:: FDIFF_3 semantic presentation

REAL is V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() set
NAT is V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() Element of K19(REAL)
K19(REAL) is set
COMPLEX is V11() V12() V49() V51() V71() V77() set
omega is V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() set
K19(omega) is set
K20(NAT,REAL) is complex-valued ext-real-valued real-valued set
K19(K20(NAT,REAL)) is set
K19(K19(REAL)) is set
0 is set
the V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() set is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() set
1 is V11() epsilon-transitive epsilon-connected ordinal natural V30() real ext-real positive non negative V71() V72() V73() V74() V75() V76() V91() V92() V97() V99() Element of NAT
{0,1} is V51() set
K20(REAL,REAL) is complex-valued ext-real-valued real-valued set
K19(K20(REAL,REAL)) is set
RAT is V11() V12() V49() V51() V71() V72() V73() V74() V77() set
INT is V11() V12() V49() V51() V71() V72() V73() V74() V75() V77() set
K20(COMPLEX,COMPLEX) is complex-valued set
K19(K20(COMPLEX,COMPLEX)) is set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is complex-valued set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is set
K20(K20(REAL,REAL),REAL) is complex-valued ext-real-valued real-valued set
K19(K20(K20(REAL,REAL),REAL)) is set
K20(RAT,RAT) is V5( RAT ) complex-valued ext-real-valued real-valued set
K19(K20(RAT,RAT)) is set
K20(K20(RAT,RAT),RAT) is V5( RAT ) complex-valued ext-real-valued real-valued set
K19(K20(K20(RAT,RAT),RAT)) is set
K20(INT,INT) is V5( RAT ) V5( INT ) complex-valued ext-real-valued real-valued set
K19(K20(INT,INT)) is set
K20(K20(INT,INT),INT) is V5( RAT ) V5( INT ) complex-valued ext-real-valued real-valued set
K19(K20(K20(INT,INT),INT)) is set
K20(NAT,NAT) is V5( RAT ) V5( INT ) complex-valued ext-real-valued real-valued natural-valued set
K20(K20(NAT,NAT),NAT) is V5( RAT ) V5( INT ) complex-valued ext-real-valued real-valued natural-valued set
K19(K20(K20(NAT,NAT),NAT)) is set
K19(NAT) is set
0 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
{0} is V51() V71() V72() V73() V74() V75() V76() V99() V100() V101() set
K19(K20(NAT,NAT)) is set
2 is V11() epsilon-transitive epsilon-connected ordinal natural V30() real ext-real positive non negative V71() V72() V73() V74() V75() V76() V91() V92() V97() V99() Element of NAT
-infty is V11() non real ext-real non positive negative set
+infty is V11() non real ext-real positive non negative set
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V30() real ext-real Element of REAL
x0 - N is V30() real ext-real Element of REAL
[.(x0 - N),x0.] is V71() V72() V73() V102() Element of K19(REAL)
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- N is V30() real ext-real Element of REAL
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 + 2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N / (q2 + 2) is V30() real ext-real Element of REAL
- 0 is V30() real ext-real Element of REAL
- (N / (q2 + 2)) is V30() real ext-real Element of REAL
(- N) / (q2 + 2) is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . q2 is V30() real ext-real Element of REAL
lim q1 is V30() real ext-real Element of REAL
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
q2 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + r) is V71() V72() V73() Element of K19(REAL)
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . h1 is V30() real ext-real Element of REAL
h1 is set
r . 0 is V30() real ext-real Element of REAL
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 + r) . h1 is V30() real ext-real Element of REAL
h1 + 2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N * (h1 + 2) is V30() real ext-real Element of REAL
N * 1 is V30() real ext-real Element of REAL
(h1 + 2) " is V30() real ext-real Element of REAL
(N * (h1 + 2)) * ((h1 + 2) ") is V30() real ext-real Element of REAL
N * ((h1 + 2) ") is V30() real ext-real Element of REAL
(h1 + 2) * ((h1 + 2) ") is V30() real ext-real Element of REAL
N * ((h1 + 2) * ((h1 + 2) ")) is V30() real ext-real Element of REAL
N / (h1 + 2) is V30() real ext-real Element of REAL
x0 - (N / (h1 + 2)) is V30() real ext-real Element of REAL
- (N / (h1 + 2)) is V30() real ext-real Element of REAL
x0 + (- (N / (h1 + 2))) is V30() real ext-real Element of REAL
(- N) / (h1 + 2) is V30() real ext-real Element of REAL
x0 + ((- N) / (h1 + 2)) is V30() real ext-real Element of REAL
x0 - 0 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - N <= b₁ & b₁ <= x0 ) } is set
q2 . h1 is V30() real ext-real Element of REAL
r . h1 is V30() real ext-real Element of REAL
(q2 . h1) + (r . h1) is V30() real ext-real Element of REAL
((- N) / (h1 + 2)) + (r . h1) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 + 2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N / (q2 + 2) is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . q2 is V30() real ext-real Element of REAL
lim q1 is V30() real ext-real Element of REAL
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
q2 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + r) is V71() V72() V73() Element of K19(REAL)
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . h1 is V30() real ext-real Element of REAL
h1 is set
r . 0 is V30() real ext-real Element of REAL
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 + r) . h1 is V30() real ext-real Element of REAL
h1 + 2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N * (h1 + 2) is V30() real ext-real Element of REAL
N * 1 is V30() real ext-real Element of REAL
(h1 + 2) " is V30() real ext-real Element of REAL
(N * (h1 + 2)) * ((h1 + 2) ") is V30() real ext-real Element of REAL
N * ((h1 + 2) ") is V30() real ext-real Element of REAL
(h1 + 2) * ((h1 + 2) ") is V30() real ext-real Element of REAL
N * ((h1 + 2) * ((h1 + 2) ")) is V30() real ext-real Element of REAL
N / (h1 + 2) is V30() real ext-real Element of REAL
x0 + (N / (h1 + 2)) is V30() real ext-real Element of REAL
x0 + 0 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + N ) } is set
q2 . h1 is V30() real ext-real Element of REAL
r . h1 is V30() real ext-real Element of REAL
(q2 . h1) + (r . h1) is V30() real ext-real Element of REAL
(N / (h1 + 2)) + (r . h1) is V30() real ext-real Element of REAL
(N / (h1 + 2)) + x0 is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
N + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + r) is V71() V72() V73() Element of K19(REAL)
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
f /* (N + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (N + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((f /* (N + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((f /* (N + r)) - (f /* r))) is V30() real ext-real Element of REAL
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * h1) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . (2 * h1) is V30() real ext-real Element of REAL
N . h1 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q1 . ((2 * h1) + 1) is V30() real ext-real Element of REAL
g1 . h1 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
lim g1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
lim N is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
lim q1 is V30() real ext-real Element of REAL
h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + r) is V71() V72() V73() Element of K19(REAL)
h2 is set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 + r) . c2 is V30() real ext-real Element of REAL
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * c2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . c2 is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(q2 . c2) + (r . c2) is V30() real ext-real Element of REAL
N . c2 is V30() real ext-real Element of REAL
r . (2 * c2) is V30() real ext-real Element of REAL
(N . c2) + (r . (2 * c2)) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(N . c2) + (r . c2) is V30() real ext-real Element of REAL
(N + r) . c2 is V30() real ext-real Element of REAL
g1 . c2 is V30() real ext-real Element of REAL
r . ((2 * c2) + 1) is V30() real ext-real Element of REAL
(g1 . c2) + (r . ((2 * c2) + 1)) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(g1 . c2) + (r . c2) is V30() real ext-real Element of REAL
(g1 + r) . c2 is V30() real ext-real Element of REAL
h2 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (q2 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((f /* (q2 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (q2 ") (#) ((f /* (q2 + r)) - (f /* r))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2) . h2 is V30() real ext-real Element of REAL
h2 . h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (h2 . h2) is V30() real ext-real Element of REAL
2 * h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * h2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(q2 ") . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((q2 ") . ((2 * h2) + 1)) * (((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(f /* (q2 + r)) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(f /* r) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((q2 ") . ((2 * h2) + 1)) * (((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
q2 . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(q2 . ((2 * h2) + 1)) " is V30() real ext-real Element of REAL
((q2 . ((2 * h2) + 1)) ") * (((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
g1 . h2 is V30() real ext-real Element of REAL
(g1 . h2) " is V30() real ext-real Element of REAL
((g1 . h2) ") * (((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(g1 ") . h2 is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(q2 + r) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
f . ((q2 + r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . ((2 * h2) + 1))) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((q2 + r) . ((2 * h2) + 1))) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
r . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)))) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)))) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(g1 . h2) + (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
f . ((g1 . h2) + (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
r . h2 is V30() real ext-real Element of REAL
(g1 . h2) + (r . h2) is V30() real ext-real Element of REAL
f . ((g1 . h2) + (r . h2)) is V30() real ext-real Element of REAL
(f . ((g1 . h2) + (r . h2))) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 . h2) + (r . h2))) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(g1 + r) . h2 is V30() real ext-real Element of REAL
f . ((g1 + r) . h2) is V30() real ext-real Element of REAL
(f . ((g1 + r) . h2)) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 + r) . h2)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(f /* (g1 + r)) . h2 is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
f . (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - (f . (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - (f . (r . ((2 * h2) + 1)))) is V30() real ext-real Element of REAL
f . (r . h2) is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - (f . (r . h2)) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - (f . (r . h2))) is V30() real ext-real Element of REAL
(f /* r) . h2 is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - ((f /* r) . h2) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - ((f /* r) . h2)) is V30() real ext-real Element of REAL
((f /* (g1 + r)) - (f /* r)) . h2 is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) - (f /* r)) . h2) is V30() real ext-real Element of REAL
((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) . h2 is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * c2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . (2 * c2) is V30() real ext-real Element of REAL
N . c2 is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
q2 . ((2 * c2) + 1) is V30() real ext-real Element of REAL
g1 . c2 is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
h2 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (q2 ") (#) ((f /* (q2 + r)) - (f /* r))
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2) . c2 is V30() real ext-real Element of REAL
h2 . c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (h2 . c2) is V30() real ext-real Element of REAL
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (2 * c2) is V30() real ext-real Element of REAL
(q2 ") . (2 * c2) is V30() real ext-real Element of REAL
((f /* (q2 + r)) - (f /* r)) . (2 * c2) is V30() real ext-real Element of REAL
((q2 ") . (2 * c2)) * (((f /* (q2 + r)) - (f /* r)) . (2 * c2)) is V30() real ext-real Element of REAL
(f /* (q2 + r)) . (2 * c2) is V30() real ext-real Element of REAL
