() V5() V6() cardinal set
card { b₁ where b₁ is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):] : ( S₁[b₁,x \/ {X},{S1,S2} \/ {S2}] & rng (b₁ | x) c= {S1,S2} & b₁ . X = S2 ) } is V4() V5() V6() cardinal set
F1 is set
XF is non empty Relation-like x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):]
XF | x is Relation-like x -defined x \/ {X} -defined {S1,S2} \/ {S2} -valued Function-like finite Element of bool [:(x \/ {X}),({S1,S2} \/ {S2}):]
rng (XF | x) is finite Element of bool ({S1,S2} \/ {S2})
bool ({S1,S2} \/ {S2}) is non empty finite V60() set
XF . X is set
{S2,S2,S1} is non empty finite set
XF " {S1} is finite Element of bool (x \/ {X})
bool (x \/ {X}) is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
XF is non empty Relation-like x \/ {X} -defined {S1,S2} -valued Function-like V25(x \/ {X}) quasi_total finite Element of bool [:(x \/ {X}),{S1,S2}:]
XF " {S1} is finite Element of bool (x \/ {X})
bool (x \/ {X}) is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF . X is set
{S2,S2,S1} is non empty finite set
XF | x is Relation-like x -defined x \/ {X} -defined {S1,S2} -valued Function-like finite Element of bool [:(x \/ {X}),{S1,S2}:]
rng (XF | x) is finite Element of bool {S1,S2}
bool {S1,S2} is non empty finite V60() set
n is Relation-like Function-like set
n " {S1} is set
card (n " {S1}) is V4() V5() V6() cardinal set
F1 is set
XF is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:]
XF " {S1} is finite Element of bool x
bool x is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is Relation-like Function-like set
n " {S1} is set
card (n " {S1}) is V4() V5() V6() cardinal set
F1 is set
XF is Relation-like x -defined {S1,S2} -valued Function-like V25(x) quasi_total finite Element of bool [:x,{S1,S2}:]
XF " {S1} is finite Element of bool x
bool x is non empty finite V60() set
card (XF " {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{X,S1} is non empty finite set
S2 is set
{S2} is non empty trivial finite 1 -element set
x is finite set
x \/ {S2} is non empty finite set
[:(x \/ {S2}),{X,S1}:] is non empty Relation-like finite set
bool [:(x \/ {S2}),{X,S1}:] is non empty finite V60() set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = X ) } is set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = S1 ) } is set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = X ) } \/ { b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = S1 ) } is set
((x \/ {S2}),(h + 1),X,S1) is functional finite Element of bool (Funcs ((x \/ {S2}),{X,S1}))
Funcs ((x \/ {S2}),{X,S1}) is non empty functional finite FUNCTION_DOMAIN of x \/ {S2},{X,S1}
bool (Funcs ((x \/ {S2}),{X,S1})) is non empty finite V60() set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = X ) } /\ { b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = S1 ) } is set
F1 is set
F1 is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:]
F1 " {X} is finite Element of bool (x \/ {S2})
bool (x \/ {S2}) is non empty finite V60() set
card (F1 " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . S2 is set
XF is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:]
XF " {X} is finite Element of bool (x \/ {S2})
card (XF " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF . S2 is set
F1 is set
F1 is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:]
F1 " {X} is finite Element of bool (x \/ {S2})
bool (x \/ {S2}) is non empty finite V60() set
card (F1 " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . S2 is set
F1 is set
F1 is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:]
F1 " {X} is finite Element of bool (x \/ {S2})
bool (x \/ {S2}) is non empty finite V60() set
card (F1 " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom F1 is non empty finite Element of bool (x \/ {S2})
F1 . S2 is set
rng F1 is non empty finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
F1 is set
F1 is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:]
F1 " {X} is finite Element of bool (x \/ {S2})
bool (x \/ {S2}) is non empty finite V60() set
card (F1 " {X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . S2 is set
X is set
S1 is set
S2 is set
{S2} is non empty trivial finite 1 -element set
x is finite set
x \/ {S2} is non empty finite set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((x \/ {S2}),(h + 1),X,S1) is functional finite Element of bool (Funcs ((x \/ {S2}),{X,S1}))
{X,S1} is non empty finite set
Funcs ((x \/ {S2}),{X,S1}) is non empty functional finite FUNCTION_DOMAIN of x \/ {S2},{X,S1}
bool (Funcs ((x \/ {S2}),{X,S1})) is non empty finite V60() set
card ((x \/ {S2}),(h + 1),X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(x,(h + 1),X,S1) is functional finite Element of bool (Funcs (x,{X,S1}))
Funcs (x,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of x,{X,S1}
bool (Funcs (x,{X,S1})) is non empty finite V60() set
card (x,(h + 1),X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(x,h,X,S1) is functional finite Element of bool (Funcs (x,{X,S1}))
card (x,h,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (x,(h + 1),X,S1)) + (card (x,h,X,S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(x \/ {S2}),{X,S1}:] is non empty Relation-like finite set
bool [:(x \/ {S2}),{X,S1}:] is non empty finite V60() set
{X} is non empty trivial finite 1 -element set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = S1 ) } is set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = X ) } is set
{ b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = X ) } \/ { b₁ where b₁ is non empty Relation-like x \/ {S2} -defined {X,S1} -valued Function-like V25(x \/ {S2}) quasi_total finite Element of bool [:(x \/ {S2}),{X,S1}:] : ( card (b₁ " {X}) = h + 1 & b₁ . S2 = S1 ) } is set
F1 is finite set
card F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is finite set
F1 \/ F1 is finite set
card (F1 \/ F1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card F1) + (card F1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is set
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(S2,x,X,S1) is functional finite Element of bool (Funcs (S2,{X,S1}))
{X,S1} is non empty finite set
Funcs (S2,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of S2,{X,S1}
bool (Funcs (S2,{X,S1})) is non empty finite V60() set
card (S2,x,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S2) choose x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
c is finite set
card c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y + (card c) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(c,y,X,S1) is functional finite Element of bool (Funcs (c,{X,S1}))
Funcs (c,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of c,{X,S1}
bool (Funcs (c,{X,S1})) is non empty finite V60() set
card (c,y,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card c) choose y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card c) - 1 is V11() ext-real V53() integer Element of INT
y - 1 is V11() ext-real V53() integer Element of INT
XF is set
{XF} is non empty trivial finite 1 -element set
c \ {XF} is finite Element of bool c
bool c is non empty finite V60() set
(c \ {XF}) \/ {XF} is non empty finite set
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(c,(F1 + 1),X,S1) is functional finite Element of bool (Funcs (c,{X,S1}))
card (c,(F1 + 1),X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((c \ {XF}),(F1 + 1),X,S1) is functional finite Element of bool (Funcs ((c \ {XF}),{X,S1}))
Funcs ((c \ {XF}),{X,S1}) is non empty functional finite FUNCTION_DOMAIN of c \ {XF},{X,S1}
bool (Funcs ((c \ {XF}),{X,S1})) is non empty finite V60() set
card ((c \ {XF}),(F1 + 1),X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((c \ {XF}),F1,X,S1) is functional finite Element of bool (Funcs ((c \ {XF}),{X,S1}))
card ((c \ {XF}),F1,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card ((c \ {XF}),(F1 + 1),X,S1)) + (card ((c \ {XF}),F1,X,S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
card (c \ {XF}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y + (card (c \ {XF})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (c \ {XF})) choose (F1 + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 + (card (c \ {XF})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (c \ {XF})) choose F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y is finite set
card y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h + (card y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(y,h,X,S1) is functional finite Element of bool (Funcs (y,{X,S1}))
Funcs (y,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of y,{X,S1}
bool (Funcs (y,{X,S1})) is non empty finite V60() set
card (y,h,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card y) choose h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
0 choose 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x + (card S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y is finite set
card y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h + (card y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(y,h,X,S1) is functional finite Element of bool (Funcs (y,{X,S1}))
Funcs (y,{X,S1}) is non empty functional finite FUNCTION_DOMAIN of y,{X,S1}
bool (Funcs (y,{X,S1})) is non empty finite V60() set
card (y,h,X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card y) choose h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is set
S2 is finite set
S2 --> S1 is Relation-like S2 -defined {S1} -valued Function-like constant V25(S2) quasi_total finite Element of bool [:S2,{S1}:]
{S1} is non empty trivial finite 1 -element set
[:S2,{S1}:] is Relation-like finite set
bool [:S2,{S1}:] is non empty finite V60() set
x is finite set
x --> X is Relation-like x -defined {X} -valued Function-like constant V25(x) quasi_total finite Element of bool [:x,{X}:]
{X} is non empty trivial finite 1 -element set
[:x,{X}:] is Relation-like finite set
bool [:x,{X}:] is non empty finite V60() set
(S2 --> S1) +* (x --> X) is Relation-like Function-like finite set
x \/ S2 is finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((x \/ S2),(card x),X,S1) is functional finite Element of bool (Funcs ((x \/ S2),{X,S1}))
{X,S1} is non empty finite set
Funcs ((x \/ S2),{X,S1}) is non empty functional finite FUNCTION_DOMAIN of x \/ S2,{X,S1}
bool (Funcs ((x \/ S2),{X,S1})) is non empty finite V60() set
dom (S2 --> S1) is finite Element of bool S2
bool S2 is non empty finite V60() set
dom (x --> X) is finite Element of bool x
bool x is non empty finite V60() set
dom ((S2 --> S1) +* (x --> X)) is finite set
rng (S2 --> S1) is trivial finite Element of bool {S1}
bool {S1} is non empty finite V60() set
rng (x --> X) is trivial finite Element of bool {X}
bool {X} is non empty finite V60() set
rng ((S2 --> S1) +* (x --> X)) is finite set
(rng (x --> X)) \/ (rng (S2 --> S1)) is finite set
[:(x \/ S2),{X,S1}:] is Relation-like finite set
bool [:(x \/ S2),{X,S1}:] is non empty finite V60() set
y is Relation-like x \/ S2 -defined {X,S1} -valued Function-like V25(x \/ S2) quasi_total finite Element of bool [:(x \/ S2),{X,S1}:]
y " {X} is finite Element of bool (x \/ S2)
bool (x \/ S2) is non empty finite V60() set
c is set
y . c is set
dom y is finite Element of bool (x \/ S2)
(dom (x --> X)) \/ (dom (S2 --> S1)) is finite set
(S2 --> S1) . c is set
c is set
dom y is finite Element of bool (x \/ S2)
y . c is set
(x --> X) . c is set
X is set
S1 is set
S2 is finite set
S2 --> X is Relation-like S2 -defined {X} -valued Function-like constant V25(S2) quasi_total finite Element of bool [:S2,{X}:]
{X} is non empty trivial finite 1 -element set
[:S2,{X}:] is Relation-like finite set
bool [:S2,{X}:] is non empty finite V60() set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
x --> S1 is Relation-like x -defined {S1} -valued Function-like constant V25(x) quasi_total finite Element of bool [:x,{S1}:]
{S1} is non empty trivial finite 1 -element set
[:x,{S1}:] is Relation-like finite set
bool [:x,{S1}:] is non empty finite V60() set
(S2 --> X) +* (x --> S1) is Relation-like Function-like finite set
S2 \/ x is finite set
((S2 \/ x),(card S2),X,S1) is functional finite Element of bool (Funcs ((S2 \/ x),{X,S1}))
{X,S1} is non empty finite set
Funcs ((S2 \/ x),{X,S1}) is non empty functional finite FUNCTION_DOMAIN of S2 \/ x,{X,S1}
bool (Funcs ((S2 \/ x),{X,S1})) is non empty finite V60() set
dom (S2 --> X) is finite Element of bool S2
bool S2 is non empty finite V60() set
dom (x --> S1) is finite Element of bool x
bool x is non empty finite V60() set
(x --> S1) +* (S2 --> X) is Relation-like Function-like finite set
X is Relation-like Function-like set
rng X is set
union (rng X) is set
bool (union (rng X)) is non empty set
S1 is Relation-like Function-like set
dom S1 is set
S2 is set
x is set
h is set
h is Element of bool (union (rng X))
y is set
c is set
S1 . c is set
X . c is set
F1 is set
x is Element of bool (union (rng X))
h is Element of bool (union (rng X))
y is set
c is set
S1 . c is set
X . c is set
c is set
S1 . c is set
X . c is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
S2 is Relation-like Function-like set
dom S2 is set
x is Relation-like Function-like set
x " {X} is set
(dom S2) /\ (x " {X}) is set
(S2,x,X) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
dom x is set
h is set
x . h is set
S2 . h is set
h is set
x . h is set
S2 . h is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
dom S1 is set
S2 is Relation-like Function-like set
(S1,S2,X) is Element of bool (union (rng S1))
rng S1 is set
union (rng S1) is set
bool (union (rng S1)) is non empty set
S2 " {X} is set
x is set
h is set
S2 . h is set
dom S2 is set
S1 . h is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
S2 is Relation-like Function-like set
(S1,S2,X) is Element of bool (union (rng S1))
rng S1 is set
union (rng S1) is set
bool (union (rng S1)) is non empty set
S2 " {X} is set
x is set
h is set
S1 . h is set
y is set
S1 . y is set
S2 . y is set
S2 . h is set
dom S2 is set
X is set
S1 is set
S2 is Relation-like Function-like set
x is Relation-like Function-like set
(S2,x,S1) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
rng x is set
dom x is set
{S1} is non empty trivial finite 1 -element set
x " {S1} is set
h is set
x . h is set
S2 . h is set
S1 is Relation-like Function-like set
rng S1 is set
union (rng S1) is set
S2 is Relation-like Function-like set
X is set
(S1,S2,X) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
rng S1 is set
union (rng S1) is set
S2 is Relation-like Function-like set
S2 " {X} is set
S1 | (S2 " {X}) is Relation-like Function-like set
(S2 " {X}) --> (union (rng S1)) is Relation-like S2 " {X} -defined {(union (rng S1))} -valued Function-like constant V25(S2 " {X}) quasi_total Element of bool [:(S2 " {X}),{(union (rng S1))}:]
{(union (rng S1))} is non empty trivial finite 1 -element set
[:(S2 " {X}),{(union (rng S1))}:] is Relation-like set
bool [:(S2 " {X}),{(union (rng S1))}:] is non empty set
(S1,S2,X) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
h is set
y is set
dom S2 is set
S2 . y is set
dom ((S2 " {X}) --> (union (rng S1))) is Element of bool (S2 " {X})
bool (S2 " {X}) is non empty set
((S2 " {X}) --> (union (rng S1))) . y is set
S1 . y is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
rng S1 is set
union (rng S1) is set
S2 is Relation-like Function-like set
(S1,S2,X) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
S2 " {X} is set
S1 | (S2 " {X}) is Relation-like Function-like set
(S2 " {X}) --> (union (rng S1)) is Relation-like S2 " {X} -defined {(union (rng S1))} -valued Function-like constant V25(S2 " {X}) quasi_total Element of bool [:(S2 " {X}),{(union (rng S1))}:]
{(union (rng S1))} is non empty trivial finite 1 -element set
[:(S2 " {X}),{(union (rng S1))}:] is Relation-like set
bool [:(S2 " {X}),{(union (rng S1))}:] is non empty set
dom S1 is set
(dom S1) /\ (S2 " {X}) is set
dom (S1 | (S2 " {X})) is set
dom ((S2 " {X}) --> (union (rng S1))) is Element of bool (S2 " {X})
bool (S2 " {X}) is non empty set
h is set
(S1 | (S2 " {X})) . h is set
((S2 " {X}) --> (union (rng S1))) . h is set
S2 . h is set
S1 . h is set
y is set
dom S2 is set
X is set
S1 is Relation-like Function-like set
(S1,{},X) is Element of bool (union (rng S1))
rng S1 is set
union (rng S1) is set
bool (union (rng S1)) is non empty set
{X} is non empty trivial finite 1 -element set
{} " {X} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
S1 | ({} " {X}) is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() set
({} " {X}) --> (union (rng S1)) is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding {} " {X} -defined RAT -valued {(union (rng S1))} -valued Function-like one-to-one constant functional V25({} " {X}) quasi_total V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool [:({} " {X}),{(union (rng S1))}:]
{(union (rng S1))} is non empty trivial finite 1 -element set
[:({} " {X}),{(union (rng S1))}:] is Relation-like finite set
bool [:({} " {X}),{(union (rng S1))}:] is non empty finite V60() set
X is set
S1 is set
S2 is Relation-like Function-like set
x is Relation-like Function-like set
(S2,x,X) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
x | S1 is Relation-like Function-like set
(S2,(x | S1),X) is Element of bool (union (rng S2))
h is set
y is set
dom (x | S1) is set
(x | S1) . y is set
dom x is set
(dom x) /\ S1 is set
x . y is set
S2 . y is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
S2 is Relation-like Function-like set
S2 " {X} is set
S2 | S1 is Relation-like Function-like set
(S2 | S1) " {X} is set
x is Relation-like Function-like set
(x,S2,X) is Element of bool (union (rng x))
rng x is set
union (rng x) is set
bool (union (rng x)) is non empty set
(x,(S2 | S1),X) is Element of bool (union (rng x))
h is set
y is set
dom S2 is set
S2 . y is set
(S2 | S1) . y is set
dom (S2 | S1) is set
x . y is set
X is set
S1 is set
S2 is Relation-like Function-like set
S2 | X is Relation-like Function-like set
x is Relation-like Function-like set
((S2 | X),x,S1) is Element of bool (union (rng (S2 | X)))
rng (S2 | X) is set
union (rng (S2 | X)) is set
bool (union (rng (S2 | X))) is non empty set
(S2,x,S1) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
h is set
{S1} is non empty trivial finite 1 -element set
x " {S1} is set
dom (S2 | X) is set
y is set
dom x is set
x . y is set
(S2 | X) . y is set
S2 . y is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
S2 is Relation-like Function-like set
rng S2 is set
S2 " {X} is set
x is Relation-like Function-like set
x | S1 is Relation-like Function-like set
((x | S1),S2,X) is Element of bool (union (rng (x | S1)))
rng (x | S1) is set
