REAL is non empty V12() non finite complex-membered ext-real-membered real-membered V80() set
NAT is V12() ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() Element of K19(REAL)
K19(REAL) is V12() non finite set
{} is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set
the Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set
COMPLEX is non empty V12() non finite complex-membered V80() set
RAT is non empty V12() non finite complex-membered ext-real-membered real-membered rational-membered V80() set
INT is non empty V12() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V80() set
K20(COMPLEX,COMPLEX) is V12() non finite complex-valued set
K19(K20(COMPLEX,COMPLEX)) is V12() non finite set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is V12() non finite complex-valued set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is V12() non finite set
K20(REAL,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(REAL,REAL)) is V12() non finite set
K20(K20(REAL,REAL),REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(K20(REAL,REAL),REAL)) is V12() non finite set
K20(RAT,RAT) is RAT -valued V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(RAT,RAT)) is V12() non finite set
K20(K20(RAT,RAT),RAT) is RAT -valued V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(K20(RAT,RAT),RAT)) is V12() non finite set
K20(INT,INT) is RAT -valued INT -valued V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(INT,INT)) is V12() non finite set
K20(K20(INT,INT),INT) is RAT -valued INT -valued V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(K20(INT,INT),INT)) is V12() non finite set
K20(NAT,NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K20(K20(NAT,NAT),NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K19(K20(K20(NAT,NAT),NAT)) is set
omega is V12() ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() set
K19(omega) is V12() non finite set
K20(NAT,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(NAT,REAL)) is V12() non finite set
K19(K19(REAL)) is V12() non finite set
K19(NAT) is V12() non finite set
K20(COMPLEX,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(COMPLEX,REAL)) is V12() non finite set
K101(NAT) is V27() set
1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
{{},1} is non empty finite V51() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
2 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
3 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative V41() V42() finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of NAT
Seg 1 is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
{1} is non empty V12() finite V51() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
{1,2} is non empty finite V51() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card {} is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set
<*> REAL is Relation-like NAT -defined REAL -valued Function-like one-to-one constant functional empty ordinal natural V36() real ext-real non positive non negative finite finite-yielding V51() cardinal {} -element complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of REAL
Sum (<*> REAL) is V36() real ext-real Element of REAL
K197() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K197() $$ (<*> REAL) is V36() real ext-real Element of REAL
{0} is non empty V12() finite V51() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
dom {} is set
rng {} is set
-infty is non empty non real ext-real non positive negative set
+infty is non empty non real ext-real positive non negative set
D is V36() real ext-real set
max (D,0) is V36() real ext-real set
- D is V36() real ext-real set
max ((- D),0) is V36() real ext-real set
D is V36() real ext-real set
(D) is V36() real ext-real Element of REAL
max (D,0) is V36() real ext-real set
(D) is V36() real ext-real Element of REAL
- D is V36() real ext-real set
max ((- D),0) is V36() real ext-real set
(D) - (D) is V36() real ext-real Element of REAL
D is V36() real ext-real set
abs D is V36() real ext-real Element of REAL
(D) is V36() real ext-real Element of REAL
max (D,0) is V36() real ext-real set
(D) is V36() real ext-real Element of REAL
- D is V36() real ext-real set
max ((- D),0) is V36() real ext-real set
(D) + (D) is V36() real ext-real Element of REAL
D is V36() real ext-real set
(D) is V36() real ext-real Element of REAL
max (D,0) is V36() real ext-real set
2 * (D) is V36() real ext-real Element of REAL
abs D is V36() real ext-real Element of REAL
D + (abs D) is V36() real ext-real Element of REAL
(D) is V36() real ext-real Element of REAL
- D is V36() real ext-real set
max ((- D),0) is V36() real ext-real set
(D) - (D) is V36() real ext-real Element of REAL
((D) - (D)) + (abs D) is V36() real ext-real Element of REAL
(D) + (D) is V36() real ext-real Element of REAL
((D) - (D)) + ((D) + (D)) is V36() real ext-real Element of REAL
D is V36() real ext-real set
F is V36() real ext-real set
D * F is V36() real ext-real set
((D * F)) is V36() real ext-real Element of REAL
max ((D * F),0) is V36() real ext-real set
(F) is V36() real ext-real Element of REAL
max (F,0) is V36() real ext-real set
D * (F) is V36() real ext-real Element of REAL
D is V36() real ext-real set
F is V36() real ext-real set
D + F is V36() real ext-real set
((D + F)) is V36() real ext-real Element of REAL
max ((D + F),0) is V36() real ext-real set
(D) is V36() real ext-real Element of REAL
max (D,0) is V36() real ext-real set
(F) is V36() real ext-real Element of REAL
max (F,0) is V36() real ext-real set
(D) + (F) is V36() real ext-real Element of REAL
0 + 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative V41() V42() finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of NAT
D is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
F is non empty set
X is Relation-like NAT -defined F -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
X | D is Relation-like NAT -defined F -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg D is finite D -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
X | (Seg D) is Relation-like NAT -defined F -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,F))
K20(NAT,F) is V12() non finite set
K19(K20(NAT,F)) is V12() non finite set
D is Relation-like Function-like set
rng D is set
F is set
{F} is non empty V12() finite 1 -element set
D " {F} is set
(rng D) /\ {F} is finite set
the Element of (rng D) /\ {F} is Element of (rng D) /\ {F}
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real set
Y is V36() real ext-real set
Y / X is V36() real ext-real Element of COMPLEX
{(Y / X)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
F " {(Y / X)} is Element of K19(D)
K19(D) is set
X (#) F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(X (#) F) " {Y} is Element of K19(D)
fx is set
Y is Element of D
dom F is Element of K19(D)
dom (X (#) F) is Element of K19(D)
F . Y is V36() real ext-real Element of REAL
X * (F . Y) is V36() real ext-real Element of REAL
(X (#) F) . Y is V36() real ext-real Element of REAL
fx is set
Y is Element of D
dom (X (#) F) is Element of K19(D)
(X (#) F) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
X * (F . Y) is V36() real ext-real Element of REAL
dom F is Element of K19(D)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
0 (#) F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(0 (#) F) " {0} is Element of K19(D)
K19(D) is set
dom F is Element of K19(D)
X is set
Y is Element of D
dom (0 (#) F) is Element of K19(D)
X is set
Y is Element of D
dom (0 (#) F) is Element of K19(D)
(0 (#) F) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
0 * (F . Y) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
abs F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs F) " {X} is Element of K19(D)
K19(D) is set
- X is V36() real ext-real Element of REAL
{(- X),X} is non empty finite complex-membered ext-real-membered real-membered set
F " {(- X),X} is Element of K19(D)
dom (abs F) is Element of K19(D)
dom F is Element of K19(D)
Y is set
fx is Element of D
(abs F) . fx is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
abs (F . fx) is V36() real ext-real Element of REAL
- (F . fx) is V36() real ext-real Element of REAL
Y is set
fx is Element of D
F . fx is V36() real ext-real Element of REAL
- (F . fx) is V36() real ext-real Element of REAL
abs (- (F . fx)) is V36() real ext-real Element of REAL
abs (F . fx) is V36() real ext-real Element of REAL
(abs F) . fx is V36() real ext-real Element of REAL
abs (F . fx) is V36() real ext-real Element of REAL
(abs F) . fx is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
abs F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(abs F) " {0} is Element of K19(D)
K19(D) is set
F " {0} is Element of K19(D)
dom (abs F) is Element of K19(D)
dom F is Element of K19(D)
X is set
Y is Element of D
(abs F) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
abs (F . Y) is V36() real ext-real Element of REAL
X is set
Y is Element of D
F . Y is V36() real ext-real Element of REAL
abs (F . Y) is V36() real ext-real Element of REAL
(abs F) . Y is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
abs F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs F) " {X} is Element of K19(D)
K19(D) is set
the Element of (abs F) " {X} is Element of (abs F) " {X}
fx is Element of D
(abs F) . fx is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
abs (F . fx) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
fx is V36() real ext-real Element of REAL
fx (#) X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
fx (#) Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
rng (fx (#) X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (fx (#) Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
Y / fx is V36() real ext-real Element of REAL
Coim (X,(Y / fx)) is set
{(Y / fx)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {(Y / fx)} is set
Coim ((fx (#) X),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(fx (#) X) " {Y} is set
Coim (Y,(Y / fx)) is set
Y " {(Y / fx)} is set
Coim ((fx (#) Y),Y) is set
(fx (#) Y) " {Y} is set
card (Coim ((fx (#) X),Y)) is ordinal cardinal set
card (Coim ((fx (#) Y),Y)) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
fx * Y is V36() real ext-real Element of REAL
(fx * Y) / fx is V36() real ext-real Element of REAL
{((fx * Y) / fx)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
Y " {((fx * Y) / fx)} is Element of K19(F)
K19(F) is set
Coim ((fx (#) Y),(fx * Y)) is set
{(fx * Y)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(fx (#) Y) " {(fx * Y)} is set
X " {((fx * Y) / fx)} is Element of K19(D)
K19(D) is set
Coim ((fx (#) X),(fx * Y)) is set
(fx (#) X) " {(fx * Y)} is set
Coim (X,Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {Y} is set
card (Coim (X,Y)) is ordinal cardinal set
Coim (Y,Y) is set
Y " {Y} is set
card (Coim (Y,Y)) is ordinal cardinal set
rng X is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng Y is complex-membered ext-real-membered real-membered Element of K19(REAL)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- 1 is V36() real ext-real non positive set
(- 1) (#) X is Relation-like Function-like set
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
- Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(- 1) (#) Y is Relation-like Function-like set
- 1 is V36() real ext-real non positive Element of REAL
(- 1) (#) X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
abs X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
abs Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
fx is V36() real ext-real Element of REAL
{fx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs X) " {fx} is Element of K19(D)
K19(D) is set
- fx is V36() real ext-real Element of REAL
{(- fx),fx} is non empty finite complex-membered ext-real-membered real-membered set
X " {(- fx),fx} is Element of K19(D)
(abs Y) " {fx} is Element of K19(F)
K19(F) is set
Y " {(- fx),fx} is Element of K19(F)
card ((abs X) " {fx}) is ordinal cardinal set
card ((abs Y) " {fx}) is ordinal cardinal set
fx is V36() real ext-real Element of REAL
{fx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs X) " {fx} is Element of K19(D)
K19(D) is set
X " {fx} is Element of K19(D)
(abs Y) " {fx} is Element of K19(F)
K19(F) is set
Y " {fx} is Element of K19(F)
card ((abs X) " {fx}) is ordinal cardinal set
card ((abs Y) " {fx}) is ordinal cardinal set
fx is V36() real ext-real Element of REAL
{fx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs X) " {fx} is Element of K19(D)
K19(D) is set
card ((abs X) " {fx}) is ordinal cardinal set
(abs Y) " {fx} is Element of K19(F)
K19(F) is set
card ((abs Y) " {fx}) is ordinal cardinal set
fx is V36() real ext-real Element of REAL
{fx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs X) " {fx} is Element of K19(D)
K19(D) is set
card ((abs X) " {fx}) is ordinal cardinal set
(abs Y) " {fx} is Element of K19(F)
K19(F) is set
card ((abs Y) " {fx}) is ordinal cardinal set
fx is V36() real ext-real Element of REAL
{fx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(abs X) " {fx} is Element of K19(D)
K19(D) is set
card ((abs X) " {fx}) is ordinal cardinal set
(abs Y) " {fx} is Element of K19(F)
K19(F) is set
card ((abs Y) " {fx}) is ordinal cardinal set
Coim ((abs X),fx) is set
(abs X) " {fx} is set
card (Coim ((abs X),fx)) is ordinal cardinal set
Coim ((abs Y),fx) is set
(abs Y) " {fx} is set
card (Coim ((abs Y),fx)) is ordinal cardinal set
rng (abs X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (abs Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
D is set
F is set
K20(D,F) is set
K19(K20(D,F)) is set
X is Relation-like D -defined F -valued Function-like Element of K19(K20(D,F))
{X} is functional non empty V12() finite 1 -element set
Y is Relation-like Function-like Element of {X}
D is set
F is set
K20(D,F) is set
K19(K20(D,F)) is set
X is Relation-like D -defined F -valued Function-like Element of K19(K20(D,F))
{X} is functional non empty V12() finite 1 -element set
Y is Relation-like Function-like Element of {X}
D is set
F is set
PFuncs (D,F) is functional non empty set
K20(D,F) is set
K19(K20(D,F)) is set
X is Relation-like Function-like Element of PFuncs (D,F)
X is non empty (D,F)
K20(D,F) is set
K19(K20(D,F)) is set
Y is Element of X
D is non empty set
K19(D) is set
F is non empty set
X is Element of K19(D)
Y is Element of F
X --> Y is set
(D,F) is functional non empty (D,F)
X --> Y is Relation-like D -defined F -valued Function-like Element of K19(K20(D,F))
K20(D,F) is set
K19(K20(D,F)) is set
D is non empty set
F is complex-membered ext-real-membered real-membered set
(D,F) is functional non empty (D,F)
X is Relation-like D -defined F -valued Function-like (D,F,(D,F))
D is non empty set
F is complex-membered ext-real-membered real-membered set
(D,F) is functional non empty (D,F)
X is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued (D,F,(D,F))
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued (D,F,(D,F))
X + Y is Relation-like Function-like set
(D,REAL) is functional non empty (D, REAL )
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
fx is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
fx + Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
X - Y is Relation-like Function-like set
- Y is Relation-like Function-like complex-valued set
- 1 is V36() real ext-real non positive set
(- 1) (#) Y is Relation-like Function-like set
X + (- Y) is Relation-like Function-like set
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
fx is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
fx - Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
- Y is Relation-like Function-like complex-valued set
(- 1) (#) Y is Relation-like Function-like set
fx + (- Y) is Relation-like Function-like set
X (#) Y is Relation-like Function-like set
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
fx is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
fx (#) Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
X / Y is Relation-like Function-like set
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
fx is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
fx / Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
D is non empty set
F is complex-membered ext-real-membered real-membered set
(D,F) is functional non empty (D,F)
X is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued (D,F,(D,F))
|.X.| is Relation-like Function-like complex-valued ext-real-valued real-valued set
(D,REAL) is functional non empty (D, REAL )
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
abs Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
- X is Relation-like Function-like complex-valued set
(- 1) (#) X is Relation-like Function-like set
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
- Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
(- 1) (#) Y is Relation-like Function-like set
X ^ is Relation-like Function-like set
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
Y is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y ^ is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
D is non empty set
F is complex-membered ext-real-membered real-membered set
(D,F) is functional non empty (D,F)
X is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued (D,F,(D,F))
Y is V36() real ext-real set
Y (#) X is Relation-like Function-like set
(D,REAL) is functional non empty (D, REAL )
K20(D,F) is complex-valued ext-real-valued real-valued set
K19(K20(D,F)) is set
fx is Relation-like D -defined F -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,F))
Y (#) fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
F is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
F is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
X is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
F . (Y,fx) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[Y,fx] is set
{Y,fx} is functional non empty finite set
{Y} is functional non empty V12() finite 1 -element set
{{Y,fx},{Y}} is non empty finite V51() set
F . [Y,fx] is Relation-like Function-like set
(D,REAL,Y,fx) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
X . (Y,fx) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
X . [Y,fx] is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (F,X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[F,X] is set
{F,X} is functional non empty finite set
{F} is functional non empty V12() finite 1 -element set
{{F,X},{F}} is non empty finite V51() set
(D) . [F,X] is Relation-like Function-like set
(D) . (X,F) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[X,F] is set
{X,F} is functional non empty finite set
{X} is functional non empty V12() finite 1 -element set
{{X,F},{X}} is non empty finite V51() set
(D) . [X,F] is Relation-like Function-like set
(D) . (F,X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,X,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (X,F) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
D is non empty set
(D,REAL) is functional non empty (D, REAL )
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (X,Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[X,Y] is set
{X,Y} is functional non empty finite set
{X} is functional non empty V12() finite 1 -element set
{{X,Y},{X}} is non empty finite V51() set
(D) . [X,Y] is Relation-like Function-like set
(D) . (F,((D) . (X,Y))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[F,((D) . (X,Y))] is set
{F,((D) . (X,Y))} is functional non empty finite set
{F} is functional non empty V12() finite 1 -element set
{{F,((D) . (X,Y))},{F}} is non empty finite V51() set
(D) . [F,((D) . (X,Y))] is Relation-like Function-like set
(D) . (F,X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[F,X] is set
{F,X} is functional non empty finite set
{{F,X},{F}} is non empty finite V51() set
(D) . [F,X] is Relation-like Function-like set
(D) . (((D) . (F,X)),Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[((D) . (F,X)),Y] is set
{((D) . (F,X)),Y} is functional non empty finite set
{((D) . (F,X))} is functional non empty V12() finite 1 -element set
{{((D) . (F,X)),Y},{((D) . (F,X))}} is non empty finite V51() set