(f /* r) . (2 * c2) is V30() real ext-real Element of REAL
((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((q2 ") . (2 * c2)) * (((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
q2 . (2 * c2) is V30() real ext-real Element of REAL
(q2 . (2 * c2)) " is V30() real ext-real Element of REAL
((q2 . (2 * c2)) ") * (((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
N . c2 is V30() real ext-real Element of REAL
(N . c2) " is V30() real ext-real Element of REAL
((N . c2) ") * (((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(N ") . c2 is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(q2 + r) . (2 * c2) is V30() real ext-real Element of REAL
f . ((q2 + r) . (2 * c2)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . (2 * c2))) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((q2 + r) . (2 * c2))) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
r . (2 * c2) is V30() real ext-real Element of REAL
(q2 . (2 * c2)) + (r . (2 * c2)) is V30() real ext-real Element of REAL
f . ((q2 . (2 * c2)) + (r . (2 * c2))) is V30() real ext-real Element of REAL
(f . ((q2 . (2 * c2)) + (r . (2 * c2)))) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((q2 . (2 * c2)) + (r . (2 * c2)))) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(N . c2) + (r . (2 * c2)) is V30() real ext-real Element of REAL
f . ((N . c2) + (r . (2 * c2))) is V30() real ext-real Element of REAL
(f . ((N . c2) + (r . (2 * c2)))) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N . c2) + (r . (2 * c2)))) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(N . c2) + (r . c2) is V30() real ext-real Element of REAL
f . ((N . c2) + (r . c2)) is V30() real ext-real Element of REAL
(f . ((N . c2) + (r . c2))) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N . c2) + (r . c2))) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(N + r) . c2 is V30() real ext-real Element of REAL
f . ((N + r) . c2) is V30() real ext-real Element of REAL
(f . ((N + r) . c2)) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N + r) . c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(f /* (N + r)) . c2 is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
f . (r . (2 * c2)) is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - (f . (r . (2 * c2))) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - (f . (r . (2 * c2)))) is V30() real ext-real Element of REAL
f . (r . c2) is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - (f . (r . c2)) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - (f . (r . c2))) is V30() real ext-real Element of REAL
(f /* r) . c2 is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - ((f /* r) . c2) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - ((f /* r) . c2)) is V30() real ext-real Element of REAL
((f /* (N + r)) - (f /* r)) . c2 is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) - (f /* r)) . c2) is V30() real ext-real Element of REAL
((N ") (#) ((f /* (N + r)) - (f /* r))) . c2 is V30() real ext-real Element of REAL
lim ((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
N + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + r) is V71() V72() V73() Element of K19(REAL)
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
f /* (N + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (N + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((f /* (N + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((f /* (N + r)) - (f /* r))) is V30() real ext-real Element of REAL
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * h1) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . q2 is V30() real ext-real Element of REAL
N . h1 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
g1 . h1 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
q1 . q2 is V30() real ext-real Element of REAL
lim g1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
lim N is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
lim q1 is V30() real ext-real Element of REAL
h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + r) is V71() V72() V73() Element of K19(REAL)
h2 is set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 + r) . c2 is V30() real ext-real Element of REAL
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * c2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . (2 * c2) is V30() real ext-real Element of REAL
r . (2 * c2) is V30() real ext-real Element of REAL
(q2 . (2 * c2)) + (r . (2 * c2)) is V30() real ext-real Element of REAL
N . c2 is V30() real ext-real Element of REAL
(N . c2) + (r . (2 * c2)) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(N . c2) + (r . c2) is V30() real ext-real Element of REAL
(N + r) . c2 is V30() real ext-real Element of REAL
q2 . ((2 * c2) + 1) is V30() real ext-real Element of REAL
r . ((2 * c2) + 1) is V30() real ext-real Element of REAL
(q2 . ((2 * c2) + 1)) + (r . ((2 * c2) + 1)) is V30() real ext-real Element of REAL
g1 . c2 is V30() real ext-real Element of REAL
(g1 . c2) + (r . ((2 * c2) + 1)) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
(g1 . c2) + (r . c2) is V30() real ext-real Element of REAL
(g1 + r) . c2 is V30() real ext-real Element of REAL
h2 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (q2 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((f /* (q2 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (q2 ") (#) ((f /* (q2 + r)) - (f /* r))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2) . h2 is V30() real ext-real Element of REAL
h2 . h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (h2 . h2) is V30() real ext-real Element of REAL
2 * h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * h2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(q2 ") . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((q2 ") . ((2 * h2) + 1)) * (((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
q2 . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(q2 . ((2 * h2) + 1)) " is V30() real ext-real Element of REAL
((q2 . ((2 * h2) + 1)) ") * (((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
g1 . h2 is V30() real ext-real Element of REAL
(g1 . h2) " is V30() real ext-real Element of REAL
((g1 . h2) ") * (((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(g1 ") . h2 is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (q2 + r)) - (f /* r)) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(f /* (q2 + r)) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