union (rng (x | S1)) is set
bool (union (rng (x | S1))) is non empty set
(x,S2,X) is Element of bool (union (rng x))
rng x is set
union (rng x) is set
bool (union (rng x)) is non empty set
h is set
y is set
dom S2 is set
S2 . y is set
x . y is set
dom x is set
(dom x) /\ S1 is set
dom (x | S1) is set
(x | S1) . y is set
y is set
S2 . y is set
x . y is set
(x | S1) . y is set
X is set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is Relation-like Function-like set
S2 " {S1} is set
x is Relation-like Function-like set
(x,S2,S1) is Element of bool (union (rng x))
rng x is set
union (rng x) is set
bool (union (rng x)) is non empty set
x . X is set
dom S2 is set
S2 . X is set
h is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is Relation-like Function-like set
S2 " {S1} is set
dom S2 is set
(dom S2) \ {X} is Element of bool (dom S2)
bool (dom S2) is non empty set
S2 | ((dom S2) \ {X}) is Relation-like Function-like set
x is Relation-like Function-like set
rng x is set
union (rng x) is set
(x,(S2 | ((dom S2) \ {X})),S1) is Element of bool (union (rng x))
bool (union (rng x)) is non empty set
x . X is set
(x,(S2 | ((dom S2) \ {X})),S1) /\ (x . X) is Element of bool (union (rng x))
(x,S2,S1) is Element of bool (union (rng x))
F1 is set
XF is set
S2 . XF is set
(dom S2) /\ ((dom S2) \ {X}) is Element of bool (dom S2)
dom (S2 | ((dom S2) \ {X})) is set
(S2 | ((dom S2) \ {X})) . XF is set
x . XF is set
x . XF is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is Relation-like Function-like set
x is Relation-like Function-like set
x " {X} is set
h is Relation-like Function-like set
h " {S1} is set
(S2,x,X) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
(S2,h,S1) is Element of bool (union (rng S2))
y is set
c is set
dom h is set
h . c is set
x . c is set
dom x is set
S2 . c is set
y is set
c is set
dom x is set
x . c is set
h . c is set
dom h is set
S2 . c is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
S1 " {X} is set
S2 is Relation-like Function-like set
(S2,S1,X) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
x is set
S1 | x is Relation-like Function-like set
(S2,{},X) is Element of bool (union (rng S2))
(S1 | x) " {X} is set
X is set
{X} is non empty trivial finite 1 -element set
S1 is set
{S1} is non empty trivial finite 1 -element set
S2 is Relation-like Function-like set
S2 " {S1} is set
x is Relation-like Function-like set
(x,S2,S1) is Element of bool (union (rng x))
rng x is set
union (rng x) is set
bool (union (rng x)) is non empty set
x . X is set
dom S2 is set
(dom S2) \ {X} is Element of bool (dom S2)
bool (dom S2) is non empty set
S2 | ((dom S2) \ {X}) is Relation-like Function-like set
(S2 | ((dom S2) \ {X})) " {S1} is set
((dom S2) \ {X}) /\ {X} is finite Element of bool (dom S2)
(x,(S2 | ((dom S2) \ {X})),S1) is Element of bool (union (rng x))
(union (rng x)) /\ (x . X) is set
dom x is set
dom x is set
dom x is set
X is set
S1 is set
{X,S1} is non empty finite set
S2 is set
{S2} is non empty trivial finite 1 -element set
x is Relation-like Function-like set
x " {S2} is set
h is Relation-like Function-like set
(h,x,S2) is Element of bool (union (rng h))
rng h is set
union (rng h) is set
bool (union (rng h)) is non empty set
h . X is set
h . S1 is set
(h . X) /\ (h . S1) is set
{X} is non empty trivial finite 1 -element set
dom x is set
{X} is non empty trivial finite 1 -element set
(dom x) \ {X} is Element of bool (dom x)
bool (dom x) is non empty set
(x " {S2}) /\ ((dom x) \ {X}) is Element of bool (dom x)
(x " {S2}) /\ (dom x) is set
((x " {S2}) /\ (dom x)) \ {X} is Element of bool ((x " {S2}) /\ (dom x))
bool ((x " {S2}) /\ (dom x)) is non empty set
{S1} is non empty trivial finite 1 -element set
x | ((dom x) \ {X}) is Relation-like Function-like set
(x | ((dom x) \ {X})) " {S2} is set
(h,(x | ((dom x) \ {X})),S2) is Element of bool (union (rng h))
(h,(x | ((dom x) \ {X})),S2) /\ (h . X) is Element of bool (union (rng h))
X is set
S1 is set
S2 is Relation-like Function-like set
dom S2 is set
(dom S2) --> S1 is Relation-like dom S2 -defined {S1} -valued Function-like constant V25( dom S2) quasi_total Element of bool [:(dom S2),{S1}:]
{S1} is non empty trivial finite 1 -element set
[:(dom S2),{S1}:] is Relation-like set
bool [:(dom S2),{S1}:] is non empty set
(S2,((dom S2) --> S1),S1) is Element of bool (union (rng S2))
rng S2 is set
union (rng S2) is set
bool (union (rng S2)) is non empty set
h is set
S2 . h is set
dom ((dom S2) --> S1) is Element of bool (dom S2)
bool (dom S2) is non empty set
((dom S2) --> S1) . h is set
((dom S2) --> S1) " {S1} is Element of bool (dom S2)
bool (dom S2) is non empty set
(dom S2) /\ (((dom S2) --> S1) " {S1}) is Element of bool (dom S2)
dom ((dom S2) --> S1) is Element of bool (dom S2)
h is set
((dom S2) --> S1) . h is set
S2 . h is set
X is Relation-like Function-like finite-yielding set
S1 is set
X | S1 is Relation-like Function-like set
S2 is set
dom (X | S1) is set
(X | S1) . S2 is set
X . S2 is finite set
S1 is Relation-like Function-like set
X is Relation-like Function-like finite-yielding set
S1 (#) X is Relation-like Function-like set
S2 is set
dom (S1 (#) X) is set
(S1 (#) X) . S2 is set
S1 . S2 is set
X . (S1 . S2) is finite set
Intersect (X,S1) is Relation-like Function-like set
S2 is set
dom (Intersect (X,S1)) is set
(Intersect (X,S1)) . S2 is set
dom X is set
dom S1 is set
(dom X) /\ (dom S1) is set
X . S2 is finite set
S1 . S2 is set
(X . S2) /\ (S1 . S2) is finite set
X is set
S1 is Relation-like Function-like set
rng S1 is set
S2 is Relation-like Function-like finite-yielding set
(S2,S1,X) is Element of bool (union (rng S2))
rng S2 is V60() set
union (rng S2) is set
bool (union (rng S2)) is non empty set
dom S1 is set
x is set
S1 . x is set
{X} is non empty trivial finite 1 -element set
S1 " {X} is set
S2 . x is finite set
X is Relation-like Function-like finite-yielding set
dom X is set
rng X is V60() set
union (rng X) is set
{0,1} is non empty finite V60() Element of bool NAT
X is set
S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(S1,S2,1,0) is functional finite Element of bool (Funcs (S1,{1,0}))
{1,0} is non empty finite V60() set
Funcs (S1,{1,0}) is non empty functional finite FUNCTION_DOMAIN of S1,{1,0}
bool (Funcs (S1,{1,0})) is non empty finite V60() set
[:S1,{0,1}:] is Relation-like finite set
bool [:S1,{0,1}:] is non empty finite V60() set
{1} is non empty trivial finite V60() 1 -element Element of bool NAT
x is Relation-like S1 -defined {0,1} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{0,1}:]
x " {1} is finite Element of bool NAT
card (x " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng x is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
dom x is finite Element of bool S1
bool S1 is non empty finite V60() set
h is V8() Relation-like NAT -defined Function-like finite V205() set
rng h is finite set
y is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
rng y is V42() V43() V44() V47() finite Element of bool NAT
Sum y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
y " {1} is finite set
card (y " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
1 * (card (y " {1})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
rng x is V42() V43() V44() V47() finite Element of bool NAT
Sum x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
{1} is non empty trivial finite V60() 1 -element Element of bool NAT
x " {1} is finite set
card (x " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
1 * (card (x " {1})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:S1,{0,1}:] is Relation-like finite set
bool [:S1,{0,1}:] is non empty finite V60() set
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card X),X:] is Relation-like finite set
bool [:(card X),X:] is non empty finite V60() set
S1 is Relation-like Function-like set
dom S1 is set
rng S1 is set
S1 is Relation-like Function-like finite-yielding set
dom S1 is set
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 is finite set
(S2,X,0,1) is functional finite Element of bool (Funcs (S2,{0,1}))
{0,1} is non empty finite V60() set
Funcs (S2,{0,1}) is non empty functional finite FUNCTION_DOMAIN of S2,{0,1}
bool (Funcs (S2,{0,1})) is non empty finite V60() set
card (S2,X,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is Relation-like Function-like set
dom h is set
rng h is set
[:(card (S2,X,0,1)),(S2,X,0,1):] is Relation-like finite set
bool [:(card (S2,X,0,1)),(S2,X,0,1):] is non empty finite V60() set
y is Relation-like card (S2,X,0,1) -defined (S2,X,0,1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S2,X,0,1)),(S2,X,0,1):]
c is set
y . c is Relation-like Function-like set
dom y is finite Element of bool (card (S2,X,0,1))
bool (card (S2,X,0,1)) is non empty finite V60() set
rng y is functional finite Element of bool (S2,X,0,1)
bool (S2,X,0,1) is non empty finite V60() set
[:S2,{0,1}:] is Relation-like finite set
bool [:S2,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
F1 is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
F1 " {0} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (F1 " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng S1 is V60() set
union (rng S1) is set
(S1,F1,0) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
F1 is finite set
card F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF is Relation-like Function-like set
(S1,XF,0) is Element of bool (union (rng S1))
card (S1,XF,0) is V4() V5() V6() cardinal set
[:(card (S2,X,0,1)),NAT:] is Relation-like set
bool [:(card (S2,X,0,1)),NAT:] is non empty set
c is Relation-like card (S2,X,0,1) -defined NAT -valued Function-like V25( card (S2,X,0,1)) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(card (S2,X,0,1)),NAT:]
dom c is finite Element of bool (card (S2,X,0,1))
bool (card (S2,X,0,1)) is non empty finite V60() set
F1 is V8() Relation-like NAT -defined Function-like finite V205() set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is set
n1 is set
u is finite set
(u,X,n,n1) is functional finite Element of bool (Funcs (u,{n,n1}))
{n,n1} is non empty finite set
Funcs (u,{n,n1}) is non empty functional finite FUNCTION_DOMAIN of u,{n,n1}
bool (Funcs (u,{n,n1})) is non empty finite V60() set
card (u,X,n,n1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (u,X,n,n1)),(u,X,n,n1):] is Relation-like finite set
bool [:(card (u,X,n,n1)),(u,X,n,n1):] is non empty finite V60() set
n is Relation-like card (u,X,n,n1) -defined (u,X,n,n1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (u,X,n,n1)),(u,X,n,n1):]
dom n is finite Element of bool (card (u,X,n,n1))
bool (card (u,X,n,n1)) is non empty finite V60() set
[:S2,{0,1}:] is Relation-like finite set
bool [:S2,{0,1}:] is non empty finite V60() set
[:u,{n,n1}:] is Relation-like finite set
bool [:u,{n,n1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
{n} is non empty trivial finite 1 -element set
card (card (u,X,n,n1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng n is functional finite Element of bool (u,X,n,n1)
bool (u,X,n,n1) is non empty finite V60() set
Fyy is set
y . Fyy is Relation-like Function-like set
rng y is functional finite Element of bool (S2,X,0,1)
bool (S2,X,0,1) is non empty finite V60() set
y is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
y " {0} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is set
y . h is set
dom y is finite Element of bool S2
rng y is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
h is Relation-like u -defined {n,n1} -valued Function-like V25(u) quasi_total finite Element of bool [:u,{n,n1}:]
dom y is finite Element of bool S2
h " {n} is finite Element of bool u
bool u is non empty finite V60() set
x is set
y . x is set
dom h is finite Element of bool u
h . x is set
x is set
h . x is set
y . x is set
x is set
n . x is Relation-like Function-like set
card u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card u) choose X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S2) choose X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
y " {0} is finite Element of bool S2
y is Relation-like u -defined {n,n1} -valued Function-like V25(u) quasi_total finite Element of bool [:u,{n,n1}:]
y " {n} is finite Element of bool u
x1 is set
y . x1 is set
y . x1 is set
[:(card (S2,X,0,1)),(card (S2,X,0,1)):] is Relation-like RAT -valued V32() V33() V34() V35() V52() finite set
bool [:(card (S2,X,0,1)),(card (S2,X,0,1)):] is non empty finite V60() set
Fyy is Relation-like card (S2,X,0,1) -defined card (S2,X,0,1) -valued Function-like V25( card (S2,X,0,1)) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(card (S2,X,0,1)),(card (S2,X,0,1)):]
y is set
dom Fyy is finite Element of bool (card (S2,X,0,1))
h is set
Fyy . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of card (S2,X,0,1)
Fyy . h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of card (S2,X,0,1)
card u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card u) choose X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S2) choose X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n . (Fyy . y) is Relation-like Function-like set
rng n is functional finite Element of bool (u,X,n,n1)
bool (u,X,n,n1) is non empty finite V60() set
x is Relation-like u -defined {n,n1} -valued Function-like V25(u) quasi_total finite Element of bool [:u,{n,n1}:]
x " {n} is finite Element of bool u
bool u is non empty finite V60() set
card (x " {n}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . h is Relation-like Function-like set
rng y is functional finite Element of bool (S2,X,0,1)
bool (S2,X,0,1) is non empty finite V60() set
y is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
y " {0} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (y " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . y is Relation-like Function-like set
y is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
y " {0} is finite Element of bool S2
card (y " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is finite Element of bool S2
x1 is set
y . x1 is set
y . x1 is set
rng y is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
x . x1 is set
x . x1 is set
dom y is finite Element of bool S2
card (card (S2,X,0,1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is set
n . y is Relation-like Function-like set
h is Relation-like u -defined {n,n1} -valued Function-like V25(u) quasi_total finite Element of bool [:u,{n,n1}:]
h " {n} is finite Element of bool u
card (h " {n}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng S1 is V60() set
union (rng S1) is set
(S1,h,n) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
x is finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is Relation-like Function-like set
(S1,y,n) is Element of bool (union (rng S1))
card (S1,y,n) is V4() V5() V6() cardinal set
[:(card (u,X,n,n1)),NAT:] is Relation-like set
bool [:(card (u,X,n,n1)),NAT:] is non empty set
y is Relation-like card (u,X,n,n1) -defined NAT -valued Function-like V25( card (u,X,n,n1)) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(card (u,X,n,n1)),NAT:]
dom y is finite Element of bool (card (u,X,n,n1))
h is V8() Relation-like NAT -defined Function-like finite V205() set
x is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
y is set
y is Relation-like Function-like set
n . y is Relation-like Function-like set
x . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,y,n) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,y,n) is V4() V5() V6() cardinal set
[:(dom x),(dom x):] is Relation-like RAT -valued V32() V33() V34() V35() V52() finite set
bool [:(dom x),(dom x):] is non empty finite V60() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
y is Relation-like dom x -defined dom x -valued Function-like one-to-one V25( dom x) quasi_total onto bijective V32() V33() V34() V35() V52() finite Element of bool [:(dom x),(dom x):]
y (#) x is Relation-like dom x -defined NAT -valued RAT -valued Function-like V32() V33() V34() V35() V52() finite set
y is set
F1 . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(y (#) x) . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
dom y is finite Element of bool (dom x)
bool (dom x) is non empty finite V60() set
y . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of dom x
n . (y . y) is Relation-like Function-like set
x1 is Relation-like u -defined {n,n1} -valued Function-like V25(u) quasi_total finite Element of bool [:u,{n,n1}:]
x1 " {n} is finite Element of bool u
card (x1 " {n}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x . (y . y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,x1,n) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,x1,n) is V4() V5() V6() cardinal set
y . y is Relation-like Function-like set
x2 is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
x2 " {0} is finite Element of bool S2
card (x2 " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,x2,0) is Element of bool (union (rng S1))
card (S1,x2,0) is V4() V5() V6() cardinal set
rng y is V42() V43() V44() V47() finite Element of bool (dom x)
bool (dom x) is non empty finite V60() set
dom y is finite Element of bool (dom x)
dom (y (#) x) is finite Element of bool (dom x)
addnat "**" x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
addnat "**" F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is finite set
(S2,X,0,1) is functional finite Element of bool (Funcs (S2,{0,1}))
{0,1} is non empty finite V60() set
Funcs (S2,{0,1}) is non empty functional finite FUNCTION_DOMAIN of S2,{0,1}
bool (Funcs (S2,{0,1})) is non empty finite V60() set
card (S2,X,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c is Relation-like Function-like set
dom c is set
rng c is set
[:(card (S2,X,0,1)),(S2,X,0,1):] is Relation-like finite set
bool [:(card (S2,X,0,1)),(S2,X,0,1):] is non empty finite V60() set
F1 is Relation-like card (S2,X,0,1) -defined (S2,X,0,1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S2,X,0,1)),(S2,X,0,1):]
dom F1 is finite Element of bool (card (S2,X,0,1))
bool (card (S2,X,0,1)) is non empty finite V60() set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
XF is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
n is set
F1 . n is Relation-like Function-like set
rng F1 is functional finite Element of bool (S2,X,0,1)
bool (S2,X,0,1) is non empty finite V60() set
[:S2,{0,1}:] is Relation-like finite set
bool [:S2,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
n1 is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
n1 " {0} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (n1 " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,n1,0) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,n1,0) is V4() V5() V6() cardinal set