(D) . [((D) . (F,X)),Y] is Relation-like Function-like set
(D) . (X,Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D) . (F,((D) . (X,Y))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[F,((D) . (X,Y))] is set
{F,((D) . (X,Y))} is functional non empty finite set
{{F,((D) . (X,Y))},{F}} is non empty finite V51() set
(D) . [F,((D) . (X,Y))] is Relation-like Function-like set
(D,REAL,X,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (F,(D,REAL,X,Y)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[F,(D,REAL,X,Y)] is set
{F,(D,REAL,X,Y)} is functional non empty finite set
{{F,(D,REAL,X,Y)},{F}} is non empty finite V51() set
(D) . [F,(D,REAL,X,Y)] is Relation-like Function-like set
(D,REAL,F,(D,REAL,X,Y)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,F,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,(D,REAL,F,X),Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (F,X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,((D) . (F,X)),Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (((D) . (F,X)),Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[((D) . (F,X)),Y] is set
{((D) . (F,X)),Y} is functional non empty finite set
{((D) . (F,X))} is functional non empty V12() finite 1 -element set
{{((D) . (F,X)),Y},{((D) . (F,X))}} is non empty finite V51() set
(D) . [((D) . (F,X)),Y] is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
[#] D is Element of K19(D)
K19(D) is set
(D,REAL,([#] D),0) is Relation-like D -defined REAL -valued RAT -valued INT -valued Function-like complex-valued ext-real-valued real-valued natural-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
dom (D,REAL,([#] D),0) is Element of K19(D)
Y is Element of D
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,X,(D,REAL,([#] D),0)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dom (D,REAL,X,(D,REAL,([#] D),0)) is Element of K19(D)
(D,REAL,X,(D,REAL,([#] D),0)) . Y is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(D,REAL,([#] D),0) . Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal Element of REAL
(X . Y) + ((D,REAL,([#] D),0) . Y) is V36() real ext-real Element of REAL
(X . Y) + 0 is V36() real ext-real Element of REAL
dom X is Element of K19(D)
(dom X) /\ D is Element of K19(D)
(D) . (X,(D,REAL,([#] D),0)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[X,(D,REAL,([#] D),0)] is set
{X,(D,REAL,([#] D),0)} is functional non empty finite set
{X} is functional non empty V12() finite 1 -element set
{{X,(D,REAL,([#] D),0)},{X}} is non empty finite V51() set
(D) . [X,(D,REAL,([#] D),0)] is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
the_unity_wrt (D) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[#] D is Element of K19(D)
K19(D) is set
(D,REAL,([#] D),0) is Relation-like D -defined REAL -valued RAT -valued INT -valued Function-like complex-valued ext-real-valued real-valued natural-valued (D, REAL ,(D,REAL))
D is non empty set
(D,REAL) is functional non empty (D, REAL )
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
[#] D is Element of K19(D)
K19(D) is set
(D,REAL,([#] D),0) is Relation-like D -defined REAL -valued RAT -valued INT -valued Function-like complex-valued ext-real-valued real-valued natural-valued (D, REAL ,(D,REAL))
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ F is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
D is non empty set
(D,REAL) is functional non empty (D, REAL )
<*> (D,REAL) is Relation-like NAT -defined (D,REAL) -valued Function-like one-to-one constant functional empty ordinal natural V36() real ext-real non positive non negative finite finite-yielding V51() cardinal {} -element complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of (D,REAL)
(D,(<*> (D,REAL))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ (<*> (D,REAL)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[#] D is Element of K19(D)
K19(D) is set
(D,REAL,([#] D),0) is Relation-like D -defined REAL -valued RAT -valued INT -valued Function-like complex-valued ext-real-valued real-valued natural-valued (D, REAL ,(D,REAL))
the_unity_wrt (D) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ F is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*X*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
F ^ <*X*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(F ^ <*X*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (F ^ <*X*>) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,F),X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . (((D) $$ F),X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[((D) $$ F),X] is set
{((D) $$ F),X} is functional non empty finite set
{((D) $$ F)} is functional non empty V12() finite 1 -element set
{{((D) $$ F),X},{((D) $$ F)}} is non empty finite V51() set
(D) . [((D) $$ F),X] is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
F ^ X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(F ^ X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ (F ^ X) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ F is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ X is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,F),(D,X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) . ((D,F),(D,X)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
[(D,F),(D,X)] is set
{(D,F),(D,X)} is functional non empty finite set
{(D,F)} is functional non empty V12() finite 1 -element set
{{(D,F),(D,X)},{(D,F)}} is non empty finite V51() set
(D) . [(D,F),(D,X)] is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ F is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*X*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*X*> ^ F is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(<*X*> ^ F)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (<*X*> ^ F) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,X,(D,F)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,<*X*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ <*X*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,<*X*>),(D,F)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*F,X*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,<*F,X*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ <*F,X*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,F,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*F*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*X*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*F*> ^ <*X*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
1 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,(<*F*> ^ <*X*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (<*F*> ^ <*X*>) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,<*F*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ <*F*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,<*F*>),X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*F,X,Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite 3 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,<*F,X,Y*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ <*F,X,Y*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,F,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,(D,REAL,F,X),Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*F,X*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*F,X*> ^ <*Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite 2 + 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
2 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,(<*F,X*> ^ <*Y*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (<*F,X*> ^ <*Y*>) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,<*F,X*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ <*F,X*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,<*F,X*>),Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ F is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ X is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
len F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,fx) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ fx is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ Y is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx . (Y + 1) is Relation-like Function-like set
rng fx is functional finite Element of K19((D,REAL))
K19((D,REAL)) is set
rng Y is functional finite Element of K19((D,REAL))
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
xf is ordinal natural V36() real ext-real non negative finite cardinal set
Y . xf is Relation-like Function-like set
Y | xf is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
Y /^ xf is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(Y | xf) ^ (Y /^ xf) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (Y | xf) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx | Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
<*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(fx | Y) ^ <*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
xf - 1 is V36() real ext-real Element of REAL
max (0,(xf - 1)) is V36() real ext-real Element of REAL
x is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
x + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg x is finite x -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg xf is finite xf -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Y | xf) . xf is Relation-like Function-like set
(Y | xf) | x is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
((Y | xf) | x) ^ <*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(Y | xf) | (Seg x) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,(D,REAL)))
K20(NAT,(D,REAL)) is V12() non finite set
K19(K20(NAT,(D,REAL))) is V12() non finite set
Y | (Seg xf) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,(D,REAL)))
(Y | (Seg xf)) | (Seg x) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,(D,REAL)))
(Seg xf) /\ (Seg x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y | ((Seg xf) /\ (Seg x)) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,(D,REAL)))
Y | (Seg x) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSubsequence-like Element of K19(K20(NAT,(D,REAL)))
Y | x is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
x is set
Coim (fx,x) is finite set
{x} is non empty V12() finite 1 -element set
fx " {x} is finite set
card (Coim (fx,x)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim (Y,x) is finite set
Y " {x} is finite set
card (Coim (Y,x)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(fx | Y) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((fx | Y) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
<*dr*> " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (<*dr*> " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card ((fx | Y) " {x})) + (card (<*dr*> " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(Y | x) ^ <*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
((Y | x) ^ <*dr*>) ^ (Y /^ xf) is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(((Y | x) ^ <*dr*>) ^ (Y /^ xf)) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((((Y | x) ^ <*dr*>) ^ (Y /^ xf)) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
((Y | x) ^ <*dr*>) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (((Y | x) ^ <*dr*>) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(Y /^ xf) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y /^ xf) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card (((Y | x) ^ <*dr*>) " {x})) + (card ((Y /^ xf) " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(Y | x) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y | x) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card ((Y | x) " {x})) + (card (<*dr*> " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
((card ((Y | x) " {x})) + (card (<*dr*> " {x}))) + (card ((Y /^ xf) " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(card ((Y | x) " {x})) + (card ((Y /^ xf) " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
((card ((Y | x) " {x})) + (card ((Y /^ xf) " {x}))) + (card (<*dr*> " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(Y | x) ^ (Y /^ xf) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
((Y | x) ^ (Y /^ xf)) " {x} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (((Y | x) ^ (Y /^ xf)) " {x}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card (((Y | x) ^ (Y /^ xf)) " {x})) + (card (<*dr*> " {x})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Coim ((fx | Y),x) is finite set
(fx | Y) " {x} is finite set
card (Coim ((fx | Y),x)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim (((Y | x) ^ (Y /^ xf)),x) is finite set
((Y | x) ^ (Y /^ xf)) " {x} is finite set
card (Coim (((Y | x) ^ (Y /^ xf)),x)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
len (fx | Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,(fx | Y)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (fx | Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,((Y | x) ^ (Y /^ xf))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ ((Y | x) ^ (Y /^ xf)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,<*dr*>) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ <*dr*> is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,((Y | x) ^ (Y /^ xf))),(D,<*dr*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,(Y | x)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (Y | x) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,(Y /^ xf)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (Y /^ xf) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,(Y | x)),(D,(Y /^ xf))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,(D,REAL,(D,(Y | x)),(D,(Y /^ xf))),(D,<*dr*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,(D,(Y | x)),(D,<*dr*>)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,REAL,(D,REAL,(D,(Y | x)),(D,<*dr*>)),(D,(Y /^ xf))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D,(Y | xf)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (Y | xf) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,(Y | xf)),(D,(Y /^ xf))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ Y is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,fx) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ fx is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
<*> (D,REAL) is Relation-like NAT -defined (D,REAL) -valued Function-like one-to-one constant functional empty ordinal natural V36() real ext-real non positive non negative finite finite-yielding V51() cardinal {} -element complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of (D,REAL)
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng X is finite set
Y is set
fx is ordinal natural V36() real ext-real non negative finite cardinal set
X . fx is set
F . fx is set
chi ((F . fx),D) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y . fx is Relation-like Function-like set
F . fx is set
chi ((F . fx),D) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len X) is finite len X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len Y) is finite len Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is ordinal natural V36() real ext-real non negative finite cardinal set
X . fx is Relation-like Function-like set
F . fx is set
chi ((F . fx),D) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
Y . fx is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
X is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
min ((len X),(len F)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
Seg (min ((len X),(len F))) is finite min ((len X),(len F)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is ordinal natural V36() real ext-real non negative finite cardinal set
X . fx is V36() real ext-real Element of REAL
F . fx is Relation-like Function-like set
dom F is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y is V36() real ext-real Element of REAL
(D,REAL,dr,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dx is V36() real ext-real Element of REAL
rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dx (#) rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
X . Y is V36() real ext-real Element of REAL
F . Y is Relation-like Function-like set
fx . Y is Relation-like Function-like set
xf is V36() real ext-real Element of REAL
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
xf (#) dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (min ((len X),(len F))) is finite min ((len X),(len F)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len fx) is finite len fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len Y) is finite len Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dr is ordinal natural V36() real ext-real non negative finite cardinal set
X . dr is V36() real ext-real Element of REAL
dom F is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
F . dr is Relation-like Function-like set
fx . dr is Relation-like Function-like set
rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
xf is V36() real ext-real Element of REAL
(D,REAL,rd,xf) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y . dr is Relation-like Function-like set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
X is Element of D
Seg (len F) is finite len F -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y is ordinal natural V36() real ext-real non negative finite cardinal set
F . Y is Relation-like Function-like set
(F . Y) . X is set
dom F is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
fx . X is V36() real ext-real Element of REAL
Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y . fx is V36() real ext-real Element of REAL
F . fx is Relation-like Function-like set
(F . fx) . X is set
Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom Y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len Y) is finite len Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len fx) is finite len fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y is ordinal natural V36() real ext-real non negative finite cardinal set
Y . Y is V36() real ext-real Element of REAL
F . Y is Relation-like Function-like set
(F . Y) . X is set
fx . Y is V36() real ext-real Element of REAL
D is non empty set
F is non empty set
(D,F) is functional non empty (D,F)
D is non empty set
F is non empty set
(D,F) is functional non empty (D,F)
X is Relation-like NAT -defined (D,F) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,F)
Y is Element of D
fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
X | fx is Relation-like NAT -defined (D,F) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,F)
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (X | fx) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(X | fx) . Y is Relation-like Function-like set
dom ((X | fx) . Y) is set
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len (X | fx) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (X | fx)) is finite len (X | fx) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
X . Y is Relation-like Function-like set
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
D is non empty set
F is non empty set
(D,F) is functional non empty (D,F)
X is Relation-like NAT -defined (D,F) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,F)
Y is Element of D
fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
X /^ fx is Relation-like NAT -defined (D,F) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,F)
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (X /^ fx) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(X /^ fx) . Y is Relation-like Function-like set
dom ((X /^ fx) . Y) is set
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (X /^ fx) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(len X) - fx is V36() real ext-real Element of REAL
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
X . (Y + fx) is Relation-like Function-like set
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Element of D
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ X is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
dom (D,X) is Element of K19(D)
K19(D) is set
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,fx) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ fx is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