(f /* r) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (q2 + r)) . ((2 * h2) + 1)) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(q2 + r) . ((2 * h2) + 1) is V30() real ext-real Element of REAL
f . ((q2 + r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . ((2 * h2) + 1))) - ((f /* r) . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((q2 + r) . ((2 * h2) + 1))) - ((f /* r) . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
r . ((2 * h2) + 1) is V30() real ext-real Element of REAL
f . (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . ((2 * h2) + 1))) - (f . (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((q2 + r) . ((2 * h2) + 1))) - (f . (r . ((2 * h2) + 1)))) is V30() real ext-real Element of REAL
(q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)))) - (f . (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((q2 . ((2 * h2) + 1)) + (r . ((2 * h2) + 1)))) - (f . (r . ((2 * h2) + 1)))) is V30() real ext-real Element of REAL
(g1 . h2) + (r . ((2 * h2) + 1)) is V30() real ext-real Element of REAL
f . ((g1 . h2) + (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
(f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - (f . (r . ((2 * h2) + 1))) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - (f . (r . ((2 * h2) + 1)))) is V30() real ext-real Element of REAL
r . h2 is V30() real ext-real Element of REAL
f . (r . h2) is V30() real ext-real Element of REAL
(f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - (f . (r . h2)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 . h2) + (r . ((2 * h2) + 1)))) - (f . (r . h2))) is V30() real ext-real Element of REAL
(g1 . h2) + (r . h2) is V30() real ext-real Element of REAL
f . ((g1 . h2) + (r . h2)) is V30() real ext-real Element of REAL
(f . ((g1 . h2) + (r . h2))) - (f . (r . h2)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 . h2) + (r . h2))) - (f . (r . h2))) is V30() real ext-real Element of REAL
(g1 + r) . h2 is V30() real ext-real Element of REAL
f . ((g1 + r) . h2) is V30() real ext-real Element of REAL
(f . ((g1 + r) . h2)) - (f . (r . h2)) is V30() real ext-real Element of REAL
((g1 ") . h2) * ((f . ((g1 + r) . h2)) - (f . (r . h2))) is V30() real ext-real Element of REAL
(f /* (g1 + r)) . h2 is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - (f . (r . h2)) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - (f . (r . h2))) is V30() real ext-real Element of REAL
(f /* r) . h2 is V30() real ext-real Element of REAL
((f /* (g1 + r)) . h2) - ((f /* r) . h2) is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) . h2) - ((f /* r) . h2)) is V30() real ext-real Element of REAL
((f /* (g1 + r)) - (f /* r)) . h2 is V30() real ext-real Element of REAL
((g1 ") . h2) * (((f /* (g1 + r)) - (f /* r)) . h2) is V30() real ext-real Element of REAL
((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) . h2 is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h2 is V30() real ext-real Element of REAL
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(2 * c2) + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . c2 is V30() real ext-real Element of REAL
g1 . c2 is V30() real ext-real Element of REAL
h2 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (q2 ") (#) ((f /* (q2 + r)) - (f /* r))
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) * h2) . c2 is V30() real ext-real Element of REAL
h2 . c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (h2 . c2) is V30() real ext-real Element of REAL
2 * c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) . (2 * c2) is V30() real ext-real Element of REAL
(q2 ") . (2 * c2) is V30() real ext-real Element of REAL
((f /* (q2 + r)) - (f /* r)) . (2 * c2) is V30() real ext-real Element of REAL
((q2 ") . (2 * c2)) * (((f /* (q2 + r)) - (f /* r)) . (2 * c2)) is V30() real ext-real Element of REAL
q2 . (2 * c2) is V30() real ext-real Element of REAL
(q2 . (2 * c2)) " is V30() real ext-real Element of REAL
((q2 . (2 * c2)) ") * (((f /* (q2 + r)) - (f /* r)) . (2 * c2)) is V30() real ext-real Element of REAL
N . c2 is V30() real ext-real Element of REAL
(N . c2) " is V30() real ext-real Element of REAL
((N . c2) ") * (((f /* (q2 + r)) - (f /* r)) . (2 * c2)) is V30() real ext-real Element of REAL
(N ") . c2 is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (q2 + r)) - (f /* r)) . (2 * c2)) is V30() real ext-real Element of REAL
(f /* (q2 + r)) . (2 * c2) is V30() real ext-real Element of REAL
(f /* r) . (2 * c2) is V30() real ext-real Element of REAL
((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (q2 + r)) . (2 * c2)) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
(q2 + r) . (2 * c2) is V30() real ext-real Element of REAL
f . ((q2 + r) . (2 * c2)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . (2 * c2))) - ((f /* r) . (2 * c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((q2 + r) . (2 * c2))) - ((f /* r) . (2 * c2))) is V30() real ext-real Element of REAL
r . (2 * c2) is V30() real ext-real Element of REAL
f . (r . (2 * c2)) is V30() real ext-real Element of REAL
(f . ((q2 + r) . (2 * c2))) - (f . (r . (2 * c2))) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((q2 + r) . (2 * c2))) - (f . (r . (2 * c2)))) is V30() real ext-real Element of REAL
(q2 . (2 * c2)) + (r . (2 * c2)) is V30() real ext-real Element of REAL
f . ((q2 . (2 * c2)) + (r . (2 * c2))) is V30() real ext-real Element of REAL
(f . ((q2 . (2 * c2)) + (r . (2 * c2)))) - (f . (r . (2 * c2))) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((q2 . (2 * c2)) + (r . (2 * c2)))) - (f . (r . (2 * c2)))) is V30() real ext-real Element of REAL
(N . c2) + (r . (2 * c2)) is V30() real ext-real Element of REAL
f . ((N . c2) + (r . (2 * c2))) is V30() real ext-real Element of REAL
(f . ((N . c2) + (r . (2 * c2)))) - (f . (r . (2 * c2))) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N . c2) + (r . (2 * c2)))) - (f . (r . (2 * c2)))) is V30() real ext-real Element of REAL
r . c2 is V30() real ext-real Element of REAL
f . (r . c2) is V30() real ext-real Element of REAL
(f . ((N . c2) + (r . (2 * c2)))) - (f . (r . c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N . c2) + (r . (2 * c2)))) - (f . (r . c2))) is V30() real ext-real Element of REAL
(N . c2) + (r . c2) is V30() real ext-real Element of REAL