F1 . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S1 is Relation-like Function-like finite-yielding set
dom S1 is set
(X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is set
x is set
h is finite set
(h,X,S2,x) is functional finite Element of bool (Funcs (h,{S2,x}))
{S2,x} is non empty finite set
Funcs (h,{S2,x}) is non empty functional finite FUNCTION_DOMAIN of h,{S2,x}
bool (Funcs (h,{S2,x})) is non empty finite V60() set
card (h,X,S2,x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (h,X,S2,x)),(h,X,S2,x):] is Relation-like finite set
bool [:(card (h,X,S2,x)),(h,X,S2,x):] is non empty finite V60() set
y is Relation-like card (h,X,S2,x) -defined (h,X,S2,x) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (h,X,S2,x)),(h,X,S2,x):]
dom y is finite Element of bool (card (h,X,S2,x))
bool (card (h,X,S2,x)) is non empty finite V60() set
c is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
F1 is set
y . F1 is Relation-like Function-like set
rng y is functional finite Element of bool (h,X,S2,x)
bool (h,X,S2,x) is non empty finite V60() set
[:h,{S2,x}:] is Relation-like finite set
bool [:h,{S2,x}:] is non empty finite V60() set
{S2} is non empty trivial finite 1 -element set
XF is Relation-like h -defined {S2,x} -valued Function-like V25(h) quasi_total finite Element of bool [:h,{S2,x}:]
XF " {S2} is finite Element of bool h
bool h is non empty finite V60() set
card (XF " {S2}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,XF,S2) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,XF,S2) is V4() V5() V6() cardinal set
c . F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S1 is Relation-like Function-like finite-yielding set
dom S1 is set
(X,S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng S1 is V60() set
union (rng S1) is set
card (union (rng S1)) is V4() V5() V6() cardinal set
S2 is finite set
(S2,X,0,1) is functional finite Element of bool (Funcs (S2,{0,1}))
{0,1} is non empty finite V60() set
Funcs (S2,{0,1}) is non empty functional finite FUNCTION_DOMAIN of S2,{0,1}
bool (Funcs (S2,{0,1})) is non empty finite V60() set
card (S2,X,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (S2,X,0,1)),(S2,X,0,1):] is Relation-like finite set
bool [:(card (S2,X,0,1)),(S2,X,0,1):] is non empty finite V60() set
h is Relation-like card (S2,X,0,1) -defined (S2,X,0,1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S2,X,0,1)),(S2,X,0,1):]
dom h is finite Element of bool (card (S2,X,0,1))
bool (card (S2,X,0,1)) is non empty finite V60() set
y is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
len y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y . 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
<%(y . 0)%> is non empty trivial V8() Relation-like NAT -defined NAT -valued Function-like constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite 1 -element V205() set
addnat "**" y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h . 0 is Relation-like Function-like set
rng h is functional finite Element of bool (S2,X,0,1)
bool (S2,X,0,1) is non empty finite V60() set
[:S2,{0,1}:] is Relation-like finite set
bool [:S2,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
c is Relation-like S2 -defined {0,1} -valued Function-like V25(S2) quasi_total finite Element of bool [:S2,{0,1}:]
c " {0} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (c " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,c,0) is Element of bool (union (rng S1))
bool (union (rng S1)) is non empty set
card (S1,c,0) is V4() V5() V6() cardinal set
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 is Relation-like Function-like finite-yielding set
dom S2 is set
(S1,S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(X,S1,0,1) is functional finite Element of bool (Funcs (X,{0,1}))
{0,1} is non empty finite V60() set
Funcs (X,{0,1}) is non empty functional finite FUNCTION_DOMAIN of X,{0,1}
bool (Funcs (X,{0,1})) is non empty finite V60() set
card (X,S1,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (X,S1,0,1)),(X,S1,0,1):] is Relation-like finite set
bool [:(card (X,S1,0,1)),(X,S1,0,1):] is non empty finite V60() set
h is Relation-like card (X,S1,0,1) -defined (X,S1,0,1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (X,S1,0,1)),(X,S1,0,1):]
dom h is finite Element of bool (card (X,S1,0,1))
bool (card (X,S1,0,1)) is non empty finite V60() set
y is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
X is Relation-like Function-like finite-yielding set
dom X is set
(1,X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card S1),S1:] is Relation-like finite set
bool [:(card S1),S1:] is non empty finite V60() set
S2 is Relation-like card S1 -defined S1 -valued Function-like quasi_total finite Element of bool [:(card S1),S1:]
S2 (#) X is Relation-like card S1 -defined Function-like finite finite-yielding set
rng {} is empty trivial proper V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V36() V37() V38() V39() V42() V43() V44() V45() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V201() V205() Element of bool RAT
bool RAT is non empty non trivial non finite set
x is V8() Relation-like NAT -defined Function-like finite V205() set
h is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
y is set
h . y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 (#) X) . y is finite set
card ((S2 (#) X) . y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is non empty set
[:x,NAT:] is non empty non trivial Relation-like non finite set
bool [:x,NAT:] is non empty non trivial non finite set
h is non empty Relation-like x -defined NAT -valued Function-like V25(x) quasi_total V32() V33() V34() V35() V52() Element of bool [:x,NAT:]
dom h is non empty Element of bool x
bool x is non empty set
y is V8() Relation-like NAT -defined Function-like finite V205() set
c is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
F1 is set
c . F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 (#) X) . F1 is finite set
card ((S2 (#) X) . F1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,1,0,1) is functional finite Element of bool (Funcs (S1,{0,1}))
{0,1} is non empty finite V60() set
Funcs (S1,{0,1}) is non empty functional finite FUNCTION_DOMAIN of S1,{0,1}
bool (Funcs (S1,{0,1})) is non empty finite V60() set
F1 is set
S2 . F1 is set
(S2 . F1) .--> 0 is trivial Relation-like {(S2 . F1)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite Function-yielding V130() set
{(S2 . F1)} is non empty trivial finite 1 -element set
{(S2 . F1)} --> 0 is non empty Relation-like {(S2 . F1)} -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant V25({(S2 . F1)}) quasi_total V32() V33() V34() V35() V52() finite Function-yielding V130() Element of bool [:{(S2 . F1)},{0}:]
{0} is non empty trivial functional finite V60() 1 -element set
[:{(S2 . F1)},{0}:] is non empty Relation-like finite set
bool [:{(S2 . F1)},{0}:] is non empty finite V60() set
S1 \ {(S2 . F1)} is finite Element of bool S1
bool S1 is non empty finite V60() set
(S1 \ {(S2 . F1)}) --> 1 is Relation-like non-empty S1 \ {(S2 . F1)} -defined NAT -valued RAT -valued INT -valued Function-like constant V25(S1 \ {(S2 . F1)}) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(S1 \ {(S2 . F1)}),NAT:]
[:(S1 \ {(S2 . F1)}),NAT:] is Relation-like set
bool [:(S1 \ {(S2 . F1)}),NAT:] is non empty set
{1} is non empty trivial finite V60() 1 -element set
[:(S1 \ {(S2 . F1)}),{1}:] is Relation-like finite set
((S2 . F1) .--> 0) +* ((S1 \ {(S2 . F1)}) --> 1) is Relation-like NAT -valued RAT -valued INT -valued Function-like V32() V33() V34() V35() V52() finite set
dom S2 is finite Element of bool (card S1)
bool (card S1) is non empty finite V60() set
rng S2 is finite Element of bool S1
{(S2 . F1)} \/ (S1 \ {(S2 . F1)}) is non empty finite set
card {(S2 . F1)} is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
[:x,(S1,1,0,1):] is Relation-like set
bool [:x,(S1,1,0,1):] is non empty set
F1 is Relation-like x -defined (S1,1,0,1) -valued Function-like quasi_total Function-yielding V130() Element of bool [:x,(S1,1,0,1):]
rng F1 is functional finite Element of bool (S1,1,0,1)
bool (S1,1,0,1) is non empty finite V60() set
card (card S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
[:S1,{0,1}:] is Relation-like finite set
bool [:S1,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
XF is Relation-like S1 -defined {0,1} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{0,1}:]
XF " {0} is finite Element of bool S1
bool S1 is non empty finite V60() set
card (XF " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is set
{n} is non empty trivial finite 1 -element set
rng S2 is finite Element of bool S1
dom S2 is finite Element of bool (card S1)
bool (card S1) is non empty finite V60() set
n1 is set
S2 . n1 is set
F1 . n1 is Relation-like Function-like set
{(S2 . n1)} is non empty trivial finite 1 -element set
S1 \ {(S2 . n1)} is finite Element of bool S1
(S1 \ {(S2 . n1)}) --> 1 is Relation-like non-empty S1 \ {(S2 . n1)} -defined NAT -valued RAT -valued INT -valued Function-like constant V25(S1 \ {(S2 . n1)}) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(S1 \ {(S2 . n1)}),NAT:]
[:(S1 \ {(S2 . n1)}),NAT:] is Relation-like set
bool [:(S1 \ {(S2 . n1)}),NAT:] is non empty set
{1} is non empty trivial finite V60() 1 -element set
[:(S1 \ {(S2 . n1)}),{1}:] is Relation-like finite set
(S2 . n1) .--> 0 is trivial Relation-like {(S2 . n1)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite Function-yielding V130() set
{(S2 . n1)} --> 0 is non empty Relation-like {(S2 . n1)} -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant V25({(S2 . n1)}) quasi_total V32() V33() V34() V35() V52() finite Function-yielding V130() Element of bool [:{(S2 . n1)},{0}:]
{0} is non empty trivial functional finite V60() 1 -element set
[:{(S2 . n1)},{0}:] is non empty Relation-like finite set
bool [:{(S2 . n1)},{0}:] is non empty finite V60() set
((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1) is Relation-like NAT -valued RAT -valued INT -valued Function-like V32() V33() V34() V35() V52() finite set
{(S2 . n1)} \/ (S1 \ {(S2 . n1)}) is non empty finite set
Fyy is set
dom XF is finite Element of bool S1
dom ((S2 . n1) .--> 0) is trivial finite Element of bool {(S2 . n1)}
bool {(S2 . n1)} is non empty finite V60() set
dom ((S1 \ {(S2 . n1)}) --> 1) is finite Element of bool (S1 \ {(S2 . n1)})
bool (S1 \ {(S2 . n1)}) is non empty finite V60() set
(((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1)) . Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
((S2 . n1) .--> 0) . Fyy is V4() V5() V6() V10() V11() ext-real non negative Relation-like Function-like V47() V53() integer finite cardinal V197() Element of NAT
XF . Fyy is set
dom ((S1 \ {(S2 . n1)}) --> 1) is finite Element of bool (S1 \ {(S2 . n1)})
bool (S1 \ {(S2 . n1)}) is non empty finite V60() set
(((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1)) . Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
((S1 \ {(S2 . n1)}) --> 1) . Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
XF . Fyy is set
rng XF is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
(((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1)) . Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
XF . Fyy is set
(((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1)) . Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
XF . Fyy is set
dom ((S2 . n1) .--> 0) is trivial finite Element of bool {(S2 . n1)}
bool {(S2 . n1)} is non empty finite V60() set
dom ((S1 \ {(S2 . n1)}) --> 1) is finite Element of bool (S1 \ {(S2 . n1)})
bool (S1 \ {(S2 . n1)}) is non empty finite V60() set
(dom ((S2 . n1) .--> 0)) \/ (dom ((S1 \ {(S2 . n1)}) --> 1)) is finite set
dom (((S2 . n1) .--> 0) +* ((S1 \ {(S2 . n1)}) --> 1)) is finite set
card (S1,1,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose 1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom F1 is Element of bool x
card (S1,1,0,1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose 1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (S1,1,0,1)),(S1,1,0,1):] is Relation-like finite set
bool [:(card (S1,1,0,1)),(S1,1,0,1):] is non empty finite V60() set
card (card S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is Relation-like card (S1,1,0,1) -defined (S1,1,0,1) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S1,1,0,1)),(S1,1,0,1):]
dom F1 is finite Element of bool (card (S1,1,0,1))
bool (card (S1,1,0,1)) is non empty finite V60() set
XF is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
n is set
XF . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 . n is set
(S2 . n) .--> 0 is trivial Relation-like {(S2 . n)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite Function-yielding V130() set
{(S2 . n)} is non empty trivial finite 1 -element set
{(S2 . n)} --> 0 is non empty Relation-like {(S2 . n)} -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant V25({(S2 . n)}) quasi_total V32() V33() V34() V35() V52() finite Function-yielding V130() Element of bool [:{(S2 . n)},{0}:]
{0} is non empty trivial functional finite V60() 1 -element set
[:{(S2 . n)},{0}:] is non empty Relation-like finite set
bool [:{(S2 . n)},{0}:] is non empty finite V60() set
S1 \ {(S2 . n)} is finite Element of bool S1
bool S1 is non empty finite V60() set
(S1 \ {(S2 . n)}) --> 1 is Relation-like non-empty S1 \ {(S2 . n)} -defined NAT -valued RAT -valued INT -valued Function-like constant V25(S1 \ {(S2 . n)}) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(S1 \ {(S2 . n)}),NAT:]
[:(S1 \ {(S2 . n)}),NAT:] is Relation-like set
bool [:(S1 \ {(S2 . n)}),NAT:] is non empty set
{1} is non empty trivial finite V60() 1 -element set
[:(S1 \ {(S2 . n)}),{1}:] is Relation-like finite set
((S2 . n) .--> 0) +* ((S1 \ {(S2 . n)}) --> 1) is Relation-like NAT -valued RAT -valued INT -valued Function-like V32() V33() V34() V35() V52() finite set
[:S1,{0,1}:] is Relation-like finite set
bool [:S1,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
n1 is Relation-like S1 -defined {0,1} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{0,1}:]
n1 " {0} is finite Element of bool S1
card (n1 " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
u is set
{u} is non empty trivial finite 1 -element set
F1 . n is Relation-like Function-like set
(X,n1,0) is Element of bool (union (rng X))
rng X is V60() set
union (rng X) is set
bool (union (rng X)) is non empty set
card (X,n1,0) is V4() V5() V6() cardinal set
dom ((S1 \ {(S2 . n)}) --> 1) is finite Element of bool (S1 \ {(S2 . n)})
bool (S1 \ {(S2 . n)}) is non empty finite V60() set
dom ((S2 . n) .--> 0) is trivial finite Element of bool {(S2 . n)}
bool {(S2 . n)} is non empty finite V60() set
(dom ((S2 . n) .--> 0)) \/ (dom ((S1 \ {(S2 . n)}) --> 1)) is finite set
((S2 . n) .--> 0) . (S2 . n) is V4() V5() V6() V10() V11() ext-real non negative Relation-like Function-like V47() V53() integer finite cardinal V197() Element of NAT
H₂(n) . (S2 . n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of NAT
dom H₂(n) is finite set
X . (S2 . n) is finite set
card (X . (S2 . n)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 (#) X) . n is finite set
card ((S2 (#) X) . n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom S2 is finite Element of bool (card S1)
bool (card S1) is non empty finite V60() set
X is set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 --> X is Relation-like S1 -defined {X} -valued Function-like constant V25(S1) quasi_total finite Element of bool [:S1,{X}:]
{X} is non empty trivial finite 1 -element set
[:S1,{X}:] is Relation-like finite set
bool [:S1,{X}:] is non empty finite V60() set
S2 is Relation-like Function-like finite-yielding set
dom S2 is set
((card S1),S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2,(S1 --> X),X) is Element of bool (union (rng S2))
rng S2 is V60() set
union (rng S2) is set
bool (union (rng S2)) is non empty set
card (S2,(S1 --> X),X) is V4() V5() V6() cardinal set
(S1,(card S1),X,{X}) is functional finite Element of bool (Funcs (S1,{X,{X}}))
{X,{X}} is non empty finite set
Funcs (S1,{X,{X}}) is non empty functional finite FUNCTION_DOMAIN of S1,{X,{X}}
bool (Funcs (S1,{X,{X}})) is non empty finite V60() set
card (S1,(card S1),X,{X}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (S1,(card S1),X,{X})),(S1,(card S1),X,{X}):] is Relation-like finite set
bool [:(card (S1,(card S1),X,{X})),(S1,(card S1),X,{X}):] is non empty finite V60() set
h is Relation-like card (S1,(card S1),X,{X}) -defined (S1,(card S1),X,{X}) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S1,(card S1),X,{X})),(S1,(card S1),X,{X}):]
dom h is finite Element of bool (card (S1,(card S1),X,{X}))
bool (card (S1,(card S1),X,{X})) is non empty finite V60() set
y is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
c is set
{c} is non empty trivial finite 1 -element set
S1 \/ {} is finite set
{} --> {X} is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding {} -defined RAT -valued {{X}} -valued Function-like one-to-one constant functional V25( {} ) quasi_total V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of bool [:{},{{X}}:]
{{X}} is non empty trivial finite V60() 1 -element set
[:{},{{X}}:] is Relation-like finite set
bool [:{},{{X}}:] is non empty finite V60() set
({} --> {X}) +* (S1 --> X) is Relation-like Function-like finite set
{} +* (S1 --> X) is Relation-like Function-like finite set
h . 0 is Relation-like Function-like set
rng h is functional finite Element of bool (S1,(card S1),X,{X})
bool (S1,(card S1),X,{X}) is non empty finite V60() set
len y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y . 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
<%(y . 0)%> is non empty trivial V8() Relation-like NAT -defined NAT -valued Function-like constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite 1 -element V205() set
addnat "**" y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is finite set
X .--> S1 is trivial Relation-like {X} -defined Function-like one-to-one constant finite set
{X} is non empty trivial finite 1 -element set
{X} --> S1 is non empty Relation-like {X} -defined {S1} -valued Function-like constant V25({X}) quasi_total finite Element of bool [:{X},{S1}:]
{S1} is non empty trivial finite V60() 1 -element set
[:{X},{S1}:] is non empty Relation-like finite set
bool [:{X},{S1}:] is non empty finite V60() set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is Relation-like Function-like finite-yielding set
(1,S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom S2 is set
X .--> X is trivial Relation-like {X} -defined Function-like one-to-one constant finite set