dom (D,fx) is Element of K19(D)
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len fx) is finite len fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx . 1 is Relation-like Function-like set
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
dr is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx . dr is Relation-like Function-like set
dom (fx . dr) is set
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx . (Y + 1) is Relation-like Function-like set
fx | Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len (fx | Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(fx | Y) ^ <*Y*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(fx | Y)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (fx | Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,(fx | Y)),Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dom (D,(fx | Y)) is Element of K19(D)
dom Y is Element of K19(D)
(dom (D,(fx | Y))) /\ (dom Y) is Element of K19(D)
dom (D,(fx | Y)) is Element of K19(D)
dom Y is Element of K19(D)
(dom (D,(fx | Y))) /\ (dom Y) is Element of K19(D)
xf is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx . xf is Relation-like Function-like set
dom (fx . xf) is set
Seg Y is finite Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (fx | Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (fx | Y)) is finite len (fx | Y) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(fx | Y) . xf is Relation-like Function-like set
dom (fx . xf) is set
dom (fx . xf) is set
dom (fx . xf) is set
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ Y is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
dom (D,Y) is Element of K19(D)
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
X is Element of D
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
F | Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(F | Y),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) | Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len (D,F,X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (D,(F | Y),X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (F | Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len ((D,F,X) | Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom F is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len F) is finite len F -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (D,F,X) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (D,F,X)) is finite len (D,F,X) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (D,(F | Y),X) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (D,(F | Y),X)) is finite len (D,(F | Y),X) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom ((D,F,X) | Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (F | Y)) is finite len (F | Y) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx is ordinal natural V36() real ext-real non negative finite cardinal set
F . fx is Relation-like Function-like set
((D,F,X) | Y) . fx is V36() real ext-real Element of REAL
(D,F,X) . fx is V36() real ext-real Element of REAL
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
Y . X is V36() real ext-real Element of REAL
(F | Y) . fx is Relation-like Function-like set
(D,(F | Y),X) . fx is V36() real ext-real Element of REAL
D is non empty set
F is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(D,F) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,REAL) is functional non empty (D, REAL )
X is Element of D
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (D,F) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(D,F) . Y is Relation-like Function-like set
dom ((D,F) . Y) is set
F . Y is set
chi ((F . Y),D) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Element of D
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,X,Y) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
min ((len Y),(len X)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (D,X,Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(D,X,Y) . Y is Relation-like Function-like set
dom ((D,X,Y) . Y) is set
len (D,X,Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom X is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
X . Y is Relation-like Function-like set
xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dom xf is Element of K19(D)
K19(D) is set
Y . Y is V36() real ext-real Element of REAL
rd is V36() real ext-real Element of REAL
(D,REAL,xf,rd) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
D is non empty set
F is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(D,F) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,REAL) is functional non empty (D, REAL )
X is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,(D,F),X) is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
Y is Element of D
D is non empty set
(D,REAL) is functional non empty (D, REAL )
F is Element of D
X is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) is Relation-like K20((D,REAL),(D,REAL)) -defined (D,REAL) -valued Function-like V18(K20((D,REAL),(D,REAL)),(D,REAL)) Element of K19(K20(K20((D,REAL),(D,REAL)),(D,REAL)))
K20((D,REAL),(D,REAL)) is set
K20(K20((D,REAL),(D,REAL)),(D,REAL)) is set
K19(K20(K20((D,REAL),(D,REAL)),(D,REAL))) is set
(D) $$ X is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,X) . F is V36() real ext-real Element of REAL
(D,X,F) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,X,F) is V36() real ext-real Element of REAL
K197() $$ (D,X,F) is V36() real ext-real Element of REAL
len X is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,fx) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ fx is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,fx) . F is V36() real ext-real Element of REAL
(D,fx,F) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,fx,F) is V36() real ext-real Element of REAL
K197() $$ (D,fx,F) is V36() real ext-real Element of REAL
fx | Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len (fx | Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom fx is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx . (Y + 1) is Relation-like Function-like set
Seg (len fx) is finite len fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (D,fx,F) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len (D,fx,F) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (D,fx,F)) is finite len (D,fx,F) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
<*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like constant non empty V12() finite 1 -element FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(fx | Y) ^ <*dr*> is Relation-like NAT -defined (D,REAL) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
(D,(fx | Y)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ (fx | Y) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,REAL,(D,(fx | Y)),dr) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
dom dr is Element of K19(D)
K19(D) is set
xf is ordinal natural V36() real ext-real non negative finite cardinal set
(D,fx,F) . xf is V36() real ext-real Element of REAL
dr . F is V36() real ext-real Element of REAL
<*(dr . F)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
<*(dr . F)*> . xf is V36() real ext-real Element of REAL
len <*(dr . F)*> is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,fx,F) . (Y + 1) is V36() real ext-real Element of REAL
dr . F is V36() real ext-real Element of REAL
dom (D,(fx | Y)) is Element of K19(D)
(dom (D,(fx | Y))) /\ (dom dr) is Element of K19(D)
dom (D,REAL,(D,(fx | Y)),dr) is Element of K19(D)
(D,(fx | Y)) . F is V36() real ext-real Element of REAL
((D,(fx | Y)) . F) + (dr . F) is V36() real ext-real Element of REAL
(D,(fx | Y),F) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(fx | Y),F) is V36() real ext-real Element of REAL
K197() $$ (D,(fx | Y),F) is V36() real ext-real Element of REAL
(Sum (D,(fx | Y),F)) + (dr . F) is V36() real ext-real Element of REAL
(D,fx,F) | Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum ((D,fx,F) | Y) is V36() real ext-real Element of REAL
K197() $$ ((D,fx,F) | Y) is V36() real ext-real Element of REAL
(Sum ((D,fx,F) | Y)) + (dr . F) is V36() real ext-real Element of REAL
<*(dr . F)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
((D,fx,F) | Y) ^ <*(dr . F)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (((D,fx,F) | Y) ^ <*(dr . F)*>) is V36() real ext-real Element of REAL
K197() $$ (((D,fx,F) | Y) ^ <*(dr . F)*>) is V36() real ext-real Element of REAL
Y is Relation-like NAT -defined (D,REAL) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (D,REAL)
len Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued (D, REAL ,(D,REAL))
(D) $$ Y is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (D,REAL)
(D,Y) . F is V36() real ext-real Element of REAL
(D,Y,F) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,Y,F) is V36() real ext-real Element of REAL
K197() $$ (D,Y,F) is V36() real ext-real Element of REAL
<*> (D,REAL) is Relation-like NAT -defined (D,REAL) -valued Function-like one-to-one constant functional empty ordinal natural V36() real ext-real non positive non negative finite finite-yielding V51() cardinal {} -element complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of (D,REAL)
[#] D is Element of K19(D)
K19(D) is set
(D,REAL,([#] D),0) is Relation-like D -defined REAL -valued RAT -valued INT -valued Function-like complex-valued ext-real-valued real-valued natural-valued (D, REAL ,(D,REAL))
(D,REAL,([#] D),0) . F is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal Element of REAL
len (D,Y,F) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom X is Element of K19(D)
Y is set
Y is set
Y is Element of D
X . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
((F . Y)) is V36() real ext-real Element of REAL
max ((F . Y),0) is V36() real ext-real set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom X is Element of K19(D)
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom Y is Element of K19(D)
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom X is Element of K19(D)
Y is set
Y is set
Y is Element of D
X . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
((F . Y)) is V36() real ext-real Element of REAL
- (F . Y) is V36() real ext-real set
max ((- (F . Y)),0) is V36() real ext-real set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom X is Element of K19(D)
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom Y is Element of K19(D)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) - (D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- (D,F) is Relation-like Function-like complex-valued set
(- 1) (#) (D,F) is Relation-like Function-like set
(D,F) + (- (D,F)) is Relation-like Function-like set
abs F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) + (D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
2 (#) (D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F + (abs F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
(dom F) /\ (dom F) is Element of K19(D)
dom (D,F) is Element of K19(D)
dom (D,F) is Element of K19(D)
- (D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (- (D,F)) is Element of K19(D)
(D,F) + (- (D,F)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((D,F) + (- (D,F))) is Element of K19(D)
X is Element of D
((D,F) - (D,F)) . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
((D,F) . X) - ((D,F) . X) is V36() real ext-real Element of REAL
F . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
max ((F . X),0) is V36() real ext-real set
((F . X)) - ((D,F) . X) is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
- (F . X) is V36() real ext-real set
max ((- (F . X)),0) is V36() real ext-real set
((F . X)) - ((F . X)) is V36() real ext-real Element of REAL
dom (abs F) is Element of K19(D)
dom ((D,F) + (D,F)) is Element of K19(D)
X is Element of D
((D,F) + (D,F)) . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
((D,F) . X) + ((D,F) . X) is V36() real ext-real Element of REAL
F . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
max ((F . X),0) is V36() real ext-real set
((F . X)) + ((D,F) . X) is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
- (F . X) is V36() real ext-real set
max ((- (F . X)),0) is V36() real ext-real set
((F . X)) + ((F . X)) is V36() real ext-real Element of REAL
abs (F . X) is V36() real ext-real Element of REAL
(abs F) . X is V36() real ext-real Element of REAL
dom (2 (#) (D,F)) is Element of K19(D)
dom (F + (abs F)) is Element of K19(D)
X is Element of D
(2 (#) (D,F)) . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
2 * ((D,F) . X) is V36() real ext-real Element of REAL
F . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
max ((F . X),0) is V36() real ext-real set
2 * ((F . X)) is V36() real ext-real Element of REAL
abs (F . X) is V36() real ext-real Element of REAL
(F . X) + (abs (F . X)) is V36() real ext-real Element of REAL
(abs F) . X is V36() real ext-real Element of REAL
(F . X) + ((abs F) . X) is V36() real ext-real Element of REAL
(F + (abs F)) . X is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
F " {X} is Element of K19(D)
K19(D) is set
(D,F) " {X} is Element of K19(D)
dom (D,F) is Element of K19(D)
dom F is Element of K19(D)
Y is set
fx is Element of D
F . fx is V36() real ext-real Element of REAL
(D,F) . fx is V36() real ext-real Element of REAL
((F . fx)) is V36() real ext-real Element of REAL
max ((F . fx),0) is V36() real ext-real set
Y is set
fx is Element of D
(D,F) . fx is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
((F . fx)) is V36() real ext-real Element of REAL
max ((F . fx),0) is V36() real ext-real set
max ((F . fx),0) is V36() real ext-real Element of REAL
left_closed_halfline 0 is complex-membered ext-real-membered real-membered Element of K19(REAL)
K421(-infty,0) is set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
X " (left_closed_halfline 0) is Element of K19(F)
K19(F) is set
(F,X) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,X) " {0} is Element of K19(F)
dom (F,X) is Element of K19(F)
dom X is Element of K19(F)
{ b1 where b1 is V36() real ext-real Element of REAL : b1 <= 0 } is set
Y is set
fx is Element of F
X . fx is V36() real ext-real Element of REAL
max ((X . fx),0) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
(F,X) . fx is V36() real ext-real Element of REAL
((X . fx)) is V36() real ext-real Element of REAL
max ((X . fx),0) is V36() real ext-real set
Y is set
fx is Element of F
(F,X) . fx is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
((X . fx)) is V36() real ext-real Element of REAL
max ((X . fx),0) is V36() real ext-real set
max ((X . fx),0) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is Element of D
(D,F) . X is V36() real ext-real Element of REAL
dom F is Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
F . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
max ((F . X),0) is V36() real ext-real set
max ((F . X),0) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
- X is V36() real ext-real Element of REAL
{(- X)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
F " {(- X)} is Element of K19(D)
K19(D) is set
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(D,F) " {X} is Element of K19(D)
dom (D,F) is Element of K19(D)
dom F is Element of K19(D)
Y is set
fx is Element of D
F . fx is V36() real ext-real Element of REAL
(D,F) . fx is V36() real ext-real Element of REAL
((F . fx)) is V36() real ext-real Element of REAL
- (F . fx) is V36() real ext-real set
max ((- (F . fx)),0) is V36() real ext-real set
Y is set
fx is Element of D
(D,F) . fx is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
((F . fx)) is V36() real ext-real Element of REAL
- (F . fx) is V36() real ext-real set
max ((- (F . fx)),0) is V36() real ext-real set
- (F . fx) is V36() real ext-real Element of REAL
max ((- (F . fx)),0) is V36() real ext-real Element of REAL
right_closed_halfline 0 is complex-membered ext-real-membered real-membered Element of K19(REAL)
K420(0,+infty) is set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
X " (right_closed_halfline 0) is Element of K19(F)
K19(F) is set
(F,X) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,X) " {0} is Element of K19(F)
dom (F,X) is Element of K19(F)
dom X is Element of K19(F)
{ b1 where b1 is V36() real ext-real Element of REAL : 0 <= b1 } is set
Y is set
fx is Element of F
X . fx is V36() real ext-real Element of REAL
- (X . fx) is V36() real ext-real Element of REAL
max ((- (X . fx)),0) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
(F,X) . fx is V36() real ext-real Element of REAL
((X . fx)) is V36() real ext-real Element of REAL
- (X . fx) is V36() real ext-real set
max ((- (X . fx)),0) is V36() real ext-real set
Y is set
fx is Element of F
(F,X) . fx is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
((X . fx)) is V36() real ext-real Element of REAL
- (X . fx) is V36() real ext-real set
max ((- (X . fx)),0) is V36() real ext-real set
- (X . fx) is V36() real ext-real Element of REAL
max ((- (X . fx)),0) is V36() real ext-real Element of REAL
- 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is Element of D
(D,F) . X is V36() real ext-real Element of REAL
dom F is Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
F . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
- (F . X) is V36() real ext-real set
max ((- (F . X)),0) is V36() real ext-real set
- (F . X) is V36() real ext-real Element of REAL
max ((- (F . X)),0) is V36() real ext-real Element of REAL
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is non empty set
K20(X,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(X,REAL)) is V12() non finite set
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,Y) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
fx is Relation-like X -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(X,REAL))
(X,fx) is Relation-like X -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(X,REAL))
Y is V36() real ext-real Element of REAL
Coim (Y,Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
Y " {Y} is set
Coim ((F,Y),Y) is set
(F,Y) " {Y} is set
Coim (fx,Y) is set
fx " {Y} is set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
card (Coim ((X,fx),Y)) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Y " (left_closed_halfline 0) is Element of K19(F)
K19(F) is set
(F,Y) " {0} is Element of K19(F)
fx " (left_closed_halfline 0) is Element of K19(X)
K19(X) is set
(X,fx) " {0} is Element of K19(X)
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is Element of K19(F)
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
rng (F,Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (F,Y) is Element of K19(F)
K19(F) is set
dr is Element of F
(F,Y) . dr is V36() real ext-real Element of REAL
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is Element of K19(F)
rng (X,fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (X,fx) is Element of K19(X)
K19(X) is set
dr is Element of X
(X,fx) . dr is V36() real ext-real Element of REAL
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Coim ((F,Y),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((X,fx),Y)) is ordinal cardinal set
(F,Y) " {Y} is Element of K19(F)
K19(F) is set
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
K19(X) is set
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Coim ((F,Y),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((X,fx),Y)) is ordinal cardinal set
(F,Y) " {Y} is Element of K19(F)
K19(F) is set
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
K19(X) is set
card ((X,fx) " {Y}) is ordinal cardinal set
rng (F,Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (X,fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is non empty set
K20(X,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(X,REAL)) is V12() non finite set
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,Y) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
fx is Relation-like X -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(X,REAL))
(X,fx) is Relation-like X -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(X,REAL))
Y is V36() real ext-real Element of REAL
- Y is V36() real ext-real Element of REAL