f . ((N . c2) + (r . c2)) is V30() real ext-real Element of REAL
(f . ((N . c2) + (r . c2))) - (f . (r . c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N . c2) + (r . c2))) - (f . (r . c2))) is V30() real ext-real Element of REAL
(N + r) . c2 is V30() real ext-real Element of REAL
f . ((N + r) . c2) is V30() real ext-real Element of REAL
(f . ((N + r) . c2)) - (f . (r . c2)) is V30() real ext-real Element of REAL
((N ") . c2) * ((f . ((N + r) . c2)) - (f . (r . c2))) is V30() real ext-real Element of REAL
(f /* (N + r)) . c2 is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - (f . (r . c2)) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - (f . (r . c2))) is V30() real ext-real Element of REAL
(f /* r) . c2 is V30() real ext-real Element of REAL
((f /* (N + r)) . c2) - ((f /* r) . c2) is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) . c2) - ((f /* r) . c2)) is V30() real ext-real Element of REAL
((f /* (N + r)) - (f /* r)) . c2 is V30() real ext-real Element of REAL
((N ") . c2) * (((f /* (N + r)) - (f /* r)) . c2) is V30() real ext-real Element of REAL
((N ") (#) ((f /* (N + r)) - (f /* r))) . c2 is V30() real ext-real Element of REAL
lim ((q2 ") (#) ((f /* (q2 + r)) - (f /* r))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
left_open_halfline x0 is V71() V72() V73() Element of K19(REAL)
K220(-infty,x0) is V97() V98() V102() set
dom f is V71() V72() V73() Element of K19(REAL)
(left_open_halfline x0) /\ (dom f) is V71() V72() V73() Element of K19(REAL)
f . x0 is V30() real ext-real Element of REAL
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
lim g1 is V30() real ext-real Element of REAL
f /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (f /* g1) is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 - r is V30() real ext-real Element of REAL
[.(x0 - r),x0.] is V71() V72() V73() V102() Element of K19(REAL)
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 - N is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- N is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) N is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
g1 + (- N) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
N ^\ q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
{x0} is V51() V71() V72() V73() V99() V100() V101() set
h1 is set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(N ^\ q1) . h2 is V30() real ext-real Element of REAL
h2 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . (h2 + q1) is V30() real ext-real Element of REAL
h1 is set
h1 . 0 is V30() real ext-real Element of REAL
0 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . (0 + q1) is V30() real ext-real Element of REAL
h1 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h1) . h2 is V30() real ext-real Element of REAL
((f /* h1) . h2) - (f . x0) is V30() real ext-real Element of REAL
abs (((f /* h1) . h2) - (f . x0)) is V30() real ext-real Element of REAL
h1 . h2 is V30() real ext-real Element of REAL
f . (h1 . h2) is V30() real ext-real Element of REAL
(f . (h1 . h2)) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (h1 . h2)) - (f . x0)) is V30() real ext-real Element of REAL
h2 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . (h2 + q1) is V30() real ext-real Element of REAL
f . (N . (h2 + q1)) is V30() real ext-real Element of REAL
(f . (N . (h2 + q1))) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (N . (h2 + q1))) - (f . x0)) is V30() real ext-real Element of REAL
(f . x0) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . x0) - (f . x0)) is V30() real ext-real Element of REAL
lim N is V30() real ext-real Element of REAL
N . 0 is V30() real ext-real Element of REAL
lim q2 is V30() real ext-real Element of REAL
x0 - x0 is V30() real ext-real Element of REAL
q2 ^\ q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of q2
lim (q2 ^\ q1) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
g1 . h1 is V30() real ext-real Element of REAL
N . h1 is V30() real ext-real Element of REAL
(g1 . h1) - (N . h1) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not x0 <= b₁ } is set
(g1 . h1) - x0 is V30() real ext-real Element of REAL
h2 is V30() real ext-real Element of REAL
h2 is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 ^\ q1) . h1 is V30() real ext-real Element of REAL
h1 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . (h1 + q1) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(q2 ^\ q1) . h1 is V30() real ext-real Element of REAL
h2 is set
h1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
h1 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (h1 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h1)) - (f /* h1)) + (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (h1 + h1)) - (f /* h1)) + (f /* h1)) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h1)) - (f /* h1)) . h2 is V30() real ext-real Element of REAL
(f /* h1) . h2 is V30() real ext-real Element of REAL
(((f /* (h1 + h1)) - (f /* h1)) . h2) + ((f /* h1) . h2) is V30() real ext-real Element of REAL
(f /* (h1 + h1)) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h1)) . h2) - ((f /* h1) . h2) is V30() real ext-real Element of REAL
(((f /* (h1 + h1)) . h2) - ((f /* h1) . h2)) + ((f /* h1) . h2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h1 + h1) . h2 is V30() real ext-real Element of REAL
(g1 - N) + N is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((g1 - N) + N) ^\ q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (g1 - N) + N
(((g1 - N) + N) ^\ q1) . h2 is V30() real ext-real Element of REAL
h2 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((g1 - N) + N) . (h2 + q1) is V30() real ext-real Element of REAL
(g1 - N) . (h2 + q1) is V30() real ext-real Element of REAL
N . (h2 + q1) is V30() real ext-real Element of REAL
((g1 - N) . (h2 + q1)) + (N . (h2 + q1)) is V30() real ext-real Element of REAL
g1 . (h2 + q1) is V30() real ext-real Element of REAL
(g1 . (h2 + q1)) - (N . (h2 + q1)) is V30() real ext-real Element of REAL
((g1 . (h2 + q1)) - (N . (h2 + q1))) + (N . (h2 + q1)) is V30() real ext-real Element of REAL
g1 ^\ q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of g1
(g1 ^\ q1) . h2 is V30() real ext-real Element of REAL
f /* (g1 ^\ q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* g1) ^\ q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* g1