{X} --> X is non empty Relation-like {X} -defined {X} -valued Function-like constant V25({X}) quasi_total finite Element of bool [:{X},{X}:]
[:{X},{X}:] is non empty Relation-like finite set
bool [:{X},{X}:] is non empty finite V60() set
(X .--> X) " {X} is finite set
S2 . X is finite set
card {X} is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(S2,(X .--> X),X) is Element of bool (union (rng S2))
rng S2 is V60() set
union (rng S2) is set
bool (union (rng S2)) is non empty set
X is set
S1 is set
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is finite set
(X,S1) --> (S2,x) is Relation-like Function-like finite set
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S2) + (card x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 /\ x is finite set
card (S2 /\ x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is Relation-like Function-like finite-yielding set
(1,h) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(2,h) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(0,1) --> (X,S1) is Relation-like NAT -defined Function-like finite set
dom ((0,1) --> (X,S1)) is V47() finite Element of bool NAT
rng ((0,1) --> (X,S1)) is finite set
{X,S1} is non empty finite set
card {X,S1} is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
[:(card {X,S1}),{X,S1}:] is non empty Relation-like finite set
bool [:(card {X,S1}),{X,S1}:] is non empty finite V60() set
card (card {X,S1}) is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
c is non empty Relation-like card {X,S1} -defined {X,S1} -valued Function-like V25( card {X,S1}) quasi_total finite Element of bool [:(card {X,S1}),{X,S1}:]
c . 0 is set
h . X is finite set
c . 1 is set
h . S1 is finite set
dom h is set
rng c is non empty finite Element of bool {X,S1}
bool {X,S1} is non empty finite V60() set
c (#) h is Relation-like card {X,S1} -defined Function-like finite finite-yielding set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
len F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 . 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . 1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
<%(F1 . 0),(F1 . 1)%> is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite 2 -element V205() set
dom c is non empty finite Element of bool (card {X,S1})
bool (card {X,S1}) is non empty finite V60() set
(c (#) h) . 1 is finite set
h . (c . 1) is finite set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
{X,S1} --> 0 is non empty Relation-like {X,S1} -defined NAT -valued RAT -valued INT -valued Function-like constant V25({X,S1}) quasi_total V32() V33() V34() V35() V52() finite Function-yielding V130() Element of bool [:{X,S1},NAT:]
[:{X,S1},NAT:] is non empty non trivial Relation-like non finite set
bool [:{X,S1},NAT:] is non empty non trivial non finite set
{0} is non empty trivial functional finite V60() 1 -element set
[:{X,S1},{0}:] is non empty Relation-like finite set
({X,S1} --> 0) " {0} is finite Element of bool {X,S1}
(h,({X,S1} --> 0),0) is Element of bool (union (rng h))
rng h is V60() set
union (rng h) is set
bool (union (rng h)) is non empty set
(c (#) h) . 0 is finite set
h . (c . 0) is finite set
addnat "**" F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
addnat . ((card S2),(card x)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is Relation-like Function-like finite-yielding set
dom X is set
(1,X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is set
{S1} is non empty trivial finite 1 -element set
(dom X) \ {S1} is Element of bool (dom X)
bool (dom X) is non empty set
X | ((dom X) \ {S1}) is Relation-like Function-like finite-yielding set
(1,(X | ((dom X) \ {S1}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X . S1 is finite set
card (X . S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(1,(X | ((dom X) \ {S1}))) + (card (X . S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S2) - 1 is V11() ext-real V53() integer Element of INT
S2 \ {S1} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (S2 \ {S1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (S2 \ {S1})),(S2 \ {S1}):] is Relation-like finite set
bool [:(card (S2 \ {S1})),(S2 \ {S1}):] is non empty finite V60() set
y is Relation-like card (S2 \ {S1}) -defined S2 \ {S1} -valued Function-like quasi_total finite Element of bool [:(card (S2 \ {S1})),(S2 \ {S1}):]
{(card (S2 \ {S1}))} is non empty trivial finite V60() 1 -element Element of bool NAT
(card (S2 \ {S1})) \/ {(card (S2 \ {S1}))} is non empty finite set
(S2 \ {S1}) \/ {S1} is non empty finite set
[:((card (S2 \ {S1})) \/ {(card (S2 \ {S1}))}),((S2 \ {S1}) \/ {S1}):] is non empty Relation-like finite set
bool [:((card (S2 \ {S1})) \/ {(card (S2 \ {S1}))}),((S2 \ {S1}) \/ {S1}):] is non empty finite V60() set
c is non empty Relation-like (card (S2 \ {S1})) \/ {(card (S2 \ {S1}))} -defined (S2 \ {S1}) \/ {S1} -valued Function-like V25((card (S2 \ {S1})) \/ {(card (S2 \ {S1}))}) quasi_total finite Element of bool [:((card (S2 \ {S1})) \/ {(card (S2 \ {S1}))}),((S2 \ {S1}) \/ {S1}):]
c | (card (S2 \ {S1})) is Relation-like card (S2 \ {S1}) -defined (card (S2 \ {S1})) \/ {(card (S2 \ {S1}))} -defined (S2 \ {S1}) \/ {S1} -valued Function-like finite Element of bool [:((card (S2 \ {S1})) \/ {(card (S2 \ {S1}))}),((S2 \ {S1}) \/ {S1}):]
c . (card (S2 \ {S1})) is set
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
[:(card S2),S2:] is Relation-like finite set
bool [:(card S2),S2:] is non empty finite V60() set
F1 is Relation-like card S2 -defined S2 -valued Function-like quasi_total finite Element of bool [:(card S2),S2:]
F1 (#) X is Relation-like card S2 -defined Function-like finite finite-yielding set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
F1 . x is set
S2 /\ (S2 \ {S1}) is finite Element of bool S2
X | (S2 \ {S1}) is Relation-like Function-like finite finite-yielding set
dom (X | (S2 \ {S1})) is finite set
y (#) (X | (S2 \ {S1})) is Relation-like card (S2 \ {S1}) -defined Function-like finite finite-yielding set
(1,(X | (S2 \ {S1}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
n is set
F1 . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(F1 (#) X) . n is finite set
card ((F1 (#) X) . n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is finite Element of bool (card (S2 \ {S1}))
bool (card (S2 \ {S1})) is non empty finite V60() set
y . n is set
rng y is finite Element of bool (S2 \ {S1})
bool (S2 \ {S1}) is non empty finite V60() set
(X | (S2 \ {S1})) . (y . n) is finite set
X . (y . n) is finite set
dom F1 is finite Element of bool (card S2)
bool (card S2) is non empty finite V60() set
F1 . n is set
X . (F1 . n) is finite set
(y (#) (X | (S2 \ {S1}))) . n is finite set
dom F1 is finite Element of bool (card S2)
bool (card S2) is non empty finite V60() set
(F1 (#) X) . x is finite set
X . (F1 . x) is finite set
(dom F1) /\ x is finite Element of bool NAT
F1 | x is V8() Relation-like x -defined NAT -defined NAT -valued RAT -valued rng F1 -valued Function-like V32() V33() V34() V35() V52() finite V205() set
rng F1 is V42() V43() V44() V47() finite set
F1 . x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(Sum XF) + (F1 . x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 | (x + 1) is V8() Relation-like x + 1 -defined NAT -defined NAT -valued RAT -valued rng F1 -valued Function-like V32() V33() V34() V35() V52() finite V205() set
Sum (F1 | (x + 1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal V197() Element of COMPLEX
card ((F1 (#) X) . x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
S1 is Relation-like Function-like set
dom S1 is set
(dom S1) --> X is Relation-like dom S1 -defined {X} -valued Function-like constant V25( dom S1) quasi_total Element of bool [:(dom S1),{X}:]
{X} is non empty trivial finite 1 -element set
[:(dom S1),{X}:] is Relation-like set
bool [:(dom S1),{X}:] is non empty set
Intersect (S1,((dom S1) --> X)) is Relation-like Function-like set
dom (Intersect (S1,((dom S1) --> X))) is set
dom ((dom S1) --> X) is Element of bool (dom S1)
bool (dom S1) is non empty set
(dom S1) /\ (dom ((dom S1) --> X)) is Element of bool (dom S1)
S2 is set
(Intersect (S1,((dom S1) --> X))) . S2 is set
S1 . S2 is set
(S1 . S2) /\ X is set
((dom S1) --> X) . S2 is set
(S1 . S2) /\ (((dom S1) --> X) . S2) is set
X is set
S1 is Relation-like Function-like set
rng S1 is set
union (rng S1) is set
(union (rng S1)) /\ X is set
dom S1 is set
(dom S1) --> X is Relation-like dom S1 -defined {X} -valued Function-like constant V25( dom S1) quasi_total Element of bool [:(dom S1),{X}:]
{X} is non empty trivial finite 1 -element set
[:(dom S1),{X}:] is Relation-like set
bool [:(dom S1),{X}:] is non empty set
Intersect (S1,((dom S1) --> X)) is Relation-like Function-like set
rng (Intersect (S1,((dom S1) --> X))) is set
union (rng (Intersect (S1,((dom S1) --> X)))) is set
x is set
h is set
y is set
S1 . y is set
dom (Intersect (S1,((dom S1) --> X))) is set
(Intersect (S1,((dom S1) --> X))) . y is set
h /\ X is set
x is set
h is set
dom (Intersect (S1,((dom S1) --> X))) is set
y is set
(Intersect (S1,((dom S1) --> X))) . y is set
S1 . y is set
(S1 . y) /\ X is set
X is set
S1 is set
S2 is Relation-like Function-like set
rng S2 is set
union (rng S2) is set
dom S2 is set
(dom S2) --> S1 is Relation-like dom S2 -defined {S1} -valued Function-like constant V25( dom S2) quasi_total Element of bool [:(dom S2),{S1}:]
{S1} is non empty trivial finite 1 -element set
[:(dom S2),{S1}:] is Relation-like set
bool [:(dom S2),{S1}:] is non empty set
Intersect (S2,((dom S2) --> S1)) is Relation-like Function-like set
x is Relation-like Function-like set
(S2,x,X) is Element of bool (union (rng S2))
bool (union (rng S2)) is non empty set
(S2,x,X) /\ S1 is Element of bool (union (rng S2))
((Intersect (S2,((dom S2) --> S1))),x,X) is Element of bool (union (rng (Intersect (S2,((dom S2) --> S1)))))
rng (Intersect (S2,((dom S2) --> S1))) is set
union (rng (Intersect (S2,((dom S2) --> S1)))) is set
bool (union (rng (Intersect (S2,((dom S2) --> S1))))) is non empty set
F1 is set
dom x is set
F1 is set
x . F1 is set
(Intersect (S2,((dom S2) --> S1))) . F1 is set
S2 . F1 is set
(S2 . F1) /\ S1 is set
(union (rng S2)) /\ S1 is set
F1 is set
(union (rng S2)) /\ S1 is set
dom x is set
dom (Intersect (S2,((dom S2) --> S1))) is set
F1 is set
x . F1 is set
S2 . F1 is set
(Intersect (S2,((dom S2) --> S1))) . F1 is set
(S2 . F1) /\ S1 is set
X is V8() Relation-like NAT -defined Function-like finite V205() set
S1 is V8() Relation-like NAT -defined Function-like finite V205() set
rng X is finite set
rng S1 is finite set
X ^ S1 is V8() Relation-like NAT -defined Function-like finite V205() set
len X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
card (len X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (rng X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
len S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
card (len S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (rng S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom (X ^ S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
rng (X ^ S1) is finite set
[:(dom (X ^ S1)),(rng (X ^ S1)):] is Relation-like finite set
bool [:(dom (X ^ S1)),(rng (X ^ S1)):] is non empty finite V60() set
(len X) + (len S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is Relation-like dom (X ^ S1) -defined rng (X ^ S1) -valued Function-like quasi_total finite Element of bool [:(dom (X ^ S1)),(rng (X ^ S1)):]
(rng X) \/ (rng S1) is finite set
card ((rng X) \/ (rng S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (rng X)) + (card (rng S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (dom (X ^ S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (rng (X ^ S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
X + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
S1 is Relation-like Function-like finite-yielding set
dom S1 is set
((X + 1),S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is finite set
x is set
{x} is non empty trivial finite 1 -element set
(dom S1) \ {x} is Element of bool (dom S1)
bool (dom S1) is non empty set
S1 | ((dom S1) \ {x}) is Relation-like Function-like finite-yielding set
((X + 1),(S1 | ((dom S1) \ {x}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 . x is finite set
((dom S1) \ {x}) --> (S1 . x) is Relation-like (dom S1) \ {x} -defined {(S1 . x)} -valued Function-like constant V25((dom S1) \ {x}) quasi_total Element of bool [:((dom S1) \ {x}),{(S1 . x)}:]
{(S1 . x)} is non empty trivial finite V60() 1 -element set
[:((dom S1) \ {x}),{(S1 . x)}:] is Relation-like set
bool [:((dom S1) \ {x}),{(S1 . x)}:] is non empty set
Intersect ((S1 | ((dom S1) \ {x})),(((dom S1) \ {x}) --> (S1 . x))) is Relation-like Function-like finite-yielding set
(X,(Intersect ((S1 | ((dom S1) \ {x})),(((dom S1) \ {x}) --> (S1 . x))))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((X + 1),(S1 | ((dom S1) \ {x}))) + (X,(Intersect ((S1 | ((dom S1) \ {x})),(((dom S1) \ {x}) --> (S1 . x))))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 \ {x} is finite Element of bool S2
bool S2 is non empty finite V60() set
(S2 \ {x}) \/ {x} is non empty finite set
{1,0} is non empty finite V60() Element of bool NAT
[:((S2 \ {x}) \/ {x}),{1,0}:] is non empty Relation-like finite set
bool [:((S2 \ {x}) \/ {x}),{1,0}:] is non empty finite V60() set
{1} is non empty trivial finite V60() 1 -element Element of bool NAT
{ b₁ where b₁ is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:] : ( card (b₁ " {1}) = X + 1 & b₁ . x = 1 ) } is set
{ b₁ where b₁ is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:] : ( card (b₁ " {1}) = X + 1 & b₁ . x = 0 ) } is set
{ b₁ where b₁ is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:] : ( card (b₁ " {1}) = X + 1 & b₁ . x = 0 ) } \/ { b₁ where b₁ is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:] : ( card (b₁ " {1}) = X + 1 & b₁ . x = 1 ) } is set
(((S2 \ {x}) \/ {x}),(X + 1),1,0) is functional finite Element of bool (Funcs (((S2 \ {x}) \/ {x}),{1,0}))
{1,0} is non empty finite V60() set
Funcs (((S2 \ {x}) \/ {x}),{1,0}) is non empty functional finite FUNCTION_DOMAIN of (S2 \ {x}) \/ {x},{1,0}
bool (Funcs (((S2 \ {x}) \/ {x}),{1,0})) is non empty finite V60() set
n is finite set
card n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card n),n:] is Relation-like finite set
bool [:(card n),n:] is non empty finite V60() set
n1 is Relation-like card n -defined n -valued Function-like quasi_total finite Element of bool [:(card n),n:]
((S2 \ {x}),X,1,0) is functional finite Element of bool (Funcs ((S2 \ {x}),{1,0}))
Funcs ((S2 \ {x}),{1,0}) is non empty functional finite FUNCTION_DOMAIN of S2 \ {x},{1,0}
bool (Funcs ((S2 \ {x}),{1,0})) is non empty finite V60() set
card ((S2 \ {x}),X,1,0) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((S2 \ {x}) \/ {x}) \ {x} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
u is set
n1 . u is set
dom n1 is finite Element of bool (card n)
bool (card n) is non empty finite V60() set
rng n1 is finite Element of bool n
bool n is non empty finite V60() set
n is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
n " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
card (n " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n . x is set
dom n is non empty finite Element of bool ((S2 \ {x}) \/ {x})
n | (S2 \ {x}) is Relation-like S2 \ {x} -defined (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
rng (n | (S2 \ {x})) is finite V60() Element of bool {1,0}
bool {1,0} is non empty finite V60() set
((S2 \ {x}) \/ {x}) /\ (S2 \ {x}) is finite Element of bool S2
dom (n | (S2 \ {x})) is finite Element of bool ((S2 \ {x}) \/ {x})
[:(S2 \ {x}),{1,0}:] is Relation-like finite set
bool [:(S2 \ {x}),{1,0}:] is non empty finite V60() set
c is Relation-like S2 \ {x} -defined {1,0} -valued Function-like V25(S2 \ {x}) quasi_total finite Element of bool [:(S2 \ {x}),{1,0}:]
c " {1} is finite Element of bool (S2 \ {x})
bool (S2 \ {x}) is non empty finite V60() set
(c " {1}) \/ {x} is non empty finite set
card (c " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (c " {1})) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
[:(card n),((S2 \ {x}),X,1,0):] is Relation-like finite set
bool [:(card n),((S2 \ {x}),X,1,0):] is non empty finite V60() set
u is Relation-like card n -defined ((S2 \ {x}),X,1,0) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card n),((S2 \ {x}),X,1,0):]
dom u is finite Element of bool (card n)
bool (card n) is non empty finite V60() set
n is set
u . n is Relation-like Function-like set
c is set
u . c is Relation-like Function-like set
n1 . c is set
Fyy is Relation-like Function-like set
Fyy | (S2 \ {x}) is Relation-like Function-like finite set
n1 . n is set
y is Relation-like Function-like set
y | (S2 \ {x}) is Relation-like Function-like finite set
dom y is set
h is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
h " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (h " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h . x is set
dom Fyy is set
h is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
h " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (h " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h . x is set
h is set
y . h is set
Fyy . h is set
(dom y) /\ (S2 \ {x}) is finite Element of bool S2
(y | (S2 \ {x})) . h is set
(dom Fyy) /\ (S2 \ {x}) is finite Element of bool S2
x is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
x " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (x " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x . x is set
y is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
y " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
card (y " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . x is set
dom n1 is finite Element of bool (card n)
(S2 \ {x}) /\ S2 is finite Element of bool S2
dom (S1 | ((dom S1) \ {x})) is set
dom (Intersect ((S1 | ((dom S1) \ {x})),(((dom S1) \ {x}) --> (S1 . x)))) is set
n is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
addnat "**" n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((S2 \ {x}),(X + 1),1,0) is functional finite Element of bool (Funcs ((S2 \ {x}),{1,0}))
card ((S2 \ {x}),(X + 1),1,0) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF is finite set
card XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2,(X + 1),1,0) is functional finite Element of bool (Funcs (S2,{1,0}))
Funcs (S2,{1,0}) is non empty functional finite FUNCTION_DOMAIN of S2,{1,0}
bool (Funcs (S2,{1,0})) is non empty finite V60() set
[:(card XF),XF:] is Relation-like finite set
bool [:(card XF),XF:] is non empty finite V60() set