Coim (Y,(- Y)) is set
{(- Y)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
Y " {(- Y)} is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is Element of K19(F)
K19(F) is set
Coim (fx,(- Y)) is set
fx " {(- Y)} is set
(X,fx) " {Y} is Element of K19(X)
K19(X) is set
Coim ((F,Y),Y) is set
(F,Y) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((X,fx),Y)) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Y " (right_closed_halfline 0) is Element of K19(F)
K19(F) is set
(F,Y) " {0} is Element of K19(F)
fx " (right_closed_halfline 0) is Element of K19(X)
K19(X) is set
(X,fx) " {0} is Element of K19(X)
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is Element of K19(F)
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
rng (F,Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (F,Y) is Element of K19(F)
K19(F) is set
dr is Element of F
(F,Y) . dr is V36() real ext-real Element of REAL
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is Element of K19(F)
rng (X,fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (X,fx) is Element of K19(X)
K19(X) is set
dr is Element of X
(X,fx) . dr is V36() real ext-real Element of REAL
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Coim ((F,Y),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((X,fx),Y)) is ordinal cardinal set
(F,Y) " {Y} is Element of K19(F)
K19(F) is set
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
K19(X) is set
card ((X,fx) " {Y}) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Coim ((F,Y),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F,Y) " {Y} is set
card (Coim ((F,Y),Y)) is ordinal cardinal set
Coim ((X,fx),Y) is set
(X,fx) " {Y} is set
card (Coim ((X,fx),Y)) is ordinal cardinal set
(F,Y) " {Y} is Element of K19(F)
K19(F) is set
card ((F,Y) " {Y}) is ordinal cardinal set
(X,fx) " {Y} is Element of K19(X)
K19(X) is set
card ((X,fx) " {Y}) is ordinal cardinal set
rng (F,Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (X,fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is finite Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is finite Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,X) is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,X) is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like F -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,Y) is Relation-like F -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(F,Y) is Relation-like F -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(D,X) " {0} is finite Element of K19(D)
K19(D) is set
(D,X) " {0} is finite Element of K19(D)
(F,Y) " {0} is finite Element of K19(F)
K19(F) is set
(F,Y) " {0} is finite Element of K19(F)
(left_closed_halfline 0) /\ (right_closed_halfline 0) is complex-membered ext-real-membered real-membered Element of K19(REAL)
[.0,0.] is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng X is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
X " (rng X) is finite Element of K19(D)
X " REAL is finite Element of K19(D)
dom X is finite Element of K19(D)
X " (left_closed_halfline 0) is finite Element of K19(D)
X " (right_closed_halfline 0) is finite Element of K19(D)
rng Y is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
Y " (rng Y) is finite Element of K19(F)
Y " REAL is finite Element of K19(F)
dom Y is finite Element of K19(F)
Y " (left_closed_halfline 0) is finite Element of K19(F)
Y " (right_closed_halfline 0) is finite Element of K19(F)
(left_closed_halfline 0) \/ (right_closed_halfline 0) is complex-membered ext-real-membered real-membered Element of K19(REAL)
].0,0.[ is complex-membered ext-real-membered real-membered Element of K19(REAL)
REAL \ ].0,0.[ is complex-membered ext-real-membered real-membered Element of K19(REAL)
REAL \ {} is complex-membered ext-real-membered real-membered Element of K19(REAL)
gx is finite set
x is finite set
gx \/ x is finite set
x is finite set
Y is finite set
x \/ Y is finite set
Y " ((left_closed_halfline 0) \/ (right_closed_halfline 0)) is finite Element of K19(F)
card (gx \/ x) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card gx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card x is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card gx) + (card x) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
gx /\ x is finite set
card (gx /\ x) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
((card gx) + (card x)) - (card (gx /\ x)) is V36() real ext-real Element of REAL
((card gx) + (card x)) - (card (gx \/ x)) is V36() real ext-real Element of REAL
card (x \/ Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card x is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card x) + (card Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
x /\ Y is finite set
card (x /\ Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
((card x) + (card Y)) - (card (x /\ Y)) is V36() real ext-real Element of REAL
((card x) + (card Y)) - (card (x \/ Y)) is V36() real ext-real Element of REAL
dom (D,X) is finite Element of K19(D)
dom (F,Y) is finite Element of K19(F)
dy is V36() real ext-real Element of REAL
Coim (X,dy) is set
{dy} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {dy} is finite set
Coim ((D,X),dy) is set
(D,X) " {dy} is finite set
Coim (Y,dy) is set
Y " {dy} is finite set
Coim ((F,Y),dy) is set
(F,Y) " {dy} is finite set
card (Coim (X,dy)) is ordinal cardinal set
card (Coim (Y,dy)) is ordinal cardinal set
dy is V36() real ext-real Element of REAL
{dy} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {dy} is finite Element of K19(D)
(X " (left_closed_halfline 0)) /\ (X " (right_closed_halfline 0)) is finite Element of K19(D)
Y " {dy} is finite Element of K19(F)
(Y " (left_closed_halfline 0)) /\ (Y " (right_closed_halfline 0)) is finite Element of K19(F)
card (X " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card (Y " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dy is V36() real ext-real Element of REAL
- dy is V36() real ext-real Element of REAL
- (- dy) is V36() real ext-real Element of REAL
Coim (X,dy) is set
{dy} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {dy} is finite set
Coim ((D,X),(- dy)) is set
{(- dy)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(D,X) " {(- dy)} is finite set
Coim (Y,dy) is set
Y " {dy} is finite set
Coim ((F,Y),(- dy)) is set
(F,Y) " {(- dy)} is finite set
card (Coim (X,dy)) is ordinal cardinal set
card (Coim (Y,dy)) is ordinal cardinal set
dy is V36() real ext-real Element of REAL
Coim (X,dy) is set
{dy} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {dy} is finite set
card (Coim (X,dy)) is ordinal cardinal set
Coim (Y,dy) is set
Y " {dy} is finite set
card (Coim (Y,dy)) is ordinal cardinal set
X " {dy} is finite Element of K19(D)
card (X " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Y " {dy} is finite Element of K19(F)
card (Y " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dy is V36() real ext-real Element of REAL
Coim (X,dy) is set
{dy} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {dy} is finite set
card (Coim (X,dy)) is ordinal cardinal set
Coim (Y,dy) is set
Y " {dy} is finite set
card (Coim (Y,dy)) is ordinal cardinal set
X " {dy} is finite Element of K19(D)
card (X " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Y " {dy} is finite Element of K19(F)
card (Y " {dy}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
(D,F) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,(F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((D,F) | X) is Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
(dom (D,F)) /\ X is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ X is Element of K19(D)
dom (F | X) is Element of K19(D)
dom (D,(F | X)) is Element of K19(D)
Y is Element of D
((D,F) | X) . Y is V36() real ext-real Element of REAL
(D,F) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
((F . Y)) is V36() real ext-real Element of REAL
max ((F . Y),0) is V36() real ext-real set
(F | X) . Y is V36() real ext-real Element of REAL
(((F | X) . Y)) is V36() real ext-real Element of REAL
max (((F | X) . Y),0) is V36() real ext-real set
(D,(F | X)) . Y is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
(D,F) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,(F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((D,F) | X) is Element of K19(D)
K19(D) is set
dom (D,F) is Element of K19(D)
(dom (D,F)) /\ X is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ X is Element of K19(D)
dom (F | X) is Element of K19(D)
dom (D,(F | X)) is Element of K19(D)
Y is Element of D
((D,F) | X) . Y is V36() real ext-real Element of REAL
(D,F) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
((F . Y)) is V36() real ext-real Element of REAL
- (F . Y) is V36() real ext-real set
max ((- (F . Y)),0) is V36() real ext-real set
(F | X) . Y is V36() real ext-real Element of REAL
(((F | X) . Y)) is V36() real ext-real Element of REAL
- ((F | X) . Y) is V36() real ext-real set
max ((- ((F | X) . Y)),0) is V36() real ext-real set
(D,(F | X)) . Y is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (D,F) is Element of K19(D)
X is Element of D
F . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
max ((F . X),0) is V36() real ext-real set
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(- 1) (#) F is Relation-like Function-like set
dom (D,F) is Element of K19(D)
X is Element of D
F . X is V36() real ext-real Element of REAL
(D,F) . X is V36() real ext-real Element of REAL
((F . X)) is V36() real ext-real Element of REAL
- (F . X) is V36() real ext-real set
max ((- (F . X)),0) is V36() real ext-real set
- (F . X) is V36() real ext-real Element of REAL
(- F) . X is V36() real ext-real Element of REAL
dom (- F) is Element of K19(D)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F - 0 is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set
(- 0) + F is Relation-like Function-like set
X is Element of D
dom F is Element of K19(D)
K19(D) is set
(F - 0) . X is V36() real ext-real Element of REAL
F . X is V36() real ext-real Element of REAL
(F . X) - 0 is V36() real ext-real Element of REAL
dom (F - 0) is Element of K19(D)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
F - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is V36() real ext-real set
(- X) + F is Relation-like Function-like set
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(F | Y) - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(- X) + (F | Y) is Relation-like Function-like set
(F - X) | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((F | Y) - X) is Element of K19(D)
K19(D) is set
dom (F | Y) is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
dom (F - X) is Element of K19(D)
(dom (F - X)) /\ Y is Element of K19(D)
dom ((F - X) | Y) is Element of K19(D)
fx is Element of D
((F | Y) - X) . fx is V36() real ext-real Element of REAL
(F | Y) . fx is V36() real ext-real Element of REAL
((F | Y) . fx) - X is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
(F . fx) - X is V36() real ext-real Element of REAL
(F - X) . fx is V36() real ext-real Element of REAL
((F - X) | Y) . fx is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
F - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is V36() real ext-real set
(- X) + F is Relation-like Function-like set
Y is V36() real ext-real Element of REAL
Y + X is V36() real ext-real Element of REAL
{(Y + X)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
F " {(Y + X)} is Element of K19(D)
K19(D) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(F - X) " {Y} is Element of K19(D)
fx is set
Y is Element of D
dom F is Element of K19(D)
F . Y is V36() real ext-real Element of REAL
(F . Y) - X is V36() real ext-real Element of REAL
(F - X) . Y is V36() real ext-real Element of REAL
dom (F - X) is Element of K19(D)
fx is set
Y is Element of D
dom (F - X) is Element of K19(D)
dom F is Element of K19(D)
(F - X) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
(F . Y) - X is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
fx is V36() real ext-real Element of REAL
X - fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- fx is V36() real ext-real set
(- fx) + X is Relation-like Function-like set
Y - fx is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
(- fx) + Y is Relation-like Function-like set
rng (X - fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (Y - fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
Coim ((X - fx),Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(X - fx) " {Y} is set
card (Coim ((X - fx),Y)) is ordinal cardinal set
Y + fx is V36() real ext-real Element of REAL
Coim (X,(Y + fx)) is set
{(Y + fx)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {(Y + fx)} is set
card (Coim (X,(Y + fx))) is ordinal cardinal set
Coim (Y,(Y + fx)) is set
Y " {(Y + fx)} is set
card (Coim (Y,(Y + fx))) is ordinal cardinal set
Coim ((Y - fx),Y) is set
(Y - fx) " {Y} is set
card (Coim ((Y - fx),Y)) is ordinal cardinal set
Y is V36() real ext-real Element of REAL
Y - fx is V36() real ext-real Element of REAL
(Y - fx) + fx is V36() real ext-real Element of REAL
Coim (X,Y) is set
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
X " {Y} is set
card (Coim (X,Y)) is ordinal cardinal set
Coim ((X - fx),(Y - fx)) is set
{(Y - fx)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(X - fx) " {(Y - fx)} is set
card (Coim ((X - fx),(Y - fx))) is ordinal cardinal set
Coim ((Y - fx),(Y - fx)) is set
(Y - fx) " {(Y - fx)} is set
card (Coim ((Y - fx),(Y - fx))) is ordinal cardinal set
Coim (Y,Y) is set
Y " {Y} is set
card (Coim (Y,Y)) is ordinal cardinal set
rng X is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng Y is complex-membered ext-real-membered real-membered Element of K19(REAL)
D is V36() real ext-real Element of REAL
F is V36() real ext-real Element of REAL
[.D,F.] is complex-membered ext-real-membered real-membered Element of K19(REAL)
X is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom X is complex-membered ext-real-membered real-membered Element of K19(REAL)
{ b1 where b1 is V36() real ext-real Element of REAL : ( D <= b1 & b1 <= F ) } is set
fx is V36() real ext-real Element of REAL
1 - fx is V36() real ext-real Element of REAL
fx * F is V36() real ext-real Element of REAL
(1 - fx) * F is V36() real ext-real Element of REAL
(fx * F) + ((1 - fx) * F) is V36() real ext-real Element of REAL
fx * D is V36() real ext-real Element of REAL
(1 - fx) * D is V36() real ext-real Element of REAL
(fx * D) + ((1 - fx) * D) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
dr is V36() real ext-real Element of REAL
fx * Y is V36() real ext-real Element of REAL
(1 - fx) * dr is V36() real ext-real Element of REAL
(fx * Y) + ((1 - fx) * dr) is V36() real ext-real Element of REAL
X . ((fx * Y) + ((1 - fx) * dr)) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
fx * (X . Y) is V36() real ext-real Element of REAL
X . dr is V36() real ext-real Element of REAL
(1 - fx) * (X . dr) is V36() real ext-real Element of REAL
(fx * (X . Y)) + ((1 - fx) * (X . dr)) is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
1 - fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
dr is V36() real ext-real Element of REAL
fx * Y is V36() real ext-real Element of REAL
(1 - fx) * dr is V36() real ext-real Element of REAL
(fx * Y) + ((1 - fx) * dr) is V36() real ext-real Element of REAL
X . ((fx * Y) + ((1 - fx) * dr)) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
fx * (X . Y) is V36() real ext-real Element of REAL
X . dr is V36() real ext-real Element of REAL
(1 - fx) * (X . dr) is V36() real ext-real Element of REAL
(fx * (X . Y)) + ((1 - fx) * (X . dr)) is V36() real ext-real Element of REAL
D is V36() real ext-real Element of REAL
F is V36() real ext-real Element of REAL
[.D,F.] is complex-membered ext-real-membered real-membered Element of K19(REAL)
X is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom X is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
(X . Y) - (X . fx) is V36() real ext-real Element of REAL
Y - fx is V36() real ext-real Element of REAL
((X . Y) - (X . fx)) / (Y - fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(X . fx) - (X . Y) is V36() real ext-real Element of REAL
fx - Y is V36() real ext-real Element of REAL
((X . fx) - (X . Y)) / (fx - Y) is V36() real ext-real Element of REAL
Y - Y is V36() real ext-real Element of REAL
(fx - Y) / (Y - Y) is V36() real ext-real Element of REAL
(X . fx) - (X . Y) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) * ((X . fx) - (X . Y)) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) * (X . fx) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) * (X . Y) is V36() real ext-real Element of REAL
(((fx - Y) / (Y - Y)) * (X . fx)) - (((fx - Y) / (Y - Y)) * (X . Y)) is V36() real ext-real Element of REAL
1 - ((fx - Y) / (Y - Y)) is V36() real ext-real Element of REAL
(X . Y) - (X . fx) is V36() real ext-real Element of REAL
(1 - ((fx - Y) / (Y - Y))) * ((X . Y) - (X . fx)) is V36() real ext-real Element of REAL
(1 - ((fx - Y) / (Y - Y))) * (X . Y) is V36() real ext-real Element of REAL
(1 - ((fx - Y) / (Y - Y))) * (X . fx) is V36() real ext-real Element of REAL
((1 - ((fx - Y) / (Y - Y))) * (X . Y)) - ((1 - ((fx - Y) / (Y - Y))) * (X . fx)) is V36() real ext-real Element of REAL
(Y - fx) / (Y - Y) is V36() real ext-real Element of REAL
(Y - Y) " is V36() real ext-real Element of REAL
((Y - Y) ") * (Y - fx) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) + ((Y - fx) / (Y - Y)) is V36() real ext-real Element of REAL
(Y - fx) + (fx - Y) is V36() real ext-real Element of REAL
((Y - fx) + (fx - Y)) / (Y - Y) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) * Y is V36() real ext-real Element of REAL
(1 - ((fx - Y) / (Y - Y))) * Y is V36() real ext-real Element of REAL
(((fx - Y) / (Y - Y)) * Y) + ((1 - ((fx - Y) / (Y - Y))) * Y) is V36() real ext-real Element of REAL
Y * (fx - Y) is V36() real ext-real Element of REAL
(Y * (fx - Y)) / (Y - Y) is V36() real ext-real Element of REAL
Y * ((Y - fx) / (Y - Y)) is V36() real ext-real Element of REAL
((Y * (fx - Y)) / (Y - Y)) + (Y * ((Y - fx) / (Y - Y))) is V36() real ext-real Element of REAL
Y * (Y - fx) is V36() real ext-real Element of REAL
(Y * (Y - fx)) / (Y - Y) is V36() real ext-real Element of REAL
((Y * (fx - Y)) / (Y - Y)) + ((Y * (Y - fx)) / (Y - Y)) is V36() real ext-real Element of REAL
fx * Y is V36() real ext-real Element of REAL
Y * Y is V36() real ext-real Element of REAL
(fx * Y) - (Y * Y) is V36() real ext-real Element of REAL
(Y - fx) * Y is V36() real ext-real Element of REAL
((fx * Y) - (Y * Y)) + ((Y - fx) * Y) is V36() real ext-real Element of REAL
(((fx * Y) - (Y * Y)) + ((Y - fx) * Y)) / (Y - Y) is V36() real ext-real Element of REAL
fx * (Y - Y) is V36() real ext-real Element of REAL
(fx * (Y - Y)) / (Y - Y) is V36() real ext-real Element of REAL
(((fx - Y) / (Y - Y)) * (X . fx)) + ((1 - ((fx - Y) / (Y - Y))) * (X . fx)) is V36() real ext-real Element of REAL
(((fx - Y) / (Y - Y)) * (X . Y)) + ((1 - ((fx - Y) / (Y - Y))) * (X . Y)) is V36() real ext-real Element of REAL
- ((1 - ((fx - Y) / (Y - Y))) * ((X . Y) - (X . fx))) is V36() real ext-real Element of REAL
- (((fx - Y) / (Y - Y)) * ((X . fx) - (X . Y))) is V36() real ext-real Element of REAL
- ((X . Y) - (X . fx)) is V36() real ext-real Element of REAL
(1 - ((fx - Y) / (Y - Y))) * (- ((X . Y) - (X . fx))) is V36() real ext-real Element of REAL
- ((X . fx) - (X . Y)) is V36() real ext-real Element of REAL
((fx - Y) / (Y - Y)) * (- ((X . fx) - (X . Y))) is V36() real ext-real Element of REAL
(((Y - Y) ") * (Y - fx)) * ((X . fx) - (X . Y)) is V36() real ext-real Element of REAL
((Y - Y) ") * (fx - Y) is V36() real ext-real Element of REAL
(((Y - Y) ") * (fx - Y)) * ((X . Y) - (X . fx)) is V36() real ext-real Element of REAL
(Y - Y) * ((((Y - Y) ") * (fx - Y)) * ((X . Y) - (X . fx))) is V36() real ext-real Element of REAL
(Y - Y) * ((((Y - Y) ") * (Y - fx)) * ((X . fx) - (X . Y))) is V36() real ext-real Element of REAL
(Y - fx) * ((X . fx) - (X . Y)) is V36() real ext-real Element of REAL
(fx - Y) * ((X . Y) - (X . fx)) is V36() real ext-real Element of REAL