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(g1 ^\ q1) . h2 is V30() real ext-real Element of REAL
h2 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (h2 + q1) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not x0 <= b₁ } is set
0 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 + h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - r <= b₁ & b₁ <= x0 ) } is set
h2 is V30() real ext-real Element of REAL
rng (h1 + h1) is V71() V72() V73() Element of K19(REAL)
h2 is set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h1 + h1) . h2 is V30() real ext-real Element of REAL
(((g1 - N) + N) ^\ q1) . h2 is V30() real ext-real Element of REAL
h2 + q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((g1 - N) + N) . (h2 + q1) is V30() real ext-real Element of REAL
(g1 - N) . (h2 + q1) is V30() real ext-real Element of REAL
N . (h2 + q1) is V30() real ext-real Element of REAL
((g1 - N) . (h2 + q1)) + (N . (h2 + q1)) is V30() real ext-real Element of REAL
g1 . (h2 + q1) is V30() real ext-real Element of REAL
(g1 . (h2 + q1)) - (N . (h2 + q1)) is V30() real ext-real Element of REAL
((g1 . (h2 + q1)) - (N . (h2 + q1))) + (N . (h2 + q1)) is V30() real ext-real Element of REAL
(g1 ^\ q1) . h2 is V30() real ext-real Element of REAL
h1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h1 ") (#) ((f /* (h1 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 (#) ((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (h1 (#) ((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
lim ((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
0 * (lim ((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h2 is V30() real ext-real Element of REAL
(h1 (#) ((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1)))) . h2 is V30() real ext-real Element of REAL
((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1))) . h2 is V30() real ext-real Element of REAL
(h1 . h2) * (((h1 ") (#) ((f /* (h1 + h1)) - (f /* h1))) . h2) is V30() real ext-real Element of REAL
(h1 ") . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h1)) - (f /* h1)) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * (((f /* (h1 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
(h1 . h2) * (((h1 ") . h2) * (((f /* (h1 + h1)) - (f /* h1)) . h2)) is V30() real ext-real Element of REAL
(h1 . h2) " is V30() real ext-real Element of REAL
((h1 . h2) ") * (((f /* (h1 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
(h1 . h2) * (((h1 . h2) ") * (((f /* (h1 + h1)) - (f /* h1)) . h2)) is V30() real ext-real Element of REAL
(h1 . h2) * ((h1 . h2) ") is V30() real ext-real Element of REAL
((h1 . h2) * ((h1 . h2) ")) * (((f /* (h1 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
1 * (((f /* (h1 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
lim (f /* h1) is V30() real ext-real Element of REAL
lim (((f /* (h1 + h1)) - (f /* h1)) + (f /* h1)) is V30() real ext-real Element of REAL
0 + (f . x0) is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
x0 - N is V30() real ext-real Element of REAL
[.(x0 - N),x0.] is V71() V72() V73() V102() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
g1 is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (r + 1) is V30() real ext-real Element of REAL
x0 - (g1 / (r + 1)) is V30() real ext-real Element of REAL
[.(x0 - (g1 / (r + 1))),x0.] is V71() V72() V73() V102() Element of K19(REAL)
0 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / 1 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - g1 <= b₁ & b₁ <= x0 ) } is set
q1 is V30() real ext-real Element of REAL
f . q1 is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
left_open_halfline x0 is V71() V72() V73() Element of K19(REAL)
K220(-infty,x0) is V97() V98() V102() set
(left_open_halfline x0) /\ (dom f) is V71() V72() V73() Element of K19(REAL)
q1 is set
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . q2 is V30() real ext-real Element of REAL
q2 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (q2 + 1) is V30() real ext-real Element of REAL
x0 - (g1 / (q2 + 1)) is V30() real ext-real Element of REAL
[.(x0 - (g1 / (q2 + 1))),x0.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - (g1 / (q2 + 1)) <= b₁ & b₁ <= x0 ) } is set
h1 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not x0 <= b₁ } is set
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* r) . q1 is V30() real ext-real Element of REAL
r . q1 is V30() real ext-real Element of REAL
f . (r . q1) is V30() real ext-real Element of REAL
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real set
(f /* r) . q1 is V30() real ext-real Element of REAL
q1 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* r) . (q1 + 1) is V30() real ext-real Element of REAL
lim (f /* r) is V30() real ext-real Element of REAL
(f /* r) . 0 is V30() real ext-real Element of REAL
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . h1 is V30() real ext-real Element of REAL
h1 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (h1 + 1) is V30() real ext-real Element of REAL
x0 - (g1 / (h1 + 1)) is V30() real ext-real Element of REAL
[.(x0 - (g1 / (h1 + 1))),x0.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - (g1 / (h1 + 1)) <= b₁ & b₁ <= x0 ) } is set
q2 . h1 is V30() real ext-real Element of REAL
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 . h1 is V30() real ext-real Element of REAL
h1 is V30() real ext-real Element of REAL
lim q1 is V30() real ext-real Element of REAL
q1 . 0 is V30() real ext-real Element of REAL
lim q2 is V30() real ext-real Element of REAL
lim r is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
right_open_halfline x0 is V71() V72() V73() Element of K19(REAL)
K220(x0,+infty) is V97() V98() V102() set
(right_open_halfline x0) /\ (dom f) is V71() V72() V73() Element of K19(REAL)
f . x0 is V30() real ext-real Element of REAL
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
lim r is V30() real ext-real Element of REAL
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (f /* r) is V30() real ext-real Element of REAL
q1 is V30() real ext-real Element of REAL
x0 + q1 is V30() real ext-real Element of REAL
[.x0,(x0 + q1).] is V71() V72() V73() V102() Element of K19(REAL)
x0 + 0 is V30() real ext-real Element of REAL
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
r - g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- g1 is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) g1 is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