Fyy is Relation-like card XF -defined XF -valued Function-like quasi_total finite Element of bool [:(card XF),XF:]
dom n1 is finite Element of bool (card n)
dom Fyy is finite Element of bool (card XF)
bool (card XF) is non empty finite V60() set
y is V8() Relation-like NAT -defined Function-like finite V205() set
len y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
x is set
Fyy . x is set
rng Fyy is finite Element of bool XF
bool XF is non empty finite V60() set
y is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
y " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (y " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . x is set
dom y is non empty finite Element of bool ((S2 \ {x}) \/ {x})
y | (S2 \ {x}) is Relation-like S2 \ {x} -defined (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
rng (y | (S2 \ {x})) is finite V60() Element of bool {1,0}
bool {1,0} is non empty finite V60() set
((S2 \ {x}) \/ {x}) /\ (S2 \ {x}) is finite Element of bool S2
dom (y | (S2 \ {x})) is finite Element of bool ((S2 \ {x}) \/ {x})
[:(S2 \ {x}),{1,0}:] is Relation-like finite set
bool [:(S2 \ {x}),{1,0}:] is non empty finite V60() set
((S2 \ {x}) \/ {x}) \ {x} is finite Element of bool ((S2 \ {x}) \/ {x})
y is Relation-like S2 \ {x} -defined {1,0} -valued Function-like V25(S2 \ {x}) quasi_total finite Element of bool [:(S2 \ {x}),{1,0}:]
y " {1} is finite Element of bool (S2 \ {x})
bool (S2 \ {x}) is non empty finite V60() set
[:(card XF),((S2 \ {x}),(X + 1),1,0):] is Relation-like finite set
bool [:(card XF),((S2 \ {x}),(X + 1),1,0):] is non empty finite V60() set
x is Relation-like card XF -defined ((S2 \ {x}),(X + 1),1,0) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card XF),((S2 \ {x}),(X + 1),1,0):]
rng Fyy is finite Element of bool XF
bool XF is non empty finite V60() set
rng n1 is finite Element of bool n
bool n is non empty finite V60() set
(rng Fyy) \/ (rng n1) is finite set
XF \/ n is finite set
h is V8() Relation-like NAT -defined Function-like finite V205() set
y ^ h is V8() Relation-like NAT -defined Function-like finite V205() set
rng (y ^ h) is finite set
len h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
dom x is finite Element of bool (card XF)
y is set
x . y is Relation-like Function-like set
y is set
x . y is Relation-like Function-like set
Fyy . y is set
x1 is Relation-like Function-like set
x1 | (S2 \ {x}) is Relation-like Function-like finite set
Fyy . y is set
x2 is Relation-like Function-like set
x2 | (S2 \ {x}) is Relation-like Function-like finite set
dom x2 is set
f299 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
f299 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (f299 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
f299 . x is set
dom x1 is set
f299 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
f299 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (f299 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
f299 . x is set
f299 is set
x2 . f299 is set
x1 . f299 is set
(dom x2) /\ (S2 \ {x}) is finite Element of bool S2
(x2 | (S2 \ {x})) . f299 is set
(dom x1) /\ (S2 \ {x}) is finite Element of bool S2
fx9 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx9 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (fx9 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx9 . x is set
fx1 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx1 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
card (fx1 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx1 . x is set
y is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
(card XF) + (card n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (S2,(X + 1),1,0) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom (y ^ h) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
[:(card (S2,(X + 1),1,0)),(S2,(X + 1),1,0):] is Relation-like finite set
bool [:(card (S2,(X + 1),1,0)),(S2,(X + 1),1,0):] is non empty finite V60() set
y is Relation-like card (S2,(X + 1),1,0) -defined (S2,(X + 1),1,0) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S2,(X + 1),1,0)),(S2,(X + 1),1,0):]
dom y is finite Element of bool (card (S2,(X + 1),1,0))
bool (card (S2,(X + 1),1,0)) is non empty finite V60() set
x1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom x1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum x1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
addnat "**" x1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
x1 . x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
Fyy . x2 is set
x . x2 is Relation-like Function-like set
f299 is Relation-like Function-like set
f299 | (S2 \ {x}) is Relation-like Function-like finite set
S1 | (S2 \ {x}) is Relation-like Function-like finite finite-yielding set
((S1 | (S2 \ {x})),(f299 | (S2 \ {x})),1) is finite Element of bool (union (rng (S1 | (S2 \ {x}))))
rng (S1 | (S2 \ {x})) is finite V60() set
union (rng (S1 | (S2 \ {x}))) is finite set
bool (union (rng (S1 | (S2 \ {x})))) is non empty finite V60() set
card ((S1 | (S2 \ {x})),(f299 | (S2 \ {x})),1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
f299 " {1} is set
fx9 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx9 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (fx9 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx9 . x is set
fx9 is set
f299 . fx9 is set
dom f299 is set
rng f299 is set
(dom y) + 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(dom y) + (dom n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx1 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx1 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (fx1 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx1 . x is set
(f299 | (S2 \ {x})) " {1} is finite set
fx1 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx1 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (fx1 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx1 . x is set
y . x2 is Relation-like Function-like set
y . x2 is set
(S1,f299,1) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,f299,1) is V4() V5() V6() cardinal set
dom (f299 | (S2 \ {x})) is finite set
((S1 | (S2 \ {x})),f299,1) is finite Element of bool (union (rng (S1 | (S2 \ {x}))))
len y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(len y) + x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x1 . ((len y) + x2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n . x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n1 . x2 is set
u . x2 is Relation-like Function-like set
f299 is Relation-like Function-like set
f299 | (S2 \ {x}) is Relation-like Function-like finite set
fx9 is non empty Relation-like (S2 \ {x}) \/ {x} -defined {1,0} -valued Function-like V25((S2 \ {x}) \/ {x}) quasi_total finite Element of bool [:((S2 \ {x}) \/ {x}),{1,0}:]
fx9 " {1} is finite Element of bool ((S2 \ {x}) \/ {x})
bool ((S2 \ {x}) \/ {x}) is non empty finite V60() set
card (fx9 " {1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
fx9 . x is set
S1 | (S2 \ {x}) is Relation-like Function-like finite finite-yielding set
(S2 \ {x}) --> (S1 . x) is Relation-like S2 \ {x} -defined {(S1 . x)} -valued Function-like constant V25(S2 \ {x}) quasi_total finite Element of bool [:(S2 \ {x}),{(S1 . x)}:]
[:(S2 \ {x}),{(S1 . x)}:] is Relation-like finite set
bool [:(S2 \ {x}),{(S1 . x)}:] is non empty finite V60() set
Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x))) is Relation-like Function-like finite-yielding set
((Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x)))),(f299 | (S2 \ {x})),1) is Element of bool (union (rng (Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x))))))
rng (Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x)))) is V60() set
union (rng (Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x))))) is set
bool (union (rng (Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x)))))) is non empty set
rng (S1 | (S2 \ {x})) is finite V60() set
union (rng (S1 | (S2 \ {x}))) is finite set
((S1 | (S2 \ {x})),(f299 | (S2 \ {x})),1) is finite Element of bool (union (rng (S1 | (S2 \ {x}))))
bool (union (rng (S1 | (S2 \ {x})))) is non empty finite V60() set
((S1 | (S2 \ {x})),(f299 | (S2 \ {x})),1) /\ (S1 . x) is finite Element of bool (union (rng (S1 | (S2 \ {x}))))
dom fx9 is non empty finite Element of bool ((S2 \ {x}) \/ {x})
dom f299 is set
rng (f299 | (S2 \ {x})) is finite set
(f299 | (S2 \ {x})) " {1} is finite set
dom (f299 | (S2 \ {x})) is finite set
(dom f299) /\ (S2 \ {x}) is finite Element of bool S2
fx1 is finite set
{x} \/ fx1 is non empty finite set
f299 " {1} is set
card fx1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card fx1) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
y is set
(f299 | (S2 \ {x})) . y is set
(dom y) + (dom n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(dom y) + x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x2 + (len y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y . (x2 + (len y)) is Relation-like Function-like set
x1 . ((dom y) + x2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,f299,1) is Element of bool (union (rng S1))
rng S1 is V60() set
union (rng S1) is set
bool (union (rng S1)) is non empty set
card (S1,f299,1) is V4() V5() V6() cardinal set
f299 . x is set
f299 " {1} is set
card ((Intersect ((S1 | (S2 \ {x})),((S2 \ {x}) --> (S1 . x)))),(f299 | (S2 \ {x})),1) is V4() V5() V6() cardinal set
(S1,(f299 | (S2 \ {x})),1) is Element of bool (union (rng S1))
(S1,(f299 | (S2 \ {x})),1) /\ (S1 . x) is finite Element of bool (union (rng S1))
card ((S1,(f299 | (S2 \ {x})),1) /\ (S1 . x)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
len n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(len y) + (len n) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y ^ n is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
addnat "**" y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
addnat . ((addnat "**" y),(addnat "**" n)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is set
{X} is non empty trivial finite 1 -element set
S1 is Relation-like Function-like set
dom S1 is set
rng S1 is set
union (rng S1) is set
(dom S1) \ {X} is Element of bool (dom S1)
bool (dom S1) is non empty set
S1 | ((dom S1) \ {X}) is Relation-like Function-like set
rng (S1 | ((dom S1) \ {X})) is set
union (rng (S1 | ((dom S1) \ {X}))) is set
S1 . X is set
(union (rng (S1 | ((dom S1) \ {X})))) \/ (S1 . X) is set
S1 | (dom S1) is Relation-like Function-like set
((dom S1) \ {X}) \/ {X} is non empty set
S1 | {X} is Relation-like Function-like finite set
(S1 | ((dom S1) \ {X})) \/ (S1 | {X}) is Relation-like set
rng (S1 | {X}) is finite set
(rng (S1 | ((dom S1) \ {X}))) \/ (rng (S1 | {X})) is set
Im (S1,X) is set
S1 .: {X} is finite set
{(S1 . X)} is non empty trivial finite 1 -element set
union {(S1 . X)} is set
(union (rng (S1 | ((dom S1) \ {X})))) \/ (union {(S1 . X)}) is set
- 1 is V11() ext-real non positive V53() integer Element of INT
X is Relation-like Function-like finite-yielding set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(- 1) |^ S2 is V11() ext-real V53() integer set
S2 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((S2 + 1),X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ S2) * ((S2 + 1),X) is V11() ext-real V53() integer Element of INT
x is V11() ext-real V53() integer Element of INT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(- 1) |^ h is V11() ext-real V53() integer set
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((h + 1),X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ h) * ((h + 1),X) is V11() ext-real V53() integer Element of INT
S2 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S2 . x is V11() ext-real V53() integer set
(- 1) |^ x is V11() ext-real V53() integer set
x + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((x + 1),X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ x) * ((x + 1),X) is V11() ext-real V53() integer Element of INT
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
X + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
S1 is Relation-like Function-like finite-yielding set
dom S1 is set
rng S1 is V60() set
union (rng S1) is set
card (union (rng S1)) is V4() V5() V6() cardinal set
S2 is finite set
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x is set
h is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum h is V11() ext-real V53() integer Element of COMPLEX
len h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h . 0 is V11() ext-real V53() integer set
<%(h . 0)%> is non empty trivial V8() Relation-like NAT -defined Function-like constant finite 1 -element V205() set
addint "**" h is V11() ext-real V53() integer Element of INT
(- 1) |^ 0 is V11() ext-real V53() integer set
0 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((0 + 1),S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ 0) * ((0 + 1),S1) is V11() ext-real V53() integer Element of INT
y is set
{y} is non empty trivial finite 1 -element set
S1 . y is finite set
{(S1 . y)} is non empty trivial finite V60() 1 -element set
y .--> (S1 . y) is trivial Relation-like {y} -defined Function-like one-to-one constant finite set
{y} --> (S1 . y) is non empty Relation-like {y} -defined {(S1 . y)} -valued Function-like constant V25({y}) quasi_total finite Element of bool [:{y},{(S1 . y)}:]
[:{y},{(S1 . y)}:] is non empty Relation-like finite set
bool [:{y},{(S1 . y)}:] is non empty finite V60() set
y is set
{y} is non empty trivial finite 1 -element set
S2 \ {y} is finite Element of bool S2
bool S2 is non empty finite V60() set
card (S2 \ {y}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 | (S2 \ {y}) is Relation-like Function-like finite finite-yielding set
rng (S1 | (S2 \ {y})) is finite V60() set
union (rng (S1 | (S2 \ {y}))) is finite set
S1 . y is finite set
dom (S1 | (S2 \ {y})) is finite set
(dom (S1 | (S2 \ {y}))) --> (S1 . y) is Relation-like dom (S1 | (S2 \ {y})) -defined {(S1 . y)} -valued Function-like constant V25( dom (S1 | (S2 \ {y}))) quasi_total finite Element of bool [:(dom (S1 | (S2 \ {y}))),{(S1 . y)}:]
{(S1 . y)} is non empty trivial finite V60() 1 -element set
[:(dom (S1 | (S2 \ {y}))),{(S1 . y)}:] is Relation-like finite set
bool [:(dom (S1 | (S2 \ {y}))),{(S1 . y)}:] is non empty finite V60() set
Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))) is Relation-like Function-like finite-yielding set
n1 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
F1 is finite set
F1 /\ (S1 . y) is finite set
rng (Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y)))) is V60() set
union (rng (Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is set
n is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
(- 1) (#) n is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
card (S1 . y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
<%(card (S1 . y))%> is non empty trivial V8() Relation-like NAT -defined NAT -valued Function-like constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite 1 -element V205() set
<%0%> is non empty trivial V8() Relation-like NAT -defined NAT -valued Function-like constant V32() V33() V34() V35() V36() V37() V38() V39() V52() finite 1 -element V205() set
Fyy is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
<%(card (S1 . y))%> ^ Fyy is non empty V8() Relation-like NAT -defined Function-like finite V205() set
n1 ^ <%0%> is non empty V8() Relation-like NAT -defined Function-like finite V205() set
h is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
addint "**" h is V11() ext-real V53() integer Element of INT
y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
addint "**" y is V11() ext-real V53() integer Element of INT
Sum n is V11() ext-real V53() integer Element of COMPLEX
(- 1) * (Sum n) is V11() ext-real V53() integer Element of INT
Sum Fyy is V11() ext-real V53() integer Element of COMPLEX
x is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
addint "**" x is V11() ext-real V53() integer Element of INT
addint "**" Fyy is V11() ext-real V53() integer Element of INT
addint . ((card (S1 . y)),(addint "**" Fyy)) is set
(card (S1 . y)) + (addint "**" Fyy) is V11() ext-real V53() integer Element of INT
(card (S1 . y)) + (Sum Fyy) is V11() ext-real V53() integer Element of INT
y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
addint "**" y is V11() ext-real V53() integer Element of INT
addint "**" n1 is V11() ext-real V53() integer Element of INT
addint . ((addint "**" n1),0) is set
(addint "**" n1) + 0 is V11() ext-real V53() integer Element of INT
Sum n1 is V11() ext-real V53() integer Element of COMPLEX
x ^ y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
Sum (x ^ y) is V11() ext-real V53() integer Element of COMPLEX
Sum x is V11() ext-real V53() integer Element of COMPLEX
Sum y is V11() ext-real V53() integer Element of COMPLEX
(Sum x) + (Sum y) is V11() ext-real V53() integer Element of INT
(addint "**" x) + (Sum y) is V11() ext-real V53() integer Element of INT
(card (S1 . y)) + ((- 1) * (Sum n)) is V11() ext-real V53() integer Element of INT
((card (S1 . y)) + ((- 1) * (Sum n))) + (Sum n1) is V11() ext-real V53() integer Element of INT
F1 \/ (S1 . y) is finite set
x is finite set
u is finite set
dom (Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y)))) is set
card u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
len <%(card (S1 . y))%> is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
dom <%(card (S1 . y))%> is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element set
len Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom Fyy is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
len x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h . y is V11() ext-real V53() integer set
x . y is V11() ext-real V53() integer set
y . y is V11() ext-real V53() integer set
addint . ((x . y),(y . y)) is set
y . 0 is V11() ext-real V53() integer set
n1 . 0 is V11() ext-real V53() integer set
(- 1) |^ 0 is V11() ext-real V53() integer set
0 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((0 + 1),(S1 | (S2 \ {y}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ 0) * ((0 + 1),(S1 | (S2 \ {y}))) is V11() ext-real V53() integer Element of INT
x . 0 is V11() ext-real V53() integer set
addint . ((x . 0),(y . 0)) is set
(card (S1 . y)) + ((0 + 1),(S1 | (S2 \ {y}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h . 0 is V11() ext-real V53() integer set
((0 + 1),S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ 0) * ((0 + 1),S1) is V11() ext-real V53() integer Element of INT
y - 1 is V11() ext-real V53() integer Element of INT
x2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x2 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