(Y - Y) * ((Y - Y) ") is V36() real ext-real Element of REAL
((Y - Y) * ((Y - Y) ")) * ((Y - fx) * ((X . fx) - (X . Y))) is V36() real ext-real Element of REAL
1 * ((Y - fx) * ((X . fx) - (X . Y))) is V36() real ext-real Element of REAL
((Y - Y) * ((Y - Y) ")) * ((fx - Y) * ((X . Y) - (X . fx))) is V36() real ext-real Element of REAL
1 * ((fx - Y) * ((X . Y) - (X . fx))) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
Y * Y is V36() real ext-real Element of REAL
(Y * Y) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
{ b1 where b1 is V36() real ext-real Element of REAL : ( D <= b1 & b1 <= F ) } is set
(1 - Y) * F is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
Y * F is V36() real ext-real Element of REAL
(Y * F) + ((1 - Y) * F) is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
(1 - Y) * D is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
Y * D is V36() real ext-real Element of REAL
(Y * D) + ((1 - Y) * D) is V36() real ext-real Element of REAL
xf is V36() real ext-real Element of REAL
(1 - Y) * fx is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * fx) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
fx - ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
((Y * fx) + ((1 - Y) * Y)) - Y is V36() real ext-real Element of REAL
fx - Y is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(X . fx) - (X . ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
((X . fx) - (X . ((Y * fx) + ((1 - Y) * Y)))) / (fx - ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y)) / (((Y * fx) + ((1 - Y) * Y)) - Y) is V36() real ext-real Element of REAL
Y * (fx - Y) is V36() real ext-real Element of REAL
((X . fx) - (X . ((Y * fx) + ((1 - Y) * Y)))) * (Y * (fx - Y)) is V36() real ext-real Element of REAL
(1 - Y) * (fx - Y) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y)) * ((1 - Y) * (fx - Y)) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y)) * (1 - Y) is V36() real ext-real Element of REAL
(((X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y)) * (1 - Y)) * (fx - Y) is V36() real ext-real Element of REAL
((((X . ((Y * fx) + ((1 - Y) * Y))) - (X . Y)) * (1 - Y)) * (fx - Y)) / (fx - Y) is V36() real ext-real Element of REAL
((X . fx) - (X . ((Y * fx) + ((1 - Y) * Y)))) * Y is V36() real ext-real Element of REAL
(((X . fx) - (X . ((Y * fx) + ((1 - Y) * Y)))) * Y) * (fx - Y) is V36() real ext-real Element of REAL
((((X . fx) - (X . ((Y * fx) + ((1 - Y) * Y)))) * Y) * (fx - Y)) / (fx - Y) is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y) is V36() real ext-real Element of REAL
(X . Y) * (1 - Y) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) - ((X . Y) * (1 - Y)) is V36() real ext-real Element of REAL
(X . fx) * Y is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) * Y is V36() real ext-real Element of REAL
((X . fx) * Y) - ((X . ((Y * fx) + ((1 - Y) * Y))) * Y) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) * Y) + (((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) - ((X . Y) * (1 - Y))) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) * Y) + ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) is V36() real ext-real Element of REAL
(((X . ((Y * fx) + ((1 - Y) * Y))) * Y) + ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y))) - ((X . Y) * (1 - Y)) is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
((Y * fx) + ((1 - Y) * Y)) - fx is V36() real ext-real Element of REAL
Y - ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
Y - fx is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(X . Y) - (X . ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
((X . Y) - (X . ((Y * fx) + ((1 - Y) * Y)))) / (Y - ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx)) / (((Y * fx) + ((1 - Y) * Y)) - fx) is V36() real ext-real Element of REAL
(1 - Y) * (Y - fx) is V36() real ext-real Element of REAL
((X . Y) - (X . ((Y * fx) + ((1 - Y) * Y)))) * ((1 - Y) * (Y - fx)) is V36() real ext-real Element of REAL
Y * (Y - fx) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx)) * (Y * (Y - fx)) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx)) * Y is V36() real ext-real Element of REAL
(((X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx)) * Y) * (Y - fx) is V36() real ext-real Element of REAL
((((X . ((Y * fx) + ((1 - Y) * Y))) - (X . fx)) * Y) * (Y - fx)) / (Y - fx) is V36() real ext-real Element of REAL
((X . Y) - (X . ((Y * fx) + ((1 - Y) * Y)))) * (1 - Y) is V36() real ext-real Element of REAL
(((X . Y) - (X . ((Y * fx) + ((1 - Y) * Y)))) * (1 - Y)) * (Y - fx) is V36() real ext-real Element of REAL
((((X . Y) - (X . ((Y * fx) + ((1 - Y) * Y)))) * (1 - Y)) * (Y - fx)) / (Y - fx) is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) * Y is V36() real ext-real Element of REAL
(X . fx) * Y is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) * Y) - ((X . fx) * Y) is V36() real ext-real Element of REAL
(X . Y) * (1 - Y) is V36() real ext-real Element of REAL
(X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y) is V36() real ext-real Element of REAL
((X . Y) * (1 - Y)) - ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) is V36() real ext-real Element of REAL
(((X . ((Y * fx) + ((1 - Y) * Y))) * Y) - ((X . fx) * Y)) + ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) is V36() real ext-real Element of REAL
((X . ((Y * fx) + ((1 - Y) * Y))) * Y) + ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y)) is V36() real ext-real Element of REAL
(((X . ((Y * fx) + ((1 - Y) * Y))) * Y) + ((X . ((Y * fx) + ((1 - Y) * Y))) * (1 - Y))) - ((X . fx) * Y) is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
X . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
X . fx is V36() real ext-real Element of REAL
Y * (X . fx) is V36() real ext-real Element of REAL
X . Y is V36() real ext-real Element of REAL
(1 - Y) * (X . Y) is V36() real ext-real Element of REAL
(Y * (X . fx)) + ((1 - Y) * (X . Y)) is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
X is set
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
X is V36() real ext-real Element of REAL
D - X is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- X is V36() real ext-real set
(- X) + D is Relation-like Function-like set
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (D - X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
(D - X) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(D - X) . fx is V36() real ext-real Element of REAL
Y * ((D - X) . fx) is V36() real ext-real Element of REAL
(D - X) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((D - X) . Y) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D - X) . Y)) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
(D . ((Y * fx) + ((1 - Y) * Y))) - X is V36() real ext-real Element of REAL
((Y * (D . fx)) + ((1 - Y) * (D . Y))) - X is V36() real ext-real Element of REAL
(D . fx) - X is V36() real ext-real Element of REAL
Y * ((D . fx) - X) is V36() real ext-real Element of REAL
(D . Y) - X is V36() real ext-real Element of REAL
(1 - Y) * ((D . Y) - X) is V36() real ext-real Element of REAL
(Y * ((D . fx) - X)) + ((1 - Y) * ((D . Y) - X)) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D . Y) - X)) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
(D - X) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(D - X) . fx is V36() real ext-real Element of REAL
Y * ((D - X) . fx) is V36() real ext-real Element of REAL
(D - X) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((D - X) . Y) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D - X) . Y)) is V36() real ext-real Element of REAL
(D . ((Y * fx) + ((1 - Y) * Y))) - X is V36() real ext-real Element of REAL
(D . Y) - X is V36() real ext-real Element of REAL
(1 - Y) * ((D . Y) - X) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D . Y) - X)) is V36() real ext-real Element of REAL
(D . fx) - X is V36() real ext-real Element of REAL
Y * ((D . fx) - X) is V36() real ext-real Element of REAL
(1 - Y) * X is V36() real ext-real Element of REAL
((1 - Y) * (D . Y)) - ((1 - Y) * X) is V36() real ext-real Element of REAL
(Y * ((D . fx) - X)) + (((1 - Y) * (D . Y)) - ((1 - Y) * X)) is V36() real ext-real Element of REAL
((Y * (D . fx)) + ((1 - Y) * (D . Y))) - X is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
X is V36() real ext-real Element of REAL
X (#) D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (X (#) D) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
(X (#) D) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(X (#) D) . fx is V36() real ext-real Element of REAL
Y * ((X (#) D) . fx) is V36() real ext-real Element of REAL
(X (#) D) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((X (#) D) . Y) is V36() real ext-real Element of REAL
(Y * ((X (#) D) . fx)) + ((1 - Y) * ((X (#) D) . Y)) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
X * (D . ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
X * ((Y * (D . fx)) + ((1 - Y) * (D . Y))) is V36() real ext-real Element of REAL
X * (D . fx) is V36() real ext-real Element of REAL
Y * (X * (D . fx)) is V36() real ext-real Element of REAL
(1 - Y) * X is V36() real ext-real Element of REAL
((1 - Y) * X) * (D . Y) is V36() real ext-real Element of REAL
(Y * (X * (D . fx))) + (((1 - Y) * X) * (D . Y)) is V36() real ext-real Element of REAL
X * (D . Y) is V36() real ext-real Element of REAL
(1 - Y) * (X * (D . Y)) is V36() real ext-real Element of REAL
(Y * ((X (#) D) . fx)) + ((1 - Y) * (X * (D . Y))) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
(X (#) D) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(X (#) D) . fx is V36() real ext-real Element of REAL
Y * ((X (#) D) . fx) is V36() real ext-real Element of REAL
(X (#) D) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((X (#) D) . Y) is V36() real ext-real Element of REAL
(Y * ((X (#) D) . fx)) + ((1 - Y) * ((X (#) D) . Y)) is V36() real ext-real Element of REAL
X * (D . ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
X * (D . fx) is V36() real ext-real Element of REAL
Y * (X * (D . fx)) is V36() real ext-real Element of REAL
(Y * (X * (D . fx))) + ((1 - Y) * ((X (#) D) . Y)) is V36() real ext-real Element of REAL
X * (D . Y) is V36() real ext-real Element of REAL
(1 - Y) * (X * (D . Y)) is V36() real ext-real Element of REAL
(Y * (X * (D . fx))) + ((1 - Y) * (X * (D . Y))) is V36() real ext-real Element of REAL
(X * (D . ((Y * fx) + ((1 - Y) * Y)))) / X is V36() real ext-real Element of REAL
X * ((Y * (D . fx)) + ((1 - Y) * (D . Y))) is V36() real ext-real Element of REAL
(X * ((Y * (D . fx)) + ((1 - Y) * (D . Y)))) / X is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
0 (#) D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
dom (0 (#) D) is complex-membered ext-real-membered real-membered Element of K19(REAL)
X is V36() real ext-real Element of REAL
1 - X is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
X * Y is V36() real ext-real Element of REAL
(1 - X) * fx is V36() real ext-real Element of REAL
(X * Y) + ((1 - X) * fx) is V36() real ext-real Element of REAL
(0 (#) D) . ((X * Y) + ((1 - X) * fx)) is V36() real ext-real Element of REAL
(0 (#) D) . Y is V36() real ext-real Element of REAL
X * ((0 (#) D) . Y) is V36() real ext-real Element of REAL
(0 (#) D) . fx is V36() real ext-real Element of REAL
(1 - X) * ((0 (#) D) . fx) is V36() real ext-real Element of REAL
(X * ((0 (#) D) . Y)) + ((1 - X) * ((0 (#) D) . fx)) is V36() real ext-real Element of REAL
D . ((X * Y) + ((1 - X) * fx)) is V36() real ext-real Element of REAL
0 * (D . ((X * Y) + ((1 - X) * fx))) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
0 * (D . Y) is V36() real ext-real Element of REAL
X * (0 * (D . Y)) is V36() real ext-real Element of REAL
(X * (0 * (D . Y))) + ((1 - X) * ((0 (#) D) . fx)) is V36() real ext-real Element of REAL
X * 0 is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
0 * (D . fx) is V36() real ext-real Element of REAL
(1 - X) * (0 * (D . fx)) is V36() real ext-real Element of REAL
(X * 0) + ((1 - X) * (0 * (D . fx))) is V36() real ext-real Element of REAL
0 + 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative V41() V42() finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of NAT
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
D + F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
X is set
dom (D + F) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom F is complex-membered ext-real-membered real-membered Element of K19(REAL)
(dom D) /\ (dom F) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
(D + F) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(D + F) . fx is V36() real ext-real Element of REAL
Y * ((D + F) . fx) is V36() real ext-real Element of REAL
(D + F) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((D + F) . Y) is V36() real ext-real Element of REAL
(Y * ((D + F) . fx)) + ((1 - Y) * ((D + F) . Y)) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
F . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
Y * (F . fx) is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
(1 - Y) * (F . Y) is V36() real ext-real Element of REAL
(Y * (F . fx)) + ((1 - Y) * (F . Y)) is V36() real ext-real Element of REAL
(D . ((Y * fx) + ((1 - Y) * Y))) + (F . ((Y * fx) + ((1 - Y) * Y))) is V36() real ext-real Element of REAL
((Y * (D . fx)) + ((1 - Y) * (D . Y))) + ((Y * (F . fx)) + ((1 - Y) * (F . Y))) is V36() real ext-real Element of REAL
(D . fx) + (F . fx) is V36() real ext-real Element of REAL
Y * ((D . fx) + (F . fx)) is V36() real ext-real Element of REAL
(Y * ((D . fx) + (F . fx))) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
((Y * ((D . fx) + (F . fx))) + ((1 - Y) * (D . Y))) + ((1 - Y) * (F . Y)) is V36() real ext-real Element of REAL
(Y * ((D + F) . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
((Y * ((D + F) . fx)) + ((1 - Y) * (D . Y))) + ((1 - Y) * (F . Y)) is V36() real ext-real Element of REAL
(D . Y) + (F . Y) is V36() real ext-real Element of REAL
(1 - Y) * ((D . Y) + (F . Y)) is V36() real ext-real Element of REAL
(Y * ((D + F) . fx)) + ((1 - Y) * ((D . Y) + (F . Y))) is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
X is V36() real ext-real Element of REAL
D - X is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- X is V36() real ext-real set
(- X) + D is Relation-like Function-like set
(REAL,(D - X)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom D is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (D - X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (REAL,(D - X)) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y is V36() real ext-real Element of REAL
1 - Y is V36() real ext-real Element of REAL
fx is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y * fx is V36() real ext-real Element of REAL
(1 - Y) * Y is V36() real ext-real Element of REAL
(Y * fx) + ((1 - Y) * Y) is V36() real ext-real Element of REAL
(REAL,(D - X)) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(REAL,(D - X)) . fx is V36() real ext-real Element of REAL
Y * ((REAL,(D - X)) . fx) is V36() real ext-real Element of REAL
(REAL,(D - X)) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((REAL,(D - X)) . Y) is V36() real ext-real Element of REAL
(Y * ((REAL,(D - X)) . fx)) + ((1 - Y) * ((REAL,(D - X)) . Y)) is V36() real ext-real Element of REAL
D . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
D . fx is V36() real ext-real Element of REAL
Y * (D . fx) is V36() real ext-real Element of REAL
D . Y is V36() real ext-real Element of REAL
(1 - Y) * (D . Y) is V36() real ext-real Element of REAL
(Y * (D . fx)) + ((1 - Y) * (D . Y)) is V36() real ext-real Element of REAL
(D . ((Y * fx) + ((1 - Y) * Y))) - X is V36() real ext-real Element of REAL
((Y * (D . fx)) + ((1 - Y) * (D . Y))) - X is V36() real ext-real Element of REAL
(((D . ((Y * fx) + ((1 - Y) * Y))) - X)) is V36() real ext-real Element of REAL
max (((D . ((Y * fx) + ((1 - Y) * Y))) - X),0) is V36() real ext-real set
max ((((Y * (D . fx)) + ((1 - Y) * (D . Y))) - X),0) is V36() real ext-real Element of REAL
0 + Y is V36() real ext-real Element of REAL
(D - X) . Y is V36() real ext-real Element of REAL
(1 - Y) * ((D - X) . Y) is V36() real ext-real Element of REAL
(((1 - Y) * ((D - X) . Y))) is V36() real ext-real Element of REAL
max (((1 - Y) * ((D - X) . Y)),0) is V36() real ext-real set
(((D - X) . Y)) is V36() real ext-real Element of REAL
max (((D - X) . Y),0) is V36() real ext-real set
(1 - Y) * (((D - X) . Y)) is V36() real ext-real Element of REAL
(D - X) . fx is V36() real ext-real Element of REAL
Y * ((D - X) . fx) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D - X) . Y)) is V36() real ext-real Element of REAL
(((Y * ((D - X) . fx)) + ((1 - Y) * ((D - X) . Y)))) is V36() real ext-real Element of REAL
max (((Y * ((D - X) . fx)) + ((1 - Y) * ((D - X) . Y))),0) is V36() real ext-real set
((Y * ((D - X) . fx))) is V36() real ext-real Element of REAL
max ((Y * ((D - X) . fx)),0) is V36() real ext-real set
((Y * ((D - X) . fx))) + (((1 - Y) * ((D - X) . Y))) is V36() real ext-real Element of REAL
(((D - X) . fx)) is V36() real ext-real Element of REAL
max (((D - X) . fx),0) is V36() real ext-real set
Y * (((D - X) . fx)) is V36() real ext-real Element of REAL
(D . fx) - X is V36() real ext-real Element of REAL
Y * ((D . fx) - X) is V36() real ext-real Element of REAL
(D . Y) - X is V36() real ext-real Element of REAL
(1 - Y) * ((D . Y) - X) is V36() real ext-real Element of REAL
(Y * ((D . fx) - X)) + ((1 - Y) * ((D . Y) - X)) is V36() real ext-real Element of REAL
(Y * ((D - X) . fx)) + ((1 - Y) * ((D . Y) - X)) is V36() real ext-real Element of REAL
(Y * (((D - X) . fx))) + ((1 - Y) * (((D - X) . Y))) is V36() real ext-real Element of REAL
(D - X) . ((Y * fx) + ((1 - Y) * Y)) is V36() real ext-real Element of REAL
(((D - X) . ((Y * fx) + ((1 - Y) * Y)))) is V36() real ext-real Element of REAL
max (((D - X) . ((Y * fx) + ((1 - Y) * Y))),0) is V36() real ext-real set
(Y * ((REAL,(D - X)) . fx)) + ((1 - Y) * (((D - X) . Y))) is V36() real ext-real Element of REAL
D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
(REAL,D) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
F is set
D - 0 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered set
(- 0) + D is Relation-like Function-like set
(REAL,(D - 0)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
[#] REAL is complex-membered ext-real-membered real-membered Element of K19(REAL)
id ([#] REAL) is Relation-like REAL -defined REAL -valued Function-like one-to-one complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom (id ([#] REAL)) is complex-membered ext-real-membered real-membered Element of K19(REAL)
F is V36() real ext-real Element of REAL
1 - F is V36() real ext-real Element of REAL
X is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
F * X is V36() real ext-real Element of REAL
(1 - F) * Y is V36() real ext-real Element of REAL
(F * X) + ((1 - F) * Y) is V36() real ext-real Element of REAL
(id ([#] REAL)) . ((F * X) + ((1 - F) * Y)) is V36() real ext-real Element of REAL
(id ([#] REAL)) . X is V36() real ext-real Element of REAL
F * ((id ([#] REAL)) . X) is V36() real ext-real Element of REAL
(id ([#] REAL)) . Y is V36() real ext-real Element of REAL
(1 - F) * ((id ([#] REAL)) . Y) is V36() real ext-real Element of REAL
(F * ((id ([#] REAL)) . X)) + ((1 - F) * ((id ([#] REAL)) . Y)) is V36() real ext-real Element of REAL
D is V36() real ext-real Element of REAL
(id ([#] REAL)) - D is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- D is V36() real ext-real set
(- D) + (id ([#] REAL)) is Relation-like Function-like set
(REAL,((id ([#] REAL)) - D)) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
F | (dom (F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
fx is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (F | (dom (F | X))) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng fx is finite set
Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dr is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
dom F is Element of K19(D)
(dom F) /\ X is Element of K19(D)