r + (- g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
g1 ^\ q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of g1
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
{x0} is V51() V71() V72() V73() V99() V100() V101() set
h2 is set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(g1 ^\ q2) . h2 is V30() real ext-real Element of REAL
h2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (h2 + q2) is V30() real ext-real Element of REAL
h2 is set
h1 . 0 is V30() real ext-real Element of REAL
0 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (0 + q2) is V30() real ext-real Element of REAL
h2 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h1) . c2 is V30() real ext-real Element of REAL
((f /* h1) . c2) - (f . x0) is V30() real ext-real Element of REAL
abs (((f /* h1) . c2) - (f . x0)) is V30() real ext-real Element of REAL
h1 . c2 is V30() real ext-real Element of REAL
f . (h1 . c2) is V30() real ext-real Element of REAL
(f . (h1 . c2)) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (h1 . c2)) - (f . x0)) is V30() real ext-real Element of REAL
c2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (c2 + q2) is V30() real ext-real Element of REAL
f . (g1 . (c2 + q2)) is V30() real ext-real Element of REAL
(f . (g1 . (c2 + q2))) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (g1 . (c2 + q2))) - (f . x0)) is V30() real ext-real Element of REAL
(f . x0) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . x0) - (f . x0)) is V30() real ext-real Element of REAL
lim g1 is V30() real ext-real Element of REAL
g1 . 0 is V30() real ext-real Element of REAL
lim h1 is V30() real ext-real Element of REAL
x0 - x0 is V30() real ext-real Element of REAL
h1 ^\ q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of h1
lim (h1 ^\ q2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h2 is V30() real ext-real Element of REAL
r . h2 is V30() real ext-real Element of REAL
g1 . h2 is V30() real ext-real Element of REAL
(r . h2) - (g1 . h2) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not b₁ <= x0 } is set
(r . h2) - x0 is V30() real ext-real Element of REAL
h2 is V30() real ext-real Element of REAL
h2 is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h1 ^\ q2) . h2 is V30() real ext-real Element of REAL
h2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . (h2 + q2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h1 ^\ q2) . h2 is V30() real ext-real Element of REAL
h2 is set
h2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
h2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (h2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h2 + h1)) - (f /* h1)) + (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (h2 + h1)) - (f /* h1)) + (f /* h1)) . h2 is V30() real ext-real Element of REAL
((f /* (h2 + h1)) - (f /* h1)) . h2 is V30() real ext-real Element of REAL
(f /* h1) . h2 is V30() real ext-real Element of REAL
(((f /* (h2 + h1)) - (f /* h1)) . h2) + ((f /* h1) . h2) is V30() real ext-real Element of REAL
(f /* (h2 + h1)) . h2 is V30() real ext-real Element of REAL
((f /* (h2 + h1)) . h2) - ((f /* h1) . h2) is V30() real ext-real Element of REAL
(((f /* (h2 + h1)) . h2) - ((f /* h1) . h2)) + ((f /* h1) . h2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h2 + h1) . h2 is V30() real ext-real Element of REAL
(r - g1) + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((r - g1) + g1) ^\ q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (r - g1) + g1
(((r - g1) + g1) ^\ q2) . h2 is V30() real ext-real Element of REAL
h2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((r - g1) + g1) . (h2 + q2) is V30() real ext-real Element of REAL
(r - g1) . (h2 + q2) is V30() real ext-real Element of REAL
g1 . (h2 + q2) is V30() real ext-real Element of REAL
((r - g1) . (h2 + q2)) + (g1 . (h2 + q2)) is V30() real ext-real Element of REAL
r . (h2 + q2) is V30() real ext-real Element of REAL
(r . (h2 + q2)) - (g1 . (h2 + q2)) is V30() real ext-real Element of REAL
((r . (h2 + q2)) - (g1 . (h2 + q2))) + (g1 . (h2 + q2)) is V30() real ext-real Element of REAL
r ^\ q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of r
(r ^\ q2) . h2 is V30() real ext-real Element of REAL
f /* (r ^\ q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* r) ^\ q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* r
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(r ^\ q2) . h2 is V30() real ext-real Element of REAL
h2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . (h2 + q2) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not b₁ <= x0 } is set
0 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 + h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + q1 ) } is set
c2 is V30() real ext-real Element of REAL
rng (h2 + h1) is V71() V72() V73() Element of K19(REAL)
h2 is set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(h2 + h1) . c2 is V30() real ext-real Element of REAL
(((r - g1) + g1) ^\ q2) . c2 is V30() real ext-real Element of REAL
c2 + q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((r - g1) + g1) . (c2 + q2) is V30() real ext-real Element of REAL
(r - g1) . (c2 + q2) is V30() real ext-real Element of REAL
g1 . (c2 + q2) is V30() real ext-real Element of REAL
((r - g1) . (c2 + q2)) + (g1 . (c2 + q2)) is V30() real ext-real Element of REAL
r . (c2 + q2) is V30() real ext-real Element of REAL
(r . (c2 + q2)) - (g1 . (c2 + q2)) is V30() real ext-real Element of REAL
((r . (c2 + q2)) - (g1 . (c2 + q2))) + (g1 . (c2 + q2)) is V30() real ext-real Element of REAL
(r ^\ q2) . c2 is V30() real ext-real Element of REAL
h2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h2 ") (#) ((f /* (h2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 (#) ((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (h2 (#) ((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
lim ((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
0 * (lim ((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 . h2 is V30() real ext-real Element of REAL
(h2 (#) ((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1)))) . h2 is V30() real ext-real Element of REAL
((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1))) . h2 is V30() real ext-real Element of REAL
(h2 . h2) * (((h2 ") (#) ((f /* (h2 + h1)) - (f /* h1))) . h2) is V30() real ext-real Element of REAL