Fyy . x2 is V11() ext-real V53() integer set
n . x2 is V11() ext-real V53() integer set
(- 1) * (n . x2) is V11() ext-real V53() integer Element of INT
(- 1) |^ x2 is V11() ext-real V53() integer set
((x2 + 1),(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ x2) * ((x2 + 1),(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V11() ext-real V53() integer Element of INT
0 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(- 1) * ((- 1) |^ x2) is V11() ext-real V53() integer Element of INT
((- 1) * ((- 1) |^ x2)) * ((x2 + 1),(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V11() ext-real V53() integer Element of INT
(- 1) |^ (x2 + 1) is V11() ext-real V53() integer set
((- 1) |^ (x2 + 1)) * ((x2 + 1),(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V11() ext-real V53() integer Element of INT
(- 1) |^ y is V11() ext-real V53() integer set
y + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((y + 1),S1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ y) * ((y + 1),S1) is V11() ext-real V53() integer Element of INT
((y + 1),(S1 | (S2 \ {y}))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(x1,(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((y + 1),(S1 | (S2 \ {y}))) + (x1,(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ y) * ((y + 1),(S1 | (S2 \ {y}))) is V11() ext-real V53() integer Element of INT
((- 1) |^ y) * (x1,(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y))))) is V11() ext-real V53() integer Element of INT
(((- 1) |^ y) * ((y + 1),(S1 | (S2 \ {y})))) + (((- 1) |^ y) * (x1,(Intersect ((S1 | (S2 \ {y})),((dom (S1 | (S2 \ {y}))) --> (S1 . y)))))) is V11() ext-real V53() integer Element of INT
n1 . y is V11() ext-real V53() integer set
len n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
card F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(Sum n1) + (card (S1 . y)) is V11() ext-real V53() integer Element of INT
((Sum n1) + (card (S1 . y))) - (Sum n) is V11() ext-real V53() integer Element of INT
len <%0%> is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
dom <%0%> is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element set
len n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
len y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
len h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
addint "**" h is V11() ext-real V53() integer Element of INT
addint "**" (x ^ y) is V11() ext-real V53() integer Element of INT
X is Relation-like Function-like finite-yielding set
dom X is set
rng X is V60() set
union (rng X) is set
card (union (rng X)) is V4() V5() V6() cardinal set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum S2 is V11() ext-real V53() integer Element of COMPLEX
len S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
addint "**" S2 is V11() ext-real V53() integer Element of INT
X is Relation-like Function-like finite-yielding set
dom X is set
S1 is finite set
S2 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
rng X is V60() set
union (rng X) is set
card (union (rng X)) is V4() V5() V6() cardinal set
Sum S2 is V11() ext-real V53() integer Element of COMPLEX
X is Relation-like Function-like finite-yielding set
dom X is set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(x,X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 * ((card S1) choose x) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h is set
y is set
(S1,x,h,y) is functional finite Element of bool (Funcs (S1,{h,y}))
{h,y} is non empty finite set
Funcs (S1,{h,y}) is non empty functional finite FUNCTION_DOMAIN of S1,{h,y}
bool (Funcs (S1,{h,y})) is non empty finite V60() set
h is set
y is set
(S1,x,h,y) is functional finite Element of bool (Funcs (S1,{h,y}))
{h,y} is non empty finite set
Funcs (S1,{h,y}) is non empty functional finite FUNCTION_DOMAIN of S1,{h,y}
bool (Funcs (S1,{h,y})) is non empty finite V60() set
card (S1,x,h,y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card (S1,x,h,y)),(S1,x,h,y):] is Relation-like finite set
bool [:(card (S1,x,h,y)),(S1,x,h,y):] is non empty finite V60() set
F1 is Relation-like card (S1,x,h,y) -defined (S1,x,h,y) -valued Function-like quasi_total finite Function-yielding V130() Element of bool [:(card (S1,x,h,y)),(S1,x,h,y):]
dom F1 is finite Element of bool (card (S1,x,h,y))
bool (card (S1,x,h,y)) is non empty finite V60() set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
XF is set
F1 . XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 . XF is Relation-like Function-like set
rng F1 is functional finite Element of bool (S1,x,h,y)
bool (S1,x,h,y) is non empty finite V60() set
[:S1,{h,y}:] is Relation-like finite set
bool [:S1,{h,y}:] is non empty finite V60() set
{h} is non empty trivial finite 1 -element set
n is Relation-like S1 -defined {h,y} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{h,y}:]
n " {h} is finite Element of bool S1
bool S1 is non empty finite V60() set
card (n " {h}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(X,n,h) is Element of bool (union (rng X))
rng X is V60() set
union (rng X) is set
bool (union (rng X)) is non empty set
card (X,n,h) is V4() V5() V6() cardinal set
(dom F1) --> S2 is V8() Relation-like dom F1 -defined RAT -valued INT -valued {S2} -valued Function-like constant V25( dom F1) quasi_total V32() V33() V34() V35() V52() finite V205() Element of bool [:(dom F1),{S2}:]
{S2} is non empty trivial finite V60() 1 -element set
[:(dom F1),{S2}:] is Relation-like finite set
bool [:(dom F1),{S2}:] is non empty finite V60() set
rng F1 is V42() V43() V44() V47() finite Element of bool NAT
{0,S2} is non empty finite V60() set
F1 " {S2} is finite set
card (card (S1,x,h,y)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 * (card (card (S1,x,h,y))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X is Relation-like Function-like finite-yielding set
dom X is set
rng X is V60() set
union (rng X) is set
card (union (rng X)) is V4() V5() V6() cardinal set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
x is set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(- 1) |^ h is V11() ext-real V53() integer set
S2 . h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ h) * (S2 . h) is V11() ext-real V53() integer Element of INT
h + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose (h + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is V11() ext-real V53() integer set
c is V11() ext-real V53() integer set
y * c is V11() ext-real V53() integer Element of INT
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(- 1) |^ F1 is V11() ext-real V53() integer set
S2 . F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ F1) * (S2 . F1) is V11() ext-real V53() integer Element of INT
F1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose (F1 + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((- 1) |^ F1) * (S2 . F1)) * ((card S1) choose (F1 + 1)) is V11() ext-real V53() integer Element of INT
[:(card S1),INT:] is Relation-like set
bool [:(card S1),INT:] is non empty set
x is Relation-like card S1 -defined INT -valued Function-like V25( card S1) quasi_total V32() V33() V34() finite Element of bool [:(card S1),INT:]
dom x is finite Element of bool (card S1)
bool (card S1) is non empty finite V60() set
h is V8() Relation-like NAT -defined Function-like finite V205() set
y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V11() ext-real V53() integer Element of COMPLEX
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y . c is V11() ext-real V53() integer set
(- 1) |^ c is V11() ext-real V53() integer set
c + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
((c + 1),X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ c) * ((c + 1),X) is V11() ext-real V53() integer Element of INT
S2 . c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose (c + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S2 . c) * ((card S1) choose (c + 1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
F1 is set
(S1,(c + 1),F1,F1) is functional finite Element of bool (Funcs (S1,{F1,F1}))
{F1,F1} is non empty finite set
Funcs (S1,{F1,F1}) is non empty functional finite FUNCTION_DOMAIN of S1,{F1,F1}
bool (Funcs (S1,{F1,F1})) is non empty finite V60() set
((- 1) |^ c) * (S2 . c) is V11() ext-real V53() integer Element of INT
(((- 1) |^ c) * (S2 . c)) * ((card S1) choose (c + 1)) is V11() ext-real V53() integer Element of INT
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
y . c is V11() ext-real V53() integer set
(- 1) |^ c is V11() ext-real V53() integer set
S2 . c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ c) * (S2 . c) is V11() ext-real V53() integer Element of INT
c + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose (c + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((- 1) |^ c) * (S2 . c)) * ((card S1) choose (c + 1)) is V11() ext-real V53() integer Element of INT
X is finite set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) |^ (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is onto } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is onto } is V4() V5() V6() cardinal set
S2 is set
x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card S1) - (x + 1) is V11() ext-real V53() integer Element of INT
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h |^ (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(card S1) - y is V11() ext-real V53() integer Element of INT
((card S1) - y) - 1 is V11() ext-real V53() integer Element of INT
(((card S1) - y) - 1) |^ (card X) is V11() ext-real V53() integer set
[:(card S1),NAT:] is Relation-like set
bool [:(card S1),NAT:] is non empty set
S2 is Relation-like card S1 -defined NAT -valued Function-like V25( card S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(card S1),NAT:]
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not a₁ in rng b₁ } is set
Funcs (X,S1) is functional finite set
bool (Funcs (X,S1)) is non empty finite V60() set
h is set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not h in rng b₁ } is set
y is set
c is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
rng c is finite Element of bool S1
bool S1 is non empty finite V60() set
[:S1,(bool (Funcs (X,S1))):] is Relation-like finite set
bool [:S1,(bool (Funcs (X,S1))):] is non empty finite V60() set
h is Relation-like S1 -defined bool (Funcs (X,S1)) -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,(bool (Funcs (X,S1))):]
dom h is finite Element of bool S1
bool S1 is non empty finite V60() set
y is set
h . y is set
rng h is finite V60() Element of bool (bool (Funcs (X,S1)))
bool (bool (Funcs (X,S1))) is non empty finite V60() set
rng h is finite V60() Element of bool (bool (Funcs (X,S1)))
bool (bool (Funcs (X,S1))) is non empty finite V60() set
union (rng h) is finite set
union (bool (Funcs (X,S1))) is finite set
y is Relation-like Function-like finite-yielding set
rng y is V60() set
union (rng y) is set
dom S2 is finite Element of bool (card S1)
bool (card S1) is non empty finite V60() set
F1 is V8() Relation-like NAT -defined Function-like finite V205() set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
XF + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
F1 . XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1,(XF + 1),0,1) is functional finite Element of bool (Funcs (S1,{0,1}))
{0,1} is non empty finite V60() set
Funcs (S1,{0,1}) is non empty functional finite FUNCTION_DOMAIN of S1,{0,1}
bool (Funcs (S1,{0,1})) is non empty finite V60() set
n is Relation-like Function-like set
(y,n,0) is Element of bool (union (rng y))
bool (union (rng y)) is non empty set
card (y,n,0) is V4() V5() V6() cardinal set
[:S1,{0,1}:] is Relation-like finite set
bool [:S1,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
n1 is Relation-like S1 -defined {0,1} -valued Function-like V25(S1) quasi_total finite Element of bool [:S1,{0,1}:]
n1 " {0} is finite Element of bool S1
card (n1 " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(y,n1,0) is Element of bool (union (rng y))
S1 \ (n1 " {0}) is finite Element of bool S1
Funcs (X,(S1 \ (n1 " {0}))) is functional finite set
u is set
rng n1 is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
dom n1 is finite Element of bool S1
n is set
n1 . n is set
y . n is finite set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not n in rng b₁ } is set
c is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
rng c is finite Element of bool S1
Fyy is set
n1 . Fyy is set
y . Fyy is finite set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not Fyy in rng b₁ } is set
y is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
rng y is finite Element of bool S1
dom c is finite Element of bool X
bool X is non empty finite V60() set
card (S1 \ (n1 " {0})) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) - (XF + 1) is V11() ext-real V53() integer Element of INT
n is set
c is Relation-like Function-like set
dom c is set
rng c is set
y is set
Fyy is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not y in rng b₁ } is set
dom y is set
h . y is set
dom n1 is finite Element of bool S1
h is set
n1 . h is set
y . h is finite set
rng Fyy is finite Element of bool S1
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not h in rng b₁ } is set
u is finite set
card u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) - XF is V11() ext-real V53() integer Element of INT
((card S1) - XF) - 1 is V11() ext-real V53() integer Element of INT
(((card S1) - XF) - 1) |^ (card X) is V11() ext-real V53() integer set
(card S1) - XF is V11() ext-real V53() integer Element of INT
((card S1) - XF) - 1 is V11() ext-real V53() integer Element of INT
(((card S1) - XF) - 1) |^ (card X) is V11() ext-real V53() integer set
dom y is set
card (union (rng y)) is V4() V5() V6() cardinal set
XF is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum XF is V11() ext-real V53() integer Element of COMPLEX
((card S1) |^ (card X)) - (Sum XF) is V11() ext-real V53() integer Element of INT
c is finite set
(Funcs (X,S1)) \ c is functional finite Element of bool (Funcs (X,S1))
card ((Funcs (X,S1)) \ c) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (Funcs (X,S1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card (Funcs (X,S1))) - (card c) is V11() ext-real V53() integer Element of INT
(Funcs (X,S1)) \ (union (rng y)) is functional finite Element of bool (Funcs (X,S1))
n is set
n1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
u is set
n is set
y . n is finite set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not n in rng b₁ } is set
c is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
rng c is finite Element of bool S1
n is set
n1 is Relation-like Function-like set
dom n1 is set
rng n1 is set
u is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
rng u is finite Element of bool S1
n is set
h . n is set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : not n in rng b₁ } is set
n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
XF . n is V11() ext-real V53() integer set
(- 1) |^ n is V11() ext-real V53() integer set
n + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card S1) choose (n + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ n) * ((card S1) choose (n + 1)) is V11() ext-real V53() integer Element of INT
(card S1) - n is V11() ext-real V53() integer Element of INT
((card S1) - n) - 1 is V11() ext-real V53() integer Element of INT
(((card S1) - n) - 1) |^ (card X) is V11() ext-real V53() integer set
(((- 1) |^ n) * ((card S1) choose (n + 1))) * ((((card S1) - n) - 1) |^ (card X)) is V11() ext-real V53() integer Element of INT
F1 . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ n) * (F1 . n) is V11() ext-real V53() integer Element of INT
(((- 1) |^ n) * (F1 . n)) * ((card S1) choose (n + 1)) is V11() ext-real V53() integer Element of INT
X is finite set
S1 is finite set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is onto } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : b₁ is onto } is V4() V5() V6() cardinal set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) |^ (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S2 is V11() ext-real V53() integer Element of INT
<%S2%> is non empty trivial V8() Relation-like NAT -defined INT -valued Function-like constant V32() V33() V34() V36() V37() V38() V39() finite 1 -element V205() set
len <%S2%> is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
dom <%S2%> is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element set
h is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum h is V11() ext-real V53() integer Element of COMPLEX
((card S1) |^ (card X)) - (Sum h) is V11() ext-real V53() integer Element of INT
(- 1) (#) h is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
c is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
<%S2%> ^ c is non empty V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
F1 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum F1 is V11() ext-real V53() integer Element of COMPLEX
len c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(- 1) * (Sum h) is V11() ext-real V53() integer Element of INT
Sum c is V11() ext-real V53() integer Element of COMPLEX
S2 - (Sum h) is V11() ext-real V53() integer Element of INT
S2 + (Sum c) is V11() ext-real V53() integer Element of INT
addint . (S2,(Sum c)) is set
addint "**" <%S2%> is V11() ext-real V53() integer Element of INT
addint . ((addint "**" <%S2%>),(Sum c)) is set
addint "**" c is V11() ext-real V53() integer Element of INT
addint . ((addint "**" <%S2%>),(addint "**" c)) is V11() ext-real V53() integer Element of INT
addint "**" F1 is V11() ext-real V53() integer Element of INT
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 . F1 is V11() ext-real V53() integer set
(- 1) |^ F1 is V11() ext-real V53() integer set
(card S1) choose F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ F1) * ((card S1) choose F1) is V11() ext-real V53() integer Element of INT
(card S1) - F1 is V11() ext-real V53() integer Element of INT
((card S1) - F1) |^ (card X) is V11() ext-real V53() integer set
(((- 1) |^ F1) * ((card S1) choose F1)) * (((card S1) - F1) |^ (card X)) is V11() ext-real V53() integer Element of INT
F1 - 1 is V11() ext-real V53() integer Element of INT
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
XF + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
h . XF is V11() ext-real V53() integer set
(- 1) |^ XF is V11() ext-real V53() integer set
(card S1) choose (XF + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ XF) * ((card S1) choose (XF + 1)) is V11() ext-real V53() integer Element of INT