F | (dom F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(F | (dom F)) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
fx is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,(dom (F | X))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
F | (dom (F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
(dom F) /\ X is Element of K19(D)
F | ((dom F) /\ X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F | (dom F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(F | (dom F)) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,(F | X),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F | X) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is finite set
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X | F is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (X | F) is finite Element of K19(D)
K19(D) is set
Y is set
{Y} is non empty V12() finite 1 -element set
F \ {Y} is finite Element of K19(F)
K19(F) is finite V51() set
(D,X,(F \ {Y})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
X . Y is V36() real ext-real Element of REAL
<*(X . Y)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,X,(F \ {Y})) ^ <*(X . Y)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dom X is Element of K19(D)
(dom X) /\ F is finite Element of K19(D)
X | (F \ {Y}) is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (X | (F \ {Y})) is finite Element of K19(D)
dr is set
Coim ((X | (F \ {Y})),dr) is set
{dr} is non empty V12() finite 1 -element set
(X | (F \ {Y})) " {dr} is finite set
card (Coim ((X | (F \ {Y})),dr)) is ordinal cardinal set
Coim ((D,X,(F \ {Y})),dr) is finite set
(D,X,(F \ {Y})) " {dr} is finite set
card (Coim ((D,X,(F \ {Y})),dr)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
X | {Y} is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (X | {Y}) is finite Element of K19(D)
dom <*(X . Y)*> is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(X | {Y}) " {dr} is finite Element of K19(D)
xf is set
(X | {Y}) . xf is V36() real ext-real Element of REAL
card ((X | {Y}) " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
<*(X . Y)*> " {dr} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
xf is set
<*(X . Y)*> . xf is V36() real ext-real Element of REAL
card (<*(X . Y)*> " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
<*(X . Y)*> " {dr} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
the ordinal natural V36() real ext-real non negative V41() V42() finite cardinal Element of <*(X . Y)*> " {dr} is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal Element of <*(X . Y)*> " {dr}
<*(X . Y)*> . the ordinal natural V36() real ext-real non negative V41() V42() finite cardinal Element of <*(X . Y)*> " {dr} is V36() real ext-real Element of REAL
(X | {Y}) " {dr} is finite Element of K19(D)
the Element of (X | {Y}) " {dr} is Element of (X | {Y}) " {dr}
(X | {Y}) . the Element of (X | {Y}) " {dr} is V36() real ext-real Element of REAL
card ((X | {Y}) " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card (<*(X . Y)*> " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(X | {Y}) " {dr} is finite Element of K19(D)
card ((X | {Y}) " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
<*(X . Y)*> " {dr} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (<*(X . Y)*> " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(X | {Y}) " {dr} is finite Element of K19(D)
card ((X | {Y}) " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
<*(X . Y)*> " {dr} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (<*(X . Y)*> " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(X | (F \ {Y})) " {dr} is finite Element of K19(D)
((X | (F \ {Y})) " {dr}) \/ ((X | {Y}) " {dr}) is finite Element of K19(D)
X " {dr} is Element of K19(D)
(F \ {Y}) /\ (X " {dr}) is finite Element of K19(D)
((F \ {Y}) /\ (X " {dr})) \/ ((X | {Y}) " {dr}) is finite Element of K19(D)
{Y} /\ (X " {dr}) is finite Element of K19(D)
((F \ {Y}) /\ (X " {dr})) \/ ({Y} /\ (X " {dr})) is finite Element of K19(D)
(F \ {Y}) \/ {Y} is non empty finite set
((F \ {Y}) \/ {Y}) /\ (X " {dr}) is finite Element of K19(D)
F \/ {Y} is non empty finite set
(F \/ {Y}) /\ (X " {dr}) is finite Element of K19(D)
F /\ (X " {dr}) is finite Element of K19(D)
(X | F) " {dr} is finite Element of K19(D)
((X | (F \ {Y})) " {dr}) /\ ((X | {Y}) " {dr}) is finite Element of K19(D)
((F \ {Y}) /\ (X " {dr})) /\ ((X | {Y}) " {dr}) is finite Element of K19(D)
((F \ {Y}) /\ (X " {dr})) /\ ({Y} /\ (X " {dr})) is finite Element of K19(D)
(X " {dr}) /\ (F \ {Y}) is finite Element of K19(F)
((X " {dr}) /\ (F \ {Y})) /\ {Y} is finite Element of K19(F)
(((X " {dr}) /\ (F \ {Y})) /\ {Y}) /\ (X " {dr}) is finite Element of K19(D)
(F \ {Y}) /\ {Y} is finite Element of K19(F)
(X " {dr}) /\ ((F \ {Y}) /\ {Y}) is finite Element of K19(F)
((X " {dr}) /\ ((F \ {Y}) /\ {Y})) /\ (X " {dr}) is finite Element of K19(D)
{} /\ (X " {dr}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(D)
Coim ((X | F),dr) is set
(X | F) " {dr} is finite set
card (Coim ((X | F),dr)) is ordinal cardinal set
card ((X | (F \ {Y})) " {dr}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card ((X | (F \ {Y})) " {dr})) + (card (<*(X . Y)*> " {dr})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
card {} is Function-like functional empty ordinal natural V36() real ext-real non positive non negative V41() V42() finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of omega
((card ((X | (F \ {Y})) " {dr})) + (card (<*(X . Y)*> " {dr}))) - (card {}) is V36() real ext-real non negative Element of REAL
Coim (((D,X,(F \ {Y})) ^ <*(X . Y)*>),dr) is finite set
((D,X,(F \ {Y})) ^ <*(X . Y)*>) " {dr} is finite set
card (Coim (((D,X,(F \ {Y})) ^ <*(X . Y)*>),dr)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
X is set
F is Element of D
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (Y | X) is Element of K19(D)
K19(D) is set
{F} is non empty V12() finite 1 -element set
X \ {F} is Element of K19(X)
K19(X) is set
(D,Y,(X \ {F})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y . F is V36() real ext-real Element of REAL
<*(Y . F)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,Y,(X \ {F})) ^ <*(Y . F)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Element of D
{F} is non empty V12() finite 1 -element set
X is set
X \ {F} is Element of K19(X)
K19(X) is set
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (Y | X) is Element of K19(D)
K19(D) is set
(D,Y,(X \ {F})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y . F is V36() real ext-real Element of REAL
<*(Y . F)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,Y,(X \ {F})) ^ <*(Y . F)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Y | (X \ {F}) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (Y | (X \ {F})) is Element of K19(D)
dom Y is Element of K19(D)
(dom Y) /\ (X \ {F}) is Element of K19(X)
(dom Y) /\ X is Element of K19(D)
(dom (Y | X)) \ {F} is Element of K19(D)
xf is set
xf is set
Y | (dom (Y | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y | ((dom Y) /\ X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y | (dom Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(Y | (dom Y)) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,Y,((dom (Y | X)) \ {F})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,Y,((dom (Y | X)) \ {F})) ^ <*(Y . F)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
len (D,F,X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is finite set
card Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dom (D,F,X) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (D,F,X)) is finite len (D,F,X) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F,{}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
F | {} is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | {}) is finite Element of K19(D)
K19(D) is set
dom F is Element of K19(D)
(dom F) /\ {} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(D)
len (D,F,{}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
X is Element of D
{X} is non empty V12() finite 1 -element set
(D,F,{X}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
F . X is V36() real ext-real Element of REAL
<*(F . X)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(dom F) /\ {X} is finite Element of K19(D)
F | {X} is Relation-like D -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | {X}) is finite Element of K19(D)
rng (D,F,{X}) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (F | {X}) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
{(F . X)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
Y is set
fx is Element of D
(F | {X}) . fx is V36() real ext-real Element of REAL
Y is set
(F | {X}) . X is V36() real ext-real Element of REAL
len (D,F,{X}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
card {X} is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
len (D,F,X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,F,X) . (len (D,F,X)) is V36() real ext-real Element of REAL
Y is Element of D
F . Y is V36() real ext-real Element of REAL
{Y} is non empty V12() finite 1 -element set
X \ {Y} is Element of K19(X)
K19(X) is set
(D,F,(X \ {Y})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
<*(F . Y)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(X \ {Y})) ^ <*(F . Y)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
rng (D,F,X) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (F | X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(len (D,F,X)) - 1 is V36() real ext-real Element of REAL
max (0,((len (D,F,X)) - 1)) is V36() real ext-real Element of REAL
xf is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
xf + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,F,X) | xf is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
((D,F,X) | xf) ^ <*(F . Y)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
X \ Y is Element of K19(X)
K19(X) is set
F | (X \ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \ Y)) is Element of K19(D)
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(X \ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) ^ (D,F,(X \ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dom F is Element of K19(D)
(dom F) /\ (X \ Y) is Element of K19(X)
(dom F) /\ X is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
(dom (F | X)) \ {} is Element of K19(D)
(D,F,(dom (F | X))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,{}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
D is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
D + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
F is non empty set
K20(F,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(F,REAL)) is V12() non finite set
X is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
Y is set
X | Y is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
dom (X | Y) is Element of K19(F)
K19(F) is set
(F,X,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
fx is set
X | fx is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
dom (X | fx) is Element of K19(F)
Y \ fx is Element of K19(Y)
K19(Y) is set
X | (Y \ fx) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
dom (X | (Y \ fx)) is Element of K19(F)
(F,X,fx) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F,X,(Y \ fx)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F,X,fx) ^ (F,X,(Y \ fx)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dx is finite set
card dx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
len (F,X,fx) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
rng (F,X,fx) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (X | fx) is complex-membered ext-real-membered real-membered Element of K19(REAL)
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (F,X,fx) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(F,X,fx) . (len (F,X,fx)) is V36() real ext-real Element of REAL
gx is Element of F
(X | fx) . gx is V36() real ext-real Element of REAL
dom X is Element of K19(F)
(dom X) /\ (Y \ fx) is Element of K19(Y)
(dom X) /\ fx is Element of K19(F)
{gx} is non empty V12() finite 1 -element set
X . gx is V36() real ext-real Element of REAL
<*(X . gx)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F,X,(Y \ fx)) ^ <*(X . gx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
<*(X . gx)*> ^ (F,X,(Y \ fx)) is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
fx \ {gx} is Element of K19(fx)
K19(fx) is set
X | (fx \ {gx}) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
dom (X | (fx \ {gx})) is Element of K19(F)
Y \ (fx \ {gx}) is Element of K19(Y)
X | (Y \ (fx \ {gx})) is Relation-like F -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(F,REAL))
dom (X | (Y \ (fx \ {gx}))) is Element of K19(F)
(dom X) /\ (Y \ (fx \ {gx})) is Element of K19(Y)
(dom X) /\ (fx \ {gx}) is Element of K19(fx)
dx \ {gx} is finite Element of K19(dx)
K19(dx) is finite V51() set
dy is set
dy is set
(F,X,fx) | D is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
((F,X,fx) | D) ^ <*(X . gx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Y /\ {gx} is finite set
(Y \ fx) \/ (Y /\ {gx}) is set
(Y \ fx) \/ {gx} is non empty set
(dom X) /\ {gx} is finite Element of K19(F)
(dom (X | (Y \ fx))) \/ ((dom X) /\ {gx}) is Element of K19(F)
(dom (X | (Y \ fx))) \/ {gx} is non empty set
gy is Element of F
dy is finite set
dy is Element of F
X . dy is V36() real ext-real Element of REAL
X . gy is V36() real ext-real Element of REAL
(X | fx) . gy is V36() real ext-real Element of REAL
X . gy is V36() real ext-real Element of REAL
y is ordinal natural V36() real ext-real non negative finite cardinal set
(F,X,fx) . y is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . gy is V36() real ext-real Element of REAL
X . dy is V36() real ext-real Element of REAL
X . gy is V36() real ext-real Element of REAL
(dom X) /\ Y is Element of K19(F)
rng (F,X,(Y \ fx)) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (X | (Y \ fx)) is complex-membered ext-real-membered real-membered Element of K19(REAL)
dy is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (<*(X . gx)*> ^ (F,X,(Y \ fx))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dy + 1 is non empty ordinal natural V36() real ext-real positive non negative finite cardinal Element of REAL
(<*(X . gx)*> ^ (F,X,(Y \ fx))) . (dy + 1) is V36() real ext-real Element of REAL
(<*(X . gx)*> ^ (F,X,(Y \ fx))) . dy is V36() real ext-real Element of REAL
dy + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dy - 1 is V36() real ext-real Element of REAL
max (0,(dy - 1)) is V36() real ext-real Element of REAL
(<*(X . gx)*> ^ (F,X,(Y \ fx))) . (dy + 1) is V36() real ext-real Element of REAL
len <*(X . gx)*> is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (<*(X . gx)*> ^ (F,X,(Y \ fx))) is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (F,X,(Y \ fx)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
1 + (len (F,X,(Y \ fx))) is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(len (<*(X . gx)*> ^ (F,X,(Y \ fx)))) - 1 is V36() real ext-real Element of REAL
y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (F,X,(Y \ fx)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(F,X,(Y \ fx)) . (y + 1) is V36() real ext-real Element of REAL
s is Element of F
(X | (Y \ fx)) . s is V36() real ext-real Element of REAL
(dy + 1) - 1 is V36() real ext-real Element of REAL
(F,X,(Y \ fx)) . ((dy + 1) - 1) is V36() real ext-real Element of REAL
X . s is V36() real ext-real Element of REAL
1 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(F,X,(Y \ fx)) . y is V36() real ext-real Element of REAL
(Y \ (fx \ {gx})) \ {gx} is Element of K19(Y)
(fx \ {gx}) \/ {gx} is non empty set
Y \ ((fx \ {gx}) \/ {gx}) is Element of K19(Y)
fx \/ {gx} is non empty set
Y \ (fx \/ {gx}) is Element of K19(Y)
(F,X,(Y \ (fx \ {gx}))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
card dy is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card {gx} is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card dx) - (card {gx}) is V36() real ext-real set
(D + 1) - 1 is V36() real ext-real Element of REAL
(F,X,(fx \ {gx})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F,X,(fx \ {gx})) ^ (F,X,(Y \ (fx \ {gx}))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(F,X,(fx \ {gx})) ^ <*(X . gx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
((F,X,(fx \ {gx})) ^ <*(X . gx)*>) ^ (F,X,(Y \ fx)) is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
X \ Y is Element of K19(X)
K19(X) is set
F | (X \ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \ Y)) is Element of K19(D)
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(X \ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) ^ (D,F,(X \ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
F - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is V36() real ext-real set
(- X) + F is Relation-like Function-like set
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
K19(D) is set
(D,(F - X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
len (D,(F - X),Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,(F - X),Y) . (len (D,(F - X),Y)) is V36() real ext-real Element of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
len (D,F,Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,F,Y) . (len (D,F,Y)) is V36() real ext-real Element of REAL
fx is Element of D
(F - X) . fx is V36() real ext-real Element of REAL
F . fx is V36() real ext-real Element of REAL
(F - X) | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((F - X) | Y) is Element of K19(D)
dom (F - X) is Element of K19(D)
(dom (F - X)) /\ Y is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
dx is finite set
rng (D,F,Y) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (F | Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (D,F,Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(F | Y) . fx is V36() real ext-real Element of REAL
gx is ordinal natural V36() real ext-real non negative finite cardinal set
(D,F,Y) . gx is V36() real ext-real Element of REAL
Seg (len (D,F,Y)) is finite len (D,F,Y) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng (D,(F - X),Y) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng ((F - X) | Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
card dx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dom (D,(F - X),Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (D,(F - X),Y)) is finite len (D,(F - X),Y) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is Element of D
((F - X) | Y) . x is V36() real ext-real Element of REAL
(F | Y) . x is V36() real ext-real Element of REAL
F . x is V36() real ext-real Element of REAL
x is ordinal natural V36() real ext-real non negative finite cardinal set
(D,F,Y) . x is V36() real ext-real Element of REAL
(F - X) . x is V36() real ext-real Element of REAL
(F . x) - X is V36() real ext-real Element of REAL
(F . fx) - X is V36() real ext-real Element of REAL
Y is Element of D
(F | Y) . Y is V36() real ext-real Element of REAL
F . Y is V36() real ext-real Element of REAL
((F - X) | Y) . Y is V36() real ext-real Element of REAL
(F - X) . Y is V36() real ext-real Element of REAL
(F . Y) - X is V36() real ext-real Element of REAL
dy is ordinal natural V36() real ext-real non negative finite cardinal set
(D,(F - X),Y) . dy is V36() real ext-real Element of REAL
- X is V36() real ext-real Element of REAL
(F . Y) + (- X) is V36() real ext-real Element of REAL
((F - X) | Y) . fx is V36() real ext-real Element of REAL
Y is ordinal natural V36() real ext-real non negative finite cardinal set
(D,(F - X),Y) . Y is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is V36() real ext-real Element of REAL
F - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is V36() real ext-real set
(- X) + F is Relation-like Function-like set
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
K19(D) is set
(D,(F - X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
fx + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
(D,(F - X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
dr is finite set
card dr is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card dr) |-> X is Relation-like NAT -defined REAL -valued Function-like finite card dr -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card dr) -tuples_on REAL
(card dr) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = card dr } is set
Seg (card dr) is finite card dr -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (card dr)) --> X is Relation-like Seg (card dr) -defined {X} -valued Function-like V18( Seg (card dr),{X}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (card dr)),{X}))
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
K20((Seg (card dr)),{X}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (card dr)),{X})) is finite V51() set
(D,F,Y) - ((card dr) |-> X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K195() is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K197() * ((id REAL),K195()) is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K198() .: ((D,F,Y),((card dr) |-> X)) is set
- ((card dr) |-> X) is Relation-like Function-like complex-valued set
((card dr) |-> X) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) ((card dr) |-> X) is Relation-like Function-like set
K476((- 1)) is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K199() [;] ((- 1),(id REAL)) is set
((card dr) |-> X) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,Y) + (- ((card dr) |-> X)) is Relation-like Function-like set
(F - X) | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((F - X) | Y) is Element of K19(D)
dom (F - X) is Element of K19(D)
(dom (F - X)) /\ Y is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
len (D,(F - X),Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
rng (D,(F - X),Y) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng ((F - X) | Y) is complex-membered ext-real-membered real-membered Element of K19(REAL)
len (D,F,Y) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
0 + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (D,(F - X),Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(D,(F - X),Y) . (len (D,(F - X),Y)) is V36() real ext-real Element of REAL
dx is Element of D
((F - X) | Y) . dx is V36() real ext-real Element of REAL
{dx} is non empty V12() finite 1 -element set
Y \ {dx} is Element of K19(Y)
K19(Y) is set
F | (Y \ {dx}) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (Y \ {dx})) is Element of K19(D)
(D,(F - X),(Y \ {dx})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(Y \ {dx})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(fx + 1) |-> X is Relation-like NAT -defined REAL -valued Function-like finite fx + 1 -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (fx + 1) -tuples_on REAL
(fx + 1) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = fx + 1 } is set
Seg (fx + 1) is non empty finite fx + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (fx + 1)) --> X is Relation-like Seg (fx + 1) -defined {X} -valued Function-like V18( Seg (fx + 1),{X}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (fx + 1)),{X}))
K20((Seg (fx + 1)),{X}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (fx + 1)),{X})) is finite V51() set
fx |-> X is Relation-like NAT -defined REAL -valued Function-like finite fx -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of fx -tuples_on REAL
fx -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = fx } is set
Seg fx is finite fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg fx) --> X is Relation-like Seg fx -defined {X} -valued Function-like V18( Seg fx,{X}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg fx),{X}))
K20((Seg fx),{X}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg fx),{X})) is finite V51() set
(F - X) . dx is V36() real ext-real Element of REAL
(D,F,Y) . (len (D,F,Y)) is V36() real ext-real Element of REAL
F . dx is V36() real ext-real Element of REAL
(D,F,Y) | fx is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
<*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
((D,F,Y) | fx) ^ <*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(Y \ {dx})) ^ <*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(Y \ {dx})) - (fx |-> X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() .: ((D,F,(Y \ {dx})),(fx |-> X)) is set
- (fx |-> X) is Relation-like Function-like complex-valued set
(fx |-> X) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) (fx |-> X) is Relation-like Function-like set
(fx |-> X) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,(Y \ {dx})) + (- (fx |-> X)) is Relation-like Function-like set
dom ((D,F,(Y \ {dx})) - (fx |-> X)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len ((D,F,(Y \ {dx})) - (fx |-> X)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len ((D,F,(Y \ {dx})) - (fx |-> X))) is finite len ((D,F,(Y \ {dx})) - (fx |-> X)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(dom F) /\ (Y \ {dx}) is Element of K19(Y)
(dom (F | Y)) \ {dx} is Element of K19(D)
y is set
y is set
dy is finite set
card dy is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
y is finite set
card y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card {dx} is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card y) - (card {dx}) is V36() real ext-real set
(fx + 1) - 1 is V36() real ext-real Element of REAL
len (fx |-> X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (D,F,(Y \ {dx})) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,(F - X),Y) | fx is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
<*((F - X) . dx)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
((D,(F - X),Y) | fx) ^ <*((F - X) . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,(F - X),(Y \ {dx})) ^ <*((F - X) . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len ((fx + 1) |-> X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,F,Y) - ((fx + 1) |-> X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() .: ((D,F,Y),((fx + 1) |-> X)) is set
- ((fx + 1) |-> X) is Relation-like Function-like complex-valued set
((fx + 1) |-> X) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) ((fx + 1) |-> X) is Relation-like Function-like set
((fx + 1) |-> X) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,Y) + (- ((fx + 1) |-> X)) is Relation-like Function-like set
len ((D,F,Y) - ((fx + 1) |-> X)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom ((D,F,Y) - ((fx + 1) |-> X)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(F . dx) - X is V36() real ext-real Element of REAL
<*X*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
<*(F . dx)*> - <*X*> is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() .: (<*(F . dx)*>,<*X*>) is set
- <*X*> is Relation-like Function-like complex-valued set
<*X*> (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) <*X*> is Relation-like Function-like set
<*X*> (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
<*(F . dx)*> + (- <*X*>) is Relation-like Function-like set
dom (D,F,Y) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len (D,F,Y)) is finite len (D,F,Y) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len <*(F . dx)*> is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len <*X*> is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (<*(F . dx)*> - <*X*>) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
((D,F,(Y \ {dx})) - (fx |-> X)) ^ (<*(F . dx)*> - <*X*>) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
len (((D,F,(Y \ {dx})) - (fx |-> X)) ^ (<*(F . dx)*> - <*X*>)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dom (<*(F . dx)*> - <*X*>) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
<*(F . dx)*> . 1 is V36() real ext-real Element of REAL
<*X*> . 1 is V36() real ext-real Element of REAL
m is ordinal natural V36() real ext-real non negative finite cardinal set
((fx + 1) |-> X) . m is V36() real ext-real Element of REAL
(((D,F,(Y \ {dx})) - (fx |-> X)) ^ (<*(F . dx)*> - <*X*>)) . m is V36() real ext-real Element of REAL
(fx + 1) - fx is V36() real ext-real Element of REAL
(<*(F . dx)*> - <*X*>) . ((fx + 1) - fx) is V36() real ext-real Element of REAL
((D,F,Y) - ((fx + 1) |-> X)) . m is V36() real ext-real Element of REAL
(D,F,Y) . m is V36() real ext-real Element of REAL
((fx + 1) |-> X) . m is V36() real ext-real Element of REAL
((D,F,Y) | fx) . m is V36() real ext-real Element of REAL
(((D,F,(Y \ {dx})) - (fx |-> X)) ^ (<*(F . dx)*> - <*X*>)) . m is V36() real ext-real Element of REAL
((D,F,(Y \ {dx})) - (fx |-> X)) . m is V36() real ext-real Element of REAL
(fx |-> X) . m is V36() real ext-real Element of REAL
s is V36() real ext-real Element of REAL
s - X is V36() real ext-real Element of REAL
((D,F,Y) - ((fx + 1) |-> X)) . m is V36() real ext-real Element of REAL
fx is set
F | fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | fx) is Element of K19(D)
(D,(F - X),fx) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,fx) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y is finite set
card Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card Y) |-> X is Relation-like NAT -defined REAL -valued Function-like finite card Y -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card Y) -tuples_on REAL
(card Y) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = card Y } is set
Seg (card Y) is finite card Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (card Y)) --> X is Relation-like Seg (card Y) -defined {X} -valued Function-like V18( Seg (card Y),{X}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (card Y)),{X}))
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
K20((Seg (card Y)),{X}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (card Y)),{X})) is finite V51() set
(D,F,fx) - ((card Y) |-> X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K195() is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K197() * ((id REAL),K195()) is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K198() .: ((D,F,fx),((card Y) |-> X)) is set
- ((card Y) |-> X) is Relation-like Function-like complex-valued set
((card Y) |-> X) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) ((card Y) |-> X) is Relation-like Function-like set
K476((- 1)) is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K199() [;] ((- 1),(id REAL)) is set
((card Y) |-> X) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,fx) + (- ((card Y) |-> X)) is Relation-like Function-like set
(D,F,{}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(F - X) | fx is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((F - X) | fx) is Element of K19(D)
dom (F - X) is Element of K19(D)
(dom (F - X)) /\ fx is Element of K19(D)
dom F is Element of K19(D)
(dom F) /\ fx is Element of K19(D)
(D,(F - X),{}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card fx) |-> X is Relation-like NAT -defined REAL -valued Function-like finite card fx -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card fx) -tuples_on REAL
(card fx) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = card fx } is set
Seg (card fx) is finite card fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (card fx)) --> X is Relation-like Seg (card fx) -defined {X} -valued Function-like V18( Seg (card fx),{X}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (card fx)),{X}))
{X} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
K20((Seg (card fx)),{X}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (card fx)),{X})) is finite V51() set
(D,F,Y) - ((card fx) |-> X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K195() is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K197() * ((id REAL),K195()) is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K198() .: ((D,F,Y),((card fx) |-> X)) is set
- ((card fx) |-> X) is Relation-like Function-like complex-valued set
((card fx) |-> X) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) ((card fx) |-> X) is Relation-like Function-like set
K476((- 1)) is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K199() [;] ((- 1),(id REAL)) is set
((card fx) |-> X) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,Y) + (- ((card fx) |-> X)) is Relation-like Function-like set
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,(D,F),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y is Element of D
F . Y is V36() real ext-real Element of REAL
(F | X) . Y is V36() real ext-real Element of REAL
(D,(F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,((D,F) | X),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
rng (F | X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Y is V36() real ext-real Element of REAL
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card fx) |-> Y is Relation-like NAT -defined REAL -valued Function-like finite card fx -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card fx) -tuples_on REAL
(card fx) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = card fx } is set
Seg (card fx) is finite card fx -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (card fx)) --> Y is Relation-like Seg (card fx) -defined {Y} -valued Function-like V18( Seg (card fx),{Y}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (card fx)),{Y}))
K20((Seg (card fx)),{Y}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (card fx)),{Y})) is finite V51() set
rng (D,F,X) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (D,F,X) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len (D,F,X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (D,F,X)) is finite len (D,F,X) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
Y is set
X \/ Y is set
F | (X \/ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ Y)) is Element of K19(D)
K19(D) is set
(D,F,(X \/ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dr is set
F | dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | dr) is Element of K19(D)
X \/ dr is set
F | (X \/ dr) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ dr)) is Element of K19(D)
X /\ dr is set
(D,F,(X \/ dr)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,dr) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,dr) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
xf is finite set
card xf is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
the Element of dom (F | dr) is Element of dom (F | dr)
dx is Element of D
{dx} is non empty V12() finite 1 -element set
dr \ {dx} is Element of K19(dr)
K19(dr) is set
dom F is Element of K19(D)
(dom F) /\ dr is Element of K19(D)
X \ {dx} is Element of K19(X)
K19(X) is set
(X \/ dr) \ {dx} is Element of K19((X \/ dr))
K19((X \/ dr)) is set
X \/ (dr \ {dx}) is set
F | (dr \ {dx}) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (dr \ {dx})) is Element of K19(D)
(dom F) /\ (dr \ {dx}) is Element of K19(dr)
(dom (F | dr)) \ {dx} is Element of K19(D)
x is set
x is set
x is finite set
card x is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card {dx} is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(Y + 1) - (card {dx}) is V36() real ext-real Element of REAL
(Y + 1) - 1 is V36() real ext-real Element of REAL
(dom F) /\ (X \/ (dr \ {dx})) is Element of K19(D)
(dom F) /\ (X \/ dr) is Element of K19(D)
F | (X \/ (dr \ {dx})) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ (dr \ {dx}))) is Element of K19(D)
(dom F) /\ X is Element of K19(D)
((dom F) /\ X) \/ ((dom F) /\ dr) is Element of K19(D)
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
(dom (F | X)) \/ ((dom F) /\ dr) is Element of K19(D)
(dom (F | X)) \/ (dom (F | dr)) is Element of K19(D)
(D,F,(X \/ (dr \ {dx}))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
F . dx is V36() real ext-real Element of REAL
<*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(X \/ (dr \ {dx}))) ^ <*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
X /\ (dr \ {dx}) is Element of K19(dr)
(D,F,(dr \ {dx})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,(dr \ {dx})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
((D,F,X) ^ (D,F,(dr \ {dx}))) ^ <*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(dr \ {dx})) ^ <*(F . dx)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ ((D,F,(dr \ {dx})) ^ <*(F . dx)*>) is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
((D,F,(dr \ {dx})) ^ <*(F . dx)*>) ^ (D,F,X) is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,dr) ^ (D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
X \/ Y is set
F | (X \/ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ Y)) is Element of K19(D)
X /\ Y is set
(D,F,(X \/ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dr is finite set
card dr is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dom F is Element of K19(D)
(dom F) /\ (X \/ Y) is Element of K19(D)
(dom F) /\ X is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
((dom F) /\ X) \/ ((dom F) /\ Y) is Element of K19(D)
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
(dom (F | X)) \/ ((dom F) /\ Y) is Element of K19(D)
(dom (F | X)) \/ (dom (F | Y)) is Element of K19(D)
(D,F,(dom (F | X))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (<*> REAL) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(dom (F | Y))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,(dom (F | Y))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
X /\ Y is set
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,X) is V36() real ext-real Element of REAL
K197() $$ (D,F,X) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is V36() real ext-real Element of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,X) is V36() real ext-real Element of REAL
K197() $$ (D,F,X) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
Y (#) F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,(Y (#) F),X) is V36() real ext-real Element of REAL
(D,(Y (#) F),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(Y (#) F),X) is V36() real ext-real Element of REAL
K197() $$ (D,(Y (#) F),X) is V36() real ext-real Element of REAL
Y * (D,F,X) is V36() real ext-real Element of REAL
(Y (#) F) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((Y (#) F) | X) is Element of K19(D)
Y (#) (F | X) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (Y (#) (F | X)) is Element of K19(D)
F | (dom (F | X)) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
xf is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (F | (dom (F | X))) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng xf is finite set
dom F is Element of K19(D)
(dom F) /\ X is Element of K19(D)
F | (dom F) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(F | (dom F)) | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
rd is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
rng rd is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (F | X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Y * rd is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K476(Y) is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() [;] (Y,(id REAL)) is set
rd (#) K476(Y) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
len (Y * rd) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len rd is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dx is V36() real ext-real Element of REAL
rng (Y * rd) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng ((Y (#) F) | X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (Y (#) (F | X)) is complex-membered ext-real-membered real-membered Element of K19(REAL)
gx is Element of D
(Y (#) (F | X)) . gx is V36() real ext-real Element of REAL
(F | X) . gx is V36() real ext-real Element of REAL
dom rd is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is ordinal natural V36() real ext-real non negative finite cardinal set
rd . x is V36() real ext-real Element of REAL
Y * ((F | X) . gx) is V36() real ext-real Element of REAL
(Y * rd) . x is V36() real ext-real Element of REAL
dom (Y * rd) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dx is V36() real ext-real Element of REAL
rng (Y * rd) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
dom (Y * rd) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
gx is ordinal natural V36() real ext-real non negative finite cardinal set
(Y * rd) . gx is V36() real ext-real Element of REAL
rd . gx is V36() real ext-real Element of REAL
x is V36() real ext-real Element of REAL
Y * x is V36() real ext-real Element of REAL