(card S1) - XF is V11() ext-real V53() integer Element of INT
((card S1) - XF) - 1 is V11() ext-real V53() integer Element of INT
(((card S1) - XF) - 1) |^ (card X) is V11() ext-real V53() integer set
(((- 1) |^ XF) * ((card S1) choose (XF + 1))) * ((((card S1) - XF) - 1) |^ (card X)) is V11() ext-real V53() integer Element of INT
F1 . (XF + 1) is V11() ext-real V53() integer set
c . XF is V11() ext-real V53() integer set
(- 1) * ((- 1) |^ XF) is V11() ext-real V53() integer Element of INT
(- 1) * (h . XF) is V11() ext-real V53() integer Element of INT
X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
X block S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
1 / (S1 !) is V11() ext-real non negative V53() Element of REAL
S1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
S2 is V11() ext-real V53() integer Element of INT
<%S2%> is non empty trivial V8() Relation-like NAT -defined INT -valued Function-like constant V32() V33() V34() V36() V37() V38() V39() finite 1 -element V205() set
addint "**" <%S2%> is V11() ext-real V53() integer Element of INT
h is non empty trivial V8() Relation-like NAT -defined INT -valued Function-like constant V32() V33() V34() V36() V37() V38() V39() finite 1 -element V205() set
Sum h is V11() ext-real V53() integer Element of COMPLEX
(1 / (S1 !)) * (Sum h) is V11() ext-real V53() Element of REAL
dom h is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element Element of bool NAT
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
(- 1) |^ y is V11() ext-real V53() integer set
S1 - y is V11() ext-real V53() integer Element of INT
(S1 - y) |^ X is V11() ext-real V53() integer set
0 choose 0 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
h . y is V11() ext-real V53() integer set
S1 choose y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ y) * (S1 choose y) is V11() ext-real V53() integer Element of INT
(((- 1) |^ y) * (S1 choose y)) * ((S1 - y) |^ X) is V11() ext-real V53() integer Element of INT
(S1 + 1) --> 0 is non empty V8() Relation-like S1 + 1 -defined NAT -valued RAT -valued INT -valued Function-like constant V25(S1 + 1) quasi_total V32() V33() V34() V35() V52() finite Function-yielding V130() V205() Element of bool [:(S1 + 1),NAT:]
[:(S1 + 1),NAT:] is non empty non trivial Relation-like non finite set
bool [:(S1 + 1),NAT:] is non empty non trivial non finite set
{0} is non empty trivial functional finite V60() 1 -element set
[:(S1 + 1),{0}:] is non empty Relation-like finite set
x is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
rng ((S1 + 1) --> 0) is non empty trivial V42() V43() V44() V45() V46() V47() finite 1 -element Element of bool RAT
bool RAT is non empty non trivial non finite set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
{0,0} is non empty functional finite V60() Element of bool NAT
h is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
Sum h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of COMPLEX
h " {0} is finite set
card (h " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
0 * (card (h " {0})) is empty V4() V5() V6() V8() V9() V10() V11() ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V34() V35() V47() V52() V53() integer finite finite-yielding V60() cardinal {} -element Function-yielding V130() V205() Element of NAT
y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
Sum y is V11() ext-real V53() integer Element of COMPLEX
(1 / (S1 !)) * (Sum y) is V11() ext-real V53() Element of REAL
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
S1 - c is V11() ext-real V53() integer Element of INT
(S1 - c) |^ X is V11() ext-real V53() integer set
S1 choose c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(S1 choose c) * ((S1 - c) |^ X) is V11() ext-real V53() integer Element of INT
S1 choose c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 - c is V11() ext-real V53() integer Element of INT
(S1 - c) |^ X is V11() ext-real V53() integer set
(S1 choose c) * ((S1 - c) |^ X) is V11() ext-real V53() integer Element of INT
S1 choose c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 - c is V11() ext-real V53() integer Element of INT
(S1 - c) |^ X is V11() ext-real V53() integer set
(S1 choose c) * ((S1 - c) |^ X) is V11() ext-real V53() integer Element of INT
S1 choose c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 - c is V11() ext-real V53() integer Element of INT
(S1 - c) |^ X is V11() ext-real V53() integer set
(S1 choose c) * ((S1 - c) |^ X) is V11() ext-real V53() integer Element of INT
(- 1) |^ c is V11() ext-real V53() integer set
((- 1) |^ c) * ((S1 choose c) * ((S1 - c) |^ X)) is V11() ext-real V53() integer Element of INT
y . c is V11() ext-real V53() integer set
((- 1) |^ c) * (S1 choose c) is V11() ext-real V53() integer Element of INT
(((- 1) |^ c) * (S1 choose c)) * ((S1 - c) |^ X) is V11() ext-real V53() integer Element of INT
[:S1,S1:] is Relation-like RAT -valued V32() V33() V34() V35() V52() finite set
bool [:S1,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:] : b₁ is Relation-like S1 -defined S1 -valued Function-like one-to-one V25(S1) quasi_total onto bijective V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:] } is set
card { b₁ where b₁ is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:] : b₁ is Relation-like S1 -defined S1 -valued Function-like one-to-one V25(S1) quasi_total onto bijective V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:] } is V4() V5() V6() cardinal set
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card S1) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:X,S1:] is Relation-like RAT -valued V32() V33() V34() V35() V52() finite set
bool [:X,S1:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : ( b₁ is onto & b₁ is "increasing ) } is set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } is set
x is finite set
h is finite set
[:x,h:] is Relation-like finite set
card [:x,h:] is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card x is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card x) * (card h) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is set
XF is set
n is set
[XF,n] is set
{XF,n} is non empty finite set
{XF} is non empty trivial finite 1 -element set
{{XF,n},{XF}} is non empty finite V60() set
n1 is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
rng n1 is V42() V43() V44() V47() finite Element of bool S1
bool S1 is non empty finite V60() set
u is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:]
u * n1 is Relation-like X -defined RAT -valued S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
n is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
dom u is finite Element of bool S1
rng u is V42() V43() V44() V47() finite Element of bool S1
rng (u * n1) is V42() V43() V44() V45() V47() finite Element of bool S1
c is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:]
Fyy is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
[c,Fyy] is Element of [:(bool [:S1,S1:]),(bool [:X,S1:]):]
[:(bool [:S1,S1:]),(bool [:X,S1:]):] is non empty Relation-like finite set
{c,Fyy} is non empty functional finite V60() set
{c} is non empty trivial functional finite V60() 1 -element set
{{c,Fyy},{c}} is non empty finite V60() set
c * Fyy is Relation-like X -defined RAT -valued S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
[:[:x,h:], { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } :] is Relation-like set
bool [:[:x,h:], { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } :] is non empty set
F1 is Relation-like [:x,h:] -defined { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } -valued Function-like quasi_total finite Element of bool [:[:x,h:], { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } :]
XF is set
dom F1 is Relation-like finite set
F1 . XF is set
n is set
F1 . n is set
dom F1 is Relation-like x -defined h -valued finite Element of bool [:x,h:]
bool [:x,h:] is non empty finite V60() set
n1 is set
u is set
[n1,u] is set
{n1,u} is non empty finite set
{n1} is non empty trivial finite 1 -element set
{{n1,u},{n1}} is non empty finite V60() set
n is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:]
c is set
Fyy is set
[c,Fyy] is set
{c,Fyy} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,Fyy},{c}} is non empty finite V60() set
rng F1 is finite Element of bool { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto }
bool { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } is non empty set
c is non empty Element of bool NAT
F1 is non empty Element of bool NAT
[:c,F1:] is non empty Relation-like set
bool [:c,F1:] is non empty set
y is non empty Relation-like c -defined F1 -valued Function-like V25(c) quasi_total Element of bool [:c,F1:]
rng y is non empty Element of bool F1
bool F1 is non empty set
h is Relation-like S1 -defined S1 -valued Function-like V25(S1) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:S1,S1:]
x is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
[:(rng y),(rng y):] is non empty Relation-like set
bool [:(rng y),(rng y):] is non empty set
x1 is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
rng x is V42() V43() V44() V47() finite Element of bool S1
bool S1 is non empty finite V60() set
rng x1 is V42() V43() V44() V47() finite Element of bool S1
fx9 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
fx1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
{fx1} is non empty trivial finite V60() 1 -element set
x1 " {fx1} is finite Element of bool NAT
min* (x1 " {fx1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{fx9} is non empty trivial finite V60() 1 -element set
x1 " {fx9} is finite Element of bool NAT
min* (x1 " {fx9}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
x2 is non empty Relation-like c -defined F1 -valued Function-like V25(c) quasi_total Element of bool [:c,F1:]
fx9 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
fx1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
{fx1} is non empty trivial finite V60() 1 -element set
x " {fx1} is finite Element of bool NAT
min* (x " {fx1}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{fx9} is non empty trivial finite V60() 1 -element set
x " {fx9} is finite Element of bool NAT
min* (x " {fx9}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
f299 is non empty Relation-like c -defined F1 -valued Function-like V25(c) quasi_total Element of bool [:c,F1:]
y is non empty Relation-like rng y -defined rng y -valued Function-like one-to-one V25( rng y) quasi_total onto bijective Element of bool [:(rng y),(rng y):]
y * x2 is Relation-like c -defined rng y -valued Function-like Element of bool [:c,(rng y):]
[:c,(rng y):] is non empty Relation-like set
bool [:c,(rng y):] is non empty set
y is non empty Relation-like rng y -defined rng y -valued Function-like one-to-one V25( rng y) quasi_total onto bijective Element of bool [:(rng y),(rng y):]
y * f299 is Relation-like c -defined rng y -valued Function-like Element of bool [:c,(rng y):]
XF is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
rng F1 is finite Element of bool { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto }
bool { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } is non empty set
n is set
n1 is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
rng n1 is V42() V43() V44() V47() finite Element of bool S1
bool S1 is non empty finite V60() set
F1 is non empty Element of bool NAT
c is non empty Element of bool NAT
[:c,F1:] is non empty Relation-like set
bool [:c,F1:] is non empty set
[:F1,F1:] is non empty Relation-like set
bool [:F1,F1:] is non empty set
u is non empty Relation-like c -defined F1 -valued Function-like V25(c) quasi_total Element of bool [:c,F1:]
n is non empty Relation-like F1 -defined F1 -valued Function-like one-to-one V25(F1) quasi_total onto bijective Element of bool [:F1,F1:]
n * u is non empty Relation-like c -defined F1 -valued Function-like V25(c) quasi_total Element of bool [:c,F1:]
rng u is non empty Element of bool F1
bool F1 is non empty set
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
h is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
{h} is non empty trivial finite V60() 1 -element set
u " {h} is Element of bool c
bool c is non empty set
min* (u " {h}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
{y} is non empty trivial finite V60() 1 -element set
u " {y} is Element of bool c
min* (u " {y}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
Fyy is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:]
[n,Fyy] is Element of [:(bool [:F1,F1:]),(bool [:X,S1:]):]
[:(bool [:F1,F1:]),(bool [:X,S1:]):] is non empty Relation-like set
{n,Fyy} is non empty functional finite set
{n} is non empty trivial functional finite 1 -element set
{{n,Fyy},{n}} is non empty finite V60() set
dom F1 is Relation-like x -defined h -valued finite Element of bool [:x,h:]
bool [:x,h:] is non empty finite V60() set
F1 . [n,Fyy] is set
dom F1 is Relation-like x -defined h -valued finite Element of bool [:x,h:]
bool [:x,h:] is non empty finite V60() set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total V32() V33() V34() V35() V52() finite Element of bool [:X,S1:] : b₁ is onto } is V4() V5() V6() cardinal set
(S1 !) * (card h) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((S1 !) * (card h)) / (S1 !) is V11() ext-real non negative V53() Element of REAL
(S1 !) / (S1 !) is V11() ext-real non negative V53() Element of REAL
(card h) * ((S1 !) / (S1 !)) is V11() ext-real non negative V53() Element of REAL
(card S1) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum n is V11() ext-real V53() integer Element of COMPLEX
n1 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
Sum n1 is V11() ext-real V53() integer Element of COMPLEX
(Sum n1) * 1 is V11() ext-real V53() integer Element of INT
((Sum n1) * 1) / (S1 !) is V11() ext-real V53() Element of REAL
(1 / (S1 !)) * (Sum n1) is V11() ext-real V53() Element of REAL
dom n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n1 . u is V11() ext-real V53() integer set
(- 1) |^ u is V11() ext-real V53() integer set
S1 choose u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((- 1) |^ u) * (S1 choose u) is V11() ext-real V53() integer Element of INT
S1 - u is V11() ext-real V53() integer Element of INT
(S1 - u) |^ X is V11() ext-real V53() integer set
(((- 1) |^ u) * (S1 choose u)) * ((S1 - u) |^ X) is V11() ext-real V53() integer Element of INT
S2 is non empty trivial V8() Relation-like NAT -defined INT -valued Function-like constant V32() V33() V34() V36() V37() V38() V39() finite 1 -element V205() set
Sum S2 is V11() ext-real V53() integer Element of COMPLEX
(1 / (S1 !)) * (Sum S2) is V11() ext-real V53() Element of REAL
dom S2 is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element Element of bool NAT
S1 is finite set
X is finite set
S2 is finite set
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card S1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card S2 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) -' (card S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card X) -' (card S2)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
X \ S2 is finite Element of bool X
bool X is non empty finite V60() set
h is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
h .: (X \ S2) is finite Element of bool S1
bool S1 is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : ( b₁ is one-to-one & rng (b₁ | (X \ S2)) c= h .: (X \ S2) & ( for b₂ being set holds
( not b₂ in S2 or b₁ . b₂ = h . b₂ ) ) ) } is set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : ( b₁ is one-to-one & rng (b₁ | (X \ S2)) c= h .: (X \ S2) & ( for b₂ being set holds
( not b₂ in S2 or b₁ . b₂ = h . b₂ ) ) ) } is V4() V5() V6() cardinal set
[:(X \ S2),(h .: (X \ S2)):] is Relation-like finite set
bool [:(X \ S2),(h .: (X \ S2)):] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X \ S2 -defined h .: (X \ S2) -valued Function-like quasi_total finite Element of bool [:(X \ S2),(h .: (X \ S2)):] : b₁ is one-to-one } is set
card (X \ S2) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) - (card S2) is V11() ext-real V53() integer Element of INT
X \ (X \ S2) is finite Element of bool X
F1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
F1 | (X \ S2) is Relation-like X \ S2 -defined X -defined S1 -valued Function-like finite set
rng (F1 | (X \ S2)) is finite set
F1 | (X \ S2) is Relation-like X -defined X \ S2 -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
(F1 | (X \ S2)) | (X \ S2) is Relation-like X \ S2 -defined X -defined S1 -valued Function-like finite set
rng ((F1 | (X \ S2)) | (X \ S2)) is finite set
rng h is finite Element of bool S1
rng (F1 | (X \ S2)) is finite Element of bool S1
F1 .: (X \ S2) is finite Element of bool S1
(X \ (X \ S2)) \/ (X \ S2) is finite Element of bool X
h .: ((X \ (X \ S2)) \/ (X \ S2)) is finite Element of bool S1
h .: (X \ (X \ S2)) is finite Element of bool S1
(h .: (X \ (X \ S2))) \/ (h .: (X \ S2)) is finite Element of bool S1
h | (X \ (X \ S2)) is Relation-like X -defined X \ (X \ S2) -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
dom (h | (X \ (X \ S2))) is finite Element of bool X
dom h is finite Element of bool X
(dom h) /\ (X \ (X \ S2)) is finite Element of bool X
F1 | (X \ (X \ S2)) is Relation-like X -defined X \ (X \ S2) -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
dom (F1 | (X \ (X \ S2))) is finite Element of bool X
dom F1 is finite Element of bool X
(dom F1) /\ (X \ (X \ S2)) is finite Element of bool X
S2 /\ X is finite set
F1 is set
F1 . F1 is set
(F1 | (X \ (X \ S2))) . F1 is set
h . F1 is set
rng (F1 | (X \ (X \ S2))) is finite Element of bool S1
F1 .: (X \ (X \ S2)) is finite Element of bool S1
h .: (dom h) is finite Element of bool S1
h .: X is finite Element of bool S1
F1 .: X is finite Element of bool S1
rng F1 is finite Element of bool S1
{ b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : ( S₁[b₁,X,S1] & rng (b₁ | (X \ S2)) c= h .: (X \ S2) & ( for b₂ being set holds
( not b₂ in X \ (X \ S2) or b₁ . b₂ = H₁(b₂) ) ) ) } is set
S2 /\ X is finite set
XF is set
n is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
n | (X \ S2) is Relation-like X \ S2 -defined X -defined S1 -valued Function-like finite set
rng (n | (X \ S2)) is finite set
n | (X \ S2) is Relation-like X -defined X \ S2 -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
rng (n | (X \ S2)) is finite Element of bool S1
dom h is finite Element of bool X
card (h .: (X \ S2)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card { b₁ where b₁ is Relation-like X \ S2 -defined h .: (X \ S2) -valued Function-like quasi_total finite Element of bool [:(X \ S2),(h .: (X \ S2)):] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card (X \ S2)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is finite set
card y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card y) -' (card (X \ S2)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card y) -' (card (X \ S2))) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card (X \ S2)) !) / (((card y) -' (card (X \ S2))) !) is V11() ext-real non negative V53() Element of REAL
{ b₁ where b₁ is Relation-like X \ S2 -defined h .: (X \ S2) -valued Function-like quasi_total finite Element of bool [:(X \ S2),(h .: (X \ S2)):] : S₁[b₁,X \ S2,h .: (X \ S2)] } is set
n is set
h . n is set
rng h is finite Element of bool S1
n is set
[:(X \ S2),y:] is Relation-like finite set
bool [:(X \ S2),y:] is non empty finite V60() set
n1 is Relation-like X \ S2 -defined y -valued Function-like quasi_total finite Element of bool [:(X \ S2),y:]
n1 | (X \ S2) is Relation-like X \ S2 -defined X \ S2 -defined y -valued Function-like finite Element of bool [:(X \ S2),y:]
rng n1 is finite Element of bool y
bool y is non empty finite V60() set
n is set
[:(X \ S2),y:] is Relation-like finite set
bool [:(X \ S2),y:] is non empty finite V60() set
n1 is Relation-like X \ S2 -defined y -valued Function-like quasi_total finite Element of bool [:(X \ S2),y:]
n1 | (X \ S2) is Relation-like X \ S2 -defined X \ S2 -defined y -valued Function-like finite set
rng (n1 | (X \ S2)) is finite set
n is set
n1 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
n1 | (X \ S2) is Relation-like X -defined X \ S2 -defined X -defined S1 -valued Function-like finite Element of bool [:X,S1:]
rng (n1 | (X \ S2)) is finite Element of bool S1
dom n1 is finite Element of bool X
n1 .: (X \ S2) is finite Element of bool S1
card (n1 .: (X \ S2)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card (rng (n1 | (X \ S2))) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card { b₁ where b₁ is Relation-like X \ S2 -defined h .: (X \ S2) -valued Function-like quasi_total finite Element of bool [:(X \ S2),(h .: (X \ S2)):] : S₁[b₁,X \ S2,h .: (X \ S2)] } is V4() V5() V6() cardinal set
card { b₁ where b₁ is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:] : ( S₁[b₁,X,S1] & rng (b₁ | (X \ S2)) c= h .: (X \ S2) & ( for b₂ being set holds
( not b₂ in X \ (X \ S2) or b₁ . b₂ = H₁(b₂) ) ) ) } is V4() V5() V6() cardinal set
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is finite set
[:X,S1:] is Relation-like finite set
bool [:X,S1:] is non empty finite V60() set
S2 is Relation-like X -defined S1 -valued Function-like quasi_total finite Element of bool [:X,S1:]
dom S2 is finite Element of bool X
bool X is non empty finite V60() set
rng S2 is finite Element of bool S1
bool S1 is non empty finite V60() set
[:X,(rng S2):] is Relation-like finite set
bool [:X,(rng S2):] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = S2 . b₂ ) ) ) } is set
card { b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = S2 . b₂ ) ) ) } is V4() V5() V6() cardinal set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . a₁ = S2 . a₁ ) } is set
Funcs (X,(rng S2)) is functional finite set
bool (Funcs (X,(rng S2))) is non empty finite V60() set
x is set
S2 . x is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . x = S2 . x ) } is set
h is set
y is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
y . x is set
[:X,(bool (Funcs (X,(rng S2)))):] is Relation-like finite set
bool [:X,(bool (Funcs (X,(rng S2)))):] is non empty finite V60() set
x is Relation-like X -defined bool (Funcs (X,(rng S2))) -valued Function-like V25(X) quasi_total finite Element of bool [:X,(bool (Funcs (X,(rng S2)))):]
h is set
y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card X) - (y + 1) is V11() ext-real V53() integer Element of INT
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
F1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card X) - (F1 + 1) is V11() ext-real V53() integer Element of INT
F1 ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:(card X),NAT:] is Relation-like set
bool [:(card X),NAT:] is non empty set
h is Relation-like card X -defined NAT -valued Function-like V25( card X) quasi_total V32() V33() V34() V35() V52() finite Element of bool [:(card X),NAT:]
dom x is finite Element of bool X
y is set
x . y is set
rng x is finite V60() Element of bool (bool (Funcs (X,(rng S2))))
bool (bool (Funcs (X,(rng S2)))) is non empty finite V60() set
dom h is finite Element of bool (card X)
bool (card X) is non empty finite V60() set
F1 is V8() Relation-like NAT -defined Function-like finite V205() set
F1 is V8() Relation-like NAT -defined NAT -valued Function-like V32() V33() V34() V35() V52() finite V205() set
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
y is Relation-like Function-like finite-yielding set
XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
XF + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
F1 . XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) - (XF + 1) is V11() ext-real V53() integer Element of INT
(card X) -' (XF + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(X,(XF + 1),0,1) is functional finite Element of bool (Funcs (X,{0,1}))
{0,1} is non empty finite V60() set
Funcs (X,{0,1}) is non empty functional finite FUNCTION_DOMAIN of X,{0,1}
bool (Funcs (X,{0,1})) is non empty finite V60() set
n1 is Relation-like Function-like set
(y,n1,0) is Element of bool (union (rng y))
rng y is V60() set
union (rng y) is set
bool (union (rng y)) is non empty set
card (y,n1,0) is V4() V5() V6() cardinal set
[:X,{0,1}:] is Relation-like finite set
bool [:X,{0,1}:] is non empty finite V60() set
{0} is non empty trivial functional finite V60() 1 -element Element of bool NAT
u is Relation-like X -defined {0,1} -valued Function-like V25(X) quasi_total finite Element of bool [:X,{0,1}:]
u " {0} is finite Element of bool X
card (u " {0}) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n is finite set
X \ n is finite Element of bool X
c is finite set
[:X,c:] is Relation-like finite set
bool [:X,c:] is non empty finite V60() set
S2 .: (X \ n) is finite Element of bool S1
{ b₁ where b₁ is Relation-like X -defined c -valued Function-like quasi_total finite Element of bool [:X,c:] : ( b₁ is one-to-one & rng (b₁ | (X \ n)) c= S2 .: (X \ n) & ( for b₂ being set holds
( not b₂ in n or b₁ . b₂ = S2 . b₂ ) ) ) } is set
(y,u,0) is Element of bool (union (rng y))
y is set
h is set
u . h is set
dom u is finite Element of bool X
rng u is finite V60() Element of bool {0,1}
bool {0,1} is non empty finite V60() set
x is set
u . x is set
y . x is finite set
S2 . x is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . x = S2 . x ) } is set
y is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
y . x is set
y is set
y . y is set
S2 . y is set
u . y is set
x . y is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . y = S2 . y ) } is set
x1 is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
x1 . y is set
y | (X \ n) is Relation-like X -defined X \ n -defined X -defined rng S2 -valued Function-like finite Element of bool [:X,(rng S2):]
rng (y | (X \ n)) is finite Element of bool (rng S2)
bool (rng S2) is non empty finite V60() set
y is set
dom (y | (X \ n)) is finite Element of bool X
x1 is set
(y | (X \ n)) . x1 is set
y . x1 is set
dom y is finite Element of bool X
(dom y) /\ (X \ n) is finite Element of bool X
X \ (X \ n) is finite Element of bool X
S2 .: (X \ (X \ n)) is finite Element of bool S1
S2 .: X is finite Element of bool S1
(S2 .: X) \ (S2 .: (X \ n)) is finite Element of bool S1
x2 is set
S2 . x2 is set
X /\ (u " {0}) is finite Element of bool X
y . x2 is set
card c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
y is set
h is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
h | (X \ n) is Relation-like X -defined X \ n -defined X -defined rng S2 -valued Function-like finite Element of bool [:X,(rng S2):]
rng (h | (X \ n)) is finite Element of bool (rng S2)
bool (rng S2) is non empty finite V60() set
x is set
x . x is set
rng x is finite V60() Element of bool (bool (Funcs (X,(rng S2))))
bool (bool (Funcs (X,(rng S2)))) is non empty finite V60() set
S2 . x is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . x = S2 . x ) } is set
h . x is set
union (rng x) is finite set
dom u is finite Element of bool X
y is set
u . y is set
y . y is finite set
h . y is set
S2 . y is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . y = S2 . y ) } is set
card (y,u,0) is V4() V5() V6() cardinal set
((card X) -' (XF + 1)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c is finite set
card c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card c) -' (card X) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card c) -' (card X)) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
[:X,c:] is Relation-like finite set
bool [:X,c:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined c -valued Function-like quasi_total finite Element of bool [:X,c:] : b₁ is one-to-one } is set
card { b₁ where b₁ is Relation-like X -defined c -valued Function-like quasi_total finite Element of bool [:X,c:] : b₁ is one-to-one } is V4() V5() V6() cardinal set
(card c) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card c) !) / (((card c) -' (card X)) !) is V11() ext-real non negative V53() Element of REAL
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : b₁ is one-to-one } is set
dom y is set
rng y is V60() set
union (rng y) is set
card (union (rng y)) is V4() V5() V6() cardinal set
n1 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum n1 is V11() ext-real V53() integer Element of COMPLEX
rng x is finite V60() Element of bool (bool (Funcs (X,(rng S2))))
bool (bool (Funcs (X,(rng S2)))) is non empty finite V60() set
union (rng x) is finite set
XF is finite set
u is set
n is set
c is set
y . c is finite set
S2 . c is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . c = S2 . c ) } is set
Fyy is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
Fyy . c is set
u is finite set
XF \ u is finite Element of bool XF
bool XF is non empty finite V60() set
card (XF \ u) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card XF is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
card u is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card XF) - (card u) is V11() ext-real V53() integer Element of INT
((card X) !) - (Sum n1) is V11() ext-real V53() integer Element of INT
XF \ (union (rng y)) is finite Element of bool XF
n is set
c is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
Fyy is set
c . Fyy is set
S2 . Fyy is set
y . Fyy is finite set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . Fyy = S2 . Fyy ) } is set
n is set
c is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
Fyy is set
y is set
y . y is finite set
S2 . y is set
{ b₁ where b₁ is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):] : ( b₁ is one-to-one & b₁ . y = S2 . y ) } is set
h is Relation-like X -defined rng S2 -valued Function-like quasi_total finite Element of bool [:X,(rng S2):]
h . y is set
n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n1 . n is V11() ext-real V53() integer set
(- 1) |^ n is V11() ext-real V53() integer set
((- 1) |^ n) * ((card X) !) is V11() ext-real V53() integer Element of INT
n + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(n + 1) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((- 1) |^ n) * ((card X) !)) / ((n + 1) !) is V11() ext-real V53() Element of REAL
(card X) - (n + 1) is V11() ext-real V53() integer Element of INT
(card X) choose (n + 1) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
c ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(c !) * ((n + 1) !) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((card X) !) / ((c !) * ((n + 1) !)) is V11() ext-real non negative V53() Element of REAL
F1 . n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(F1 . n) * ((card X) choose (n + 1)) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(c !) * ((card X) !) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
((c !) * ((card X) !)) / ((c !) * ((n + 1) !)) is V11() ext-real non negative V53() Element of REAL
(c !) / ((c !) * ((n + 1) !)) is V11() ext-real non negative V53() Element of REAL
((card X) !) * ((c !) / ((c !) * ((n + 1) !))) is V11() ext-real non negative V53() Element of REAL
(c !) / (c !) is V11() ext-real non negative V53() Element of REAL
((c !) / (c !)) / ((n + 1) !) is V11() ext-real non negative V53() Element of REAL
((card X) !) * (((c !) / (c !)) / ((n + 1) !)) is V11() ext-real non negative V53() Element of REAL
1 / ((n + 1) !) is V11() ext-real non negative V53() Element of REAL
((card X) !) * (1 / ((n + 1) !)) is V11() ext-real non negative V53() Element of REAL
((card X) !) * 1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((card X) !) * 1) / ((n + 1) !) is V11() ext-real non negative V53() Element of REAL
((- 1) |^ n) * (F1 . n) is V11() ext-real V53() integer Element of INT
(((- 1) |^ n) * (F1 . n)) * ((card X) choose (n + 1)) is V11() ext-real V53() integer Element of INT
((card X) !) / ((n + 1) !) is V11() ext-real non negative V53() Element of REAL
((- 1) |^ n) * (((card X) !) / ((n + 1) !)) is V11() ext-real V53() Element of REAL
X is finite set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card X) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
S1 is Relation-like Function-like set
dom S1 is set
rng S1 is set
[:X,(rng S1):] is Relation-like set
bool [:X,(rng S1):] is non empty set
{ b₁ where b₁ is Relation-like X -defined rng S1 -valued Function-like quasi_total finite Element of bool [:X,(rng S1):] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = S1 . b₂ ) ) ) } is set
card { b₁ where b₁ is Relation-like X -defined rng S1 -valued Function-like quasi_total finite Element of bool [:X,(rng S1):] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = S1 . b₂ ) ) ) } is V4() V5() V6() cardinal set
card (rng S1) is V4() V5() V6() cardinal set
S2 is finite set
[:X,S2:] is Relation-like finite set
bool [:X,S2:] is non empty finite V60() set
x is Relation-like X -defined S2 -valued Function-like quasi_total finite Element of bool [:X,S2:]
rng x is finite Element of bool S2
bool S2 is non empty finite V60() set
y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom y is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
Sum y is V11() ext-real V53() integer Element of COMPLEX
((card X) !) - (Sum y) is V11() ext-real V53() integer Element of INT
c is V11() ext-real V53() integer Element of INT
<%c%> is non empty trivial V8() Relation-like NAT -defined INT -valued Function-like constant V32() V33() V34() V36() V37() V38() V39() finite 1 -element V205() set
len <%c%> is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
dom <%c%> is non empty trivial V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal 1 -element set
(- 1) (#) y is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
dom ((- 1) (#) y) is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
F1 is V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
<%c%> ^ F1 is non empty V8() Relation-like NAT -defined INT -valued Function-like V32() V33() V34() finite V205() set
Sum (<%c%> ^ F1) is V11() ext-real V53() integer Element of COMPLEX
dom (<%c%> ^ F1) is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of bool NAT
(- 1) * (Sum y) is V11() ext-real V53() integer Element of INT
Sum F1 is V11() ext-real V53() integer Element of COMPLEX
c - (Sum y) is V11() ext-real V53() integer Element of INT
c + (Sum F1) is V11() ext-real V53() integer Element of INT
addint . (c,(Sum F1)) is set
addint "**" <%c%> is V11() ext-real V53() integer Element of INT
addint . ((addint "**" <%c%>),(Sum F1)) is set
addint "**" F1 is V11() ext-real V53() integer Element of INT
addint . ((addint "**" <%c%>),(addint "**" F1)) is V11() ext-real V53() integer Element of INT
addint "**" (<%c%> ^ F1) is V11() ext-real V53() integer Element of INT
len F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
n is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal set
(<%c%> ^ F1) . n is V11() ext-real V53() integer set
(- 1) |^ n is V11() ext-real V53() integer set
((- 1) |^ n) * ((card X) !) is V11() ext-real V53() integer Element of INT
n ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((- 1) |^ n) * ((card X) !)) / (n !) is V11() ext-real V53() Element of REAL
n - 1 is V11() ext-real V53() integer Element of INT
n1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
n1 + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(- 1) |^ n1 is V11() ext-real V53() integer set
(- 1) * ((- 1) |^ n1) is V11() ext-real V53() integer Element of INT
dom F1 is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of bool NAT
(<%c%> ^ F1) . (n1 + 1) is V11() ext-real V53() integer set
F1 . n1 is V11() ext-real V53() integer set
y . n1 is V11() ext-real V53() integer set
(- 1) * (y . n1) is V11() ext-real V53() integer Element of INT
((- 1) |^ n1) * ((card X) !) is V11() ext-real V53() integer Element of INT
(n1 + 1) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(((- 1) |^ n1) * ((card X) !)) / ((n1 + 1) !) is V11() ext-real V53() Element of REAL
(- 1) * (((- 1) |^ n1) * ((card X) !)) is V11() ext-real V53() integer Element of INT
((- 1) * (((- 1) |^ n1) * ((card X) !))) / (n !) is V11() ext-real V53() Element of REAL
X is finite set
[:X,X:] is Relation-like finite set
bool [:X,X:] is non empty finite V60() set
{ b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = b₂ ) ) ) } is set
card { b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = b₂ ) ) ) } is V4() V5() V6() cardinal set
card X is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
(card X) + 1 is non empty V4() V5() V6() V10() V11() ext-real positive non negative V47() V53() integer finite cardinal Element of NAT
(card X) ! is V4() V5() V6() V10() V11() ext-real non negative V47() V53() integer finite cardinal Element of NAT
id X is Relation-like X -defined X -valued Function-like one-to-one V25(X) quasi_total finite Element of bool [:X,X:]
{ b₁ where b₁ is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:] : ( b₁ is one-to-one & ( for b₂ being set holds
( not b₂ in X or not b₁ . b₂ = (id X) . b₂ ) ) ) } is set
x is set
h is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:]
y is set
(id X) . y is set
h . y is set
dom (id X) is finite Element of bool X
bool X is non empty finite V60() set
rng (id X) is finite Element of bool X
x is set
h is Relation-like X -defined X -valued Function-like V25(X) quasi_total finite Element of bool [:X,X:]
y is set
(id X) . y is set
h . y is set