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom rd is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Y is Relation-like Function-like finite set
dom Y is finite set
(Y (#) (F | X)) " {dx} is Element of K19(D)
((Y (#) F) | X) " {dx} is Element of K19(D)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim ((Y * rd),dx) is finite set
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite set
dx / Y is V36() real ext-real Element of REAL
Coim (rd,(dx / Y)) is finite set
{(dx / Y)} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
rd " {(dx / Y)} is finite set
Coim (((Y (#) F) | X),dx) is set
((Y (#) F) | X) " {dx} is set
(Y (#) (F | X)) " {dx} is Element of K19(D)
Y is Relation-like Function-like finite set
Coim (Y,(dx / Y)) is set
Y " {(dx / Y)} is finite set
card (Coim ((Y * rd),dx)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
Coim (dr,dx) is set
dr " {dx} is finite set
card (Coim (dr,dx)) is ordinal cardinal set
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim ((Y * rd),dx) is finite set
(Y * rd) " {dx} is finite set
card (Coim ((Y * rd),dx)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim (dr,dx) is set
card (Coim (dr,dx)) is ordinal cardinal set
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim ((Y * rd),dx) is finite set
(Y * rd) " {dx} is finite set
card (Coim ((Y * rd),dx)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim (dr,dx) is set
card (Coim (dr,dx)) is ordinal cardinal set
dx is V36() real ext-real Element of REAL
rng (Y * rd) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dx is V36() real ext-real Element of REAL
rng (Y * rd) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
{dx} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
(Y * rd) " {dx} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card ((Y * rd) " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
dr is Relation-like Function-like finite set
dr " {dx} is finite set
card (dr " {dx}) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim ((Y * rd),dx) is finite set
(Y * rd) " {dx} is finite set
card (Coim ((Y * rd),dx)) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Coim (dr,dx) is set
card (Coim (dr,dx)) is ordinal cardinal set
rng ((Y (#) F) | X) is complex-membered ext-real-membered real-membered Element of K19(REAL)
Sum rd is V36() real ext-real Element of REAL
K197() $$ rd is V36() real ext-real Element of REAL
dom (Y (#) F) is Element of K19(D)
(dom (Y (#) F)) /\ X is Element of K19(D)
Sum (Y * rd) is V36() real ext-real Element of REAL
K197() $$ (Y * rd) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F + X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
K19(D) is set
X | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (X | Y) is Element of K19(D)
(D,(F + X),Y) is V36() real ext-real Element of REAL
(D,(F + X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(F + X),Y) is V36() real ext-real Element of REAL
K197() $$ (D,(F + X),Y) is V36() real ext-real Element of REAL
(D,F,Y) is V36() real ext-real Element of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,Y) is V36() real ext-real Element of REAL
K197() $$ (D,F,Y) is V36() real ext-real Element of REAL
(D,X,Y) is V36() real ext-real Element of REAL
(D,X,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,X,Y) is V36() real ext-real Element of REAL
K197() $$ (D,X,Y) is V36() real ext-real Element of REAL
(D,F,Y) + (D,X,Y) is V36() real ext-real Element of REAL
fx is finite set
card fx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Y + 1 is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dr + xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
rd is set
dr | rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (dr | rd) is Element of K19(D)
xf | rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (xf | rd) is Element of K19(D)
(D,(dr + xf),rd) is V36() real ext-real Element of REAL
(D,(dr + xf),rd) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(dr + xf),rd) is V36() real ext-real Element of REAL
K197() $$ (D,(dr + xf),rd) is V36() real ext-real Element of REAL
(D,dr,rd) is V36() real ext-real Element of REAL
(D,dr,rd) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,dr,rd) is V36() real ext-real Element of REAL
K197() $$ (D,dr,rd) is V36() real ext-real Element of REAL
(D,xf,rd) is V36() real ext-real Element of REAL
(D,xf,rd) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,xf,rd) is V36() real ext-real Element of REAL
K197() $$ (D,xf,rd) is V36() real ext-real Element of REAL
(D,dr,rd) + (D,xf,rd) is V36() real ext-real Element of REAL
dx is finite set
card dx is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
the Element of dx is Element of dx
dom dr is Element of K19(D)
(dom dr) /\ rd is Element of K19(D)
x is Element of D
{x} is non empty V12() finite 1 -element set
rd \ {x} is Element of K19(rd)
K19(rd) is set
dr | (rd \ {x}) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (dr | (rd \ {x})) is Element of K19(D)
xf | (rd \ {x}) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (xf | (rd \ {x})) is Element of K19(D)
dom xf is Element of K19(D)
(dom xf) /\ rd is Element of K19(D)
(dom dr) /\ (dom xf) is Element of K19(D)
dom (dr + xf) is Element of K19(D)
(dom dr) /\ (rd \ {x}) is Element of K19(rd)
dx \ {x} is finite Element of K19(dx)
K19(dx) is finite V51() set
dy is set
dy is set
(dom xf) /\ (rd \ {x}) is Element of K19(rd)
dy is finite set
y is set
y is set
card dy is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card {x} is non empty ordinal natural V36() real ext-real positive non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
(card dx) - (card {x}) is V36() real ext-real set
(Y + 1) - 1 is V36() real ext-real Element of REAL
(D,(dr + xf),(rd \ {x})) is V36() real ext-real Element of REAL
(D,(dr + xf),(rd \ {x})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(dr + xf),(rd \ {x})) is V36() real ext-real Element of REAL
K197() $$ (D,(dr + xf),(rd \ {x})) is V36() real ext-real Element of REAL
(D,dr,(rd \ {x})) is V36() real ext-real Element of REAL
(D,dr,(rd \ {x})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,dr,(rd \ {x})) is V36() real ext-real Element of REAL
K197() $$ (D,dr,(rd \ {x})) is V36() real ext-real Element of REAL
(D,xf,(rd \ {x})) is V36() real ext-real Element of REAL
(D,xf,(rd \ {x})) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,xf,(rd \ {x})) is V36() real ext-real Element of REAL
K197() $$ (D,xf,(rd \ {x})) is V36() real ext-real Element of REAL
(D,dr,(rd \ {x})) + (D,xf,(rd \ {x})) is V36() real ext-real Element of REAL
(dr + xf) | rd is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((dr + xf) | rd) is Element of K19(D)
(dr | rd) + (xf | rd) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((dr | rd) + (xf | rd)) is Element of K19(D)
dx /\ (dom (xf | rd)) is finite Element of K19(D)
(dom (dr + xf)) /\ rd is Element of K19(D)
(dr + xf) . x is V36() real ext-real Element of REAL
<*((dr + xf) . x)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,(dr + xf),(rd \ {x})) ^ <*((dr + xf) . x)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
dr . x is V36() real ext-real Element of REAL
<*(dr . x)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,dr,(rd \ {x})) ^ <*(dr . x)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum ((D,dr,(rd \ {x})) ^ <*(dr . x)*>) is V36() real ext-real Element of REAL
K197() $$ ((D,dr,(rd \ {x})) ^ <*(dr . x)*>) is V36() real ext-real Element of REAL
(D,dr,(rd \ {x})) + (dr . x) is V36() real ext-real Element of REAL
xf . x is V36() real ext-real Element of REAL
<*(xf . x)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,xf,(rd \ {x})) ^ <*(xf . x)*> is Relation-like NAT -defined REAL -valued Function-like non empty finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum ((D,xf,(rd \ {x})) ^ <*(xf . x)*>) is V36() real ext-real Element of REAL
K197() $$ ((D,xf,(rd \ {x})) ^ <*(xf . x)*>) is V36() real ext-real Element of REAL
(D,xf,(rd \ {x})) + (xf . x) is V36() real ext-real Element of REAL
(dr . x) + (xf . x) is V36() real ext-real Element of REAL
(Sum (D,(dr + xf),(rd \ {x}))) + ((dr . x) + (xf . x)) is V36() real ext-real Element of REAL
(Sum (D,(dr + xf),(rd \ {x}))) + ((dr + xf) . x) is V36() real ext-real Element of REAL
Sum ((D,(dr + xf),(rd \ {x})) ^ <*((dr + xf) . x)*>) is V36() real ext-real Element of REAL
K197() $$ ((D,(dr + xf),(rd \ {x})) ^ <*((dr + xf) . x)*>) is V36() real ext-real Element of REAL
Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
Y + dr is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
xf is set
Y | xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (Y | xf) is Element of K19(D)
dr | xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (dr | xf) is Element of K19(D)
(D,(Y + dr),xf) is V36() real ext-real Element of REAL
(D,(Y + dr),xf) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(Y + dr),xf) is V36() real ext-real Element of REAL
K197() $$ (D,(Y + dr),xf) is V36() real ext-real Element of REAL
(D,Y,xf) is V36() real ext-real Element of REAL
(D,Y,xf) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,Y,xf) is V36() real ext-real Element of REAL
K197() $$ (D,Y,xf) is V36() real ext-real Element of REAL
(D,dr,xf) is V36() real ext-real Element of REAL
(D,dr,xf) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,dr,xf) is V36() real ext-real Element of REAL
K197() $$ (D,dr,xf) is V36() real ext-real Element of REAL
(D,Y,xf) + (D,dr,xf) is V36() real ext-real Element of REAL
rd is finite set
card rd is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
rng (Y | xf) is complex-membered ext-real-membered real-membered Element of K19(REAL)
(Y + dr) | xf is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(Y | xf) + (dr | xf) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom ((Y + dr) | xf) is Element of K19(D)
(dom (Y | xf)) /\ (dom (dr | xf)) is Element of K19(D)
rng ((Y + dr) | xf) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (D,(Y + dr),xf) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (D,Y,xf) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (dr | xf) is complex-membered ext-real-membered real-membered Element of K19(REAL)
rng (D,dr,xf) is finite complex-membered ext-real-membered real-membered Element of K19(REAL)
0 + 0 is Function-like functional empty ordinal natural V36() real ext-real non positive non negative V41() V42() finite V51() cardinal {} -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V80() FinSequence-membered Element of NAT
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
F - X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- X is Relation-like Function-like complex-valued set
(- 1) (#) X is Relation-like Function-like set
F + (- X) is Relation-like Function-like set
Y is set
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
K19(D) is set
X | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (X | Y) is Element of K19(D)
(D,(F - X),Y) is V36() real ext-real Element of REAL
(D,(F - X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(F - X),Y) is V36() real ext-real Element of REAL
K197() $$ (D,(F - X),Y) is V36() real ext-real Element of REAL
(D,F,Y) is V36() real ext-real Element of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,Y) is V36() real ext-real Element of REAL
K197() $$ (D,F,Y) is V36() real ext-real Element of REAL
(D,X,Y) is V36() real ext-real Element of REAL
(D,X,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,X,Y) is V36() real ext-real Element of REAL
K197() $$ (D,X,Y) is V36() real ext-real Element of REAL
(D,F,Y) - (D,X,Y) is V36() real ext-real Element of REAL
- 1 is V36() real ext-real non positive Element of REAL
(- 1) (#) X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
((- 1) (#) X) | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (((- 1) (#) X) | Y) is Element of K19(D)
dom ((- 1) (#) X) is Element of K19(D)
(dom ((- 1) (#) X)) /\ Y is Element of K19(D)
dom X is Element of K19(D)
(dom X) /\ Y is Element of K19(D)
(D,((- 1) (#) X),Y) is V36() real ext-real Element of REAL
(D,((- 1) (#) X),Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,((- 1) (#) X),Y) is V36() real ext-real Element of REAL
K197() $$ (D,((- 1) (#) X),Y) is V36() real ext-real Element of REAL
(D,F,Y) + (D,((- 1) (#) X),Y) is V36() real ext-real Element of REAL
(- 1) * (D,X,Y) is V36() real ext-real Element of REAL
(D,F,Y) + ((- 1) * (D,X,Y)) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
K19(D) is set
(D,F,X) is V36() real ext-real Element of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,X) is V36() real ext-real Element of REAL
K197() $$ (D,F,X) is V36() real ext-real Element of REAL
Y is V36() real ext-real Element of REAL
F - Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
- Y is V36() real ext-real set
(- Y) + F is Relation-like Function-like set
(D,(F - Y),X) is V36() real ext-real Element of REAL
(D,(F - Y),X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,(F - Y),X) is V36() real ext-real Element of REAL
K197() $$ (D,(F - Y),X) is V36() real ext-real Element of REAL
Y is finite set
card Y is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
Y * (card Y) is V36() real ext-real Element of REAL
(D,F,X) - (Y * (card Y)) is V36() real ext-real Element of REAL
(card Y) |-> Y is Relation-like NAT -defined REAL -valued Function-like finite card Y -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card Y) -tuples_on REAL
(card Y) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like Element of REAL * : len b1 = card Y } is set
Seg (card Y) is finite card Y -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(Seg (card Y)) --> Y is Relation-like Seg (card Y) -defined {Y} -valued Function-like V18( Seg (card Y),{Y}) finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of K19(K20((Seg (card Y)),{Y}))
{Y} is non empty V12() finite 1 -element complex-membered ext-real-membered real-membered set
K20((Seg (card Y)),{Y}) is finite complex-valued ext-real-valued real-valued set
K19(K20((Seg (card Y)),{Y})) is finite V51() set
len (D,F,X) is ordinal natural V36() real ext-real non negative V41() V42() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(D,F,X) - ((card Y) |-> Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
K198() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K195() is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K197() * ((id REAL),K195()) is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K198() .: ((D,F,X),((card Y) |-> Y)) is set
- ((card Y) |-> Y) is Relation-like Function-like complex-valued set
((card Y) |-> Y) (#) K195() is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(- 1) (#) ((card Y) |-> Y) is Relation-like Function-like set
K476((- 1)) is Relation-like REAL -defined REAL -valued Function-like V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
K199() is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V18(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K199() [;] ((- 1),(id REAL)) is set
((card Y) |-> Y) (#) K476((- 1)) is Relation-like Function-like finite complex-valued ext-real-valued real-valued set
(D,F,X) + (- ((card Y) |-> Y)) is Relation-like Function-like set
xf is Relation-like NAT -defined REAL -valued Function-like finite card Y -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card Y) -tuples_on REAL
Sum xf is V36() real ext-real Element of REAL
K197() $$ xf is V36() real ext-real Element of REAL
rd is Relation-like NAT -defined REAL -valued Function-like finite card Y -element complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like Element of (card Y) -tuples_on REAL
Sum rd is V36() real ext-real Element of REAL
K197() $$ rd is V36() real ext-real Element of REAL
(Sum xf) - (Sum rd) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
(D,F,{}) is V36() real ext-real Element of REAL
(D,F,{}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,{}) is V36() real ext-real Element of REAL
K197() $$ (D,F,{}) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom F is Element of K19(D)
K19(D) is set
X is Element of D
{X} is non empty V12() finite 1 -element set
(D,F,{X}) is V36() real ext-real Element of REAL
(D,F,{X}) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,{X}) is V36() real ext-real Element of REAL
K197() $$ (D,F,{X}) is V36() real ext-real Element of REAL
F . X is V36() real ext-real Element of REAL
<*(F . X)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() finite 1 -element complex-valued ext-real-valued real-valued V68() decreasing non-decreasing non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum <*(F . X)*> is V36() real ext-real Element of REAL
K197() $$ <*(F . X)*> is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
Y is set
X \/ Y is set
F | (X \/ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ Y)) is Element of K19(D)
K19(D) is set
(D,F,(X \/ Y)) is V36() real ext-real Element of REAL
(D,F,(X \/ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,(X \/ Y)) is V36() real ext-real Element of REAL
K197() $$ (D,F,(X \/ Y)) is V36() real ext-real Element of REAL
(D,F,X) is V36() real ext-real Element of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,X) is V36() real ext-real Element of REAL
K197() $$ (D,F,X) is V36() real ext-real Element of REAL
(D,F,Y) is V36() real ext-real Element of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,Y) is V36() real ext-real Element of REAL
K197() $$ (D,F,Y) is V36() real ext-real Element of REAL
(D,F,X) + (D,F,Y) is V36() real ext-real Element of REAL
(D,F,X) ^ (D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum ((D,F,X) ^ (D,F,Y)) is V36() real ext-real Element of REAL
K197() $$ ((D,F,X) ^ (D,F,Y)) is V36() real ext-real Element of REAL
D is non empty set
K20(D,REAL) is V12() non finite complex-valued ext-real-valued real-valued set
K19(K20(D,REAL)) is V12() non finite set
F is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
X is set
Y is set
X \/ Y is set
F | (X \/ Y) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (X \/ Y)) is Element of K19(D)
K19(D) is set
F | X is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | X) is Element of K19(D)
F | Y is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | Y) is Element of K19(D)
(D,F,(X \/ Y)) is V36() real ext-real Element of REAL
(D,F,(X \/ Y)) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,(X \/ Y)) is V36() real ext-real Element of REAL
K197() $$ (D,F,(X \/ Y)) is V36() real ext-real Element of REAL
(D,F,X) is V36() real ext-real Element of REAL
(D,F,X) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,X) is V36() real ext-real Element of REAL
K197() $$ (D,F,X) is V36() real ext-real Element of REAL
(D,F,Y) is V36() real ext-real Element of REAL
(D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
Sum (D,F,Y) is V36() real ext-real Element of REAL
K197() $$ (D,F,Y) is V36() real ext-real Element of REAL
(D,F,X) + (D,F,Y) is V36() real ext-real Element of REAL
dom F is Element of K19(D)
(dom F) /\ (X \/ Y) is Element of K19(D)
(dom F) /\ X is Element of K19(D)
(dom F) /\ Y is Element of K19(D)
((dom F) /\ X) \/ ((dom F) /\ Y) is Element of K19(D)
(dom (F | X)) \/ ((dom F) /\ Y) is Element of K19(D)
(dom (F | X)) \/ (dom (F | Y)) is Element of K19(D)
(D,F,(dom (F | X))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,(dom (F | Y))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
F | (dom (F | (X \/ Y))) is Relation-like D -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K19(K20(D,REAL))
dom (F | (dom (F | (X \/ Y)))) is Element of K19(D)
(dom F) /\ (dom (F | (X \/ Y))) is Element of K19(D)
(dom F) /\ ((dom F) /\ (X \/ Y)) is Element of K19(D)
(dom F) /\ (dom F) is Element of K19(D)
((dom F) /\ (dom F)) /\ (X \/ Y) is Element of K19(D)
(D,F,(dom (F | (X \/ Y)))) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued non-increasing FinSequence-like FinSubsequence-like FinSequence of REAL
(D,F,X) ^ (D,F,Y) is Relation-like NAT -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued FinSequence-like FinSubsequence-like FinSequence of REAL
Sum ((D,F,X) ^ (D,F,Y)) is V36() real ext-real Element of REAL
K197() $$ ((D,F,X) ^ (D,F,Y)) is V36() real ext-real Element of REAL