(h2 ") . h2 is V30() real ext-real Element of REAL
((f /* (h2 + h1)) - (f /* h1)) . h2 is V30() real ext-real Element of REAL
((h2 ") . h2) * (((f /* (h2 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
(h2 . h2) * (((h2 ") . h2) * (((f /* (h2 + h1)) - (f /* h1)) . h2)) is V30() real ext-real Element of REAL
(h2 . h2) " is V30() real ext-real Element of REAL
((h2 . h2) ") * (((f /* (h2 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
(h2 . h2) * (((h2 . h2) ") * (((f /* (h2 + h1)) - (f /* h1)) . h2)) is V30() real ext-real Element of REAL
(h2 . h2) * ((h2 . h2) ") is V30() real ext-real Element of REAL
((h2 . h2) * ((h2 . h2) ")) * (((f /* (h2 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
1 * (((f /* (h2 + h1)) - (f /* h1)) . h2) is V30() real ext-real Element of REAL
lim (f /* h1) is V30() real ext-real Element of REAL
lim (((f /* (h2 + h1)) - (f /* h1)) + (f /* h1)) is V30() real ext-real Element of REAL
0 + (f . x0) is V30() real ext-real Element of REAL
x0 + 0 is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
g1 is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
[.x0,(x0 + g1).] is V71() V72() V73() V102() Element of K19(REAL)
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (r + 1) is V30() real ext-real Element of REAL
x0 + (g1 / (r + 1)) is V30() real ext-real Element of REAL
[.x0,(x0 + (g1 / (r + 1))).] is V71() V72() V73() V102() Element of K19(REAL)
x0 + 0 is V30() real ext-real Element of REAL
0 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / 1 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + g1 ) } is set
q1 is V30() real ext-real Element of REAL
f . q1 is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
right_open_halfline x0 is V71() V72() V73() Element of K19(REAL)
K220(x0,+infty) is V97() V98() V102() set
(right_open_halfline x0) /\ (dom f) is V71() V72() V73() Element of K19(REAL)
q1 is set
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . q2 is V30() real ext-real Element of REAL
q2 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (q2 + 1) is V30() real ext-real Element of REAL
x0 + (g1 / (q2 + 1)) is V30() real ext-real Element of REAL
[.x0,(x0 + (g1 / (q2 + 1))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + (g1 / (q2 + 1)) ) } is set
h1 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : not b₁ <= x0 } is set
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* r) . q1 is V30() real ext-real Element of REAL
r . q1 is V30() real ext-real Element of REAL
f . (r . q1) is V30() real ext-real Element of REAL
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real set
(f /* r) . q1 is V30() real ext-real Element of REAL
q1 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* r) . (q1 + 1) is V30() real ext-real Element of REAL
lim (f /* r) is V30() real ext-real Element of REAL
(f /* r) . 0 is V30() real ext-real Element of REAL
NAT --> x0 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim q1 is V30() real ext-real Element of REAL
q1 . 0 is V30() real ext-real Element of REAL
q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r . h1 is V30() real ext-real Element of REAL
h1 + 1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 / (h1 + 1) is V30() real ext-real Element of REAL
x0 + (g1 / (h1 + 1)) is V30() real ext-real Element of REAL
[.x0,(x0 + (g1 / (h1 + 1))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + (g1 / (h1 + 1)) ) } is set
q1 . h1 is V30() real ext-real Element of REAL
q2 . h1 is V30() real ext-real Element of REAL
h1 is V30() real ext-real Element of REAL
lim q2 is V30() real ext-real Element of REAL
lim r is V30() real ext-real Element of REAL
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f is V30() real ext-real Element of REAL
{f} is V51() V71() V72() V73() V99() V100() V101() set
dom x0 is V71() V72() V73() Element of K19(REAL)
N is set
g1 is V30() real ext-real Element of REAL
f - g1 is V30() real ext-real Element of REAL
[.(f - g1),f.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
N is V30() real ext-real Element of REAL
f - N is V30() real ext-real Element of REAL
[.(f - N),f.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((x0 /* (N + g1)) - (x0 /* g1))) is V30() real ext-real Element of REAL
r is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
r " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng q1 is V71() V72() V73() Element of K19(REAL)
r + q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (r + q1) is V71() V72() V73() Element of K19(REAL)
x0 /* (r + q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (r + q1)) - (x0 /* q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (r + q1)) + (- (x0 /* q1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(r ") (#) ((x0 /* (r + q1)) - (x0 /* q1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((r ") (#) ((x0 /* (r + q1)) - (x0 /* q1))) is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
f - N is V30() real ext-real Element of REAL
[.(f - N),f.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
q1 is V30() real ext-real Element of REAL
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((x0 /* (N + g1)) - (x0 /* g1))) is V30() real ext-real Element of REAL
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f is V30() real ext-real Element of REAL
{f} is V51() V71() V72() V73() V99() V100() V101() set
dom x0 is V71() V72() V73() Element of K19(REAL)
N is set
g1 is V30() real ext-real Element of REAL
f + g1 is V30() real ext-real Element of REAL
[.f,(f + g1).] is V71() V72() V73() V102() Element of K19(REAL)
f + 0 is V30() real ext-real Element of REAL
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
N is V30() real ext-real Element of REAL
f + N is V30() real ext-real Element of REAL
[.f,(f + N).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((x0 /* (N + g1)) - (x0 /* g1))) is V30() real ext-real Element of REAL
r is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
r " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng q1 is V71() V72() V73() Element of K19(REAL)
r + q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (r + q1) is V71() V72() V73() Element of K19(REAL)
x0 /* (r + q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (r + q1)) - (x0 /* q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (r + q1)) + (- (x0 /* q1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(r ") (#) ((x0 /* (r + q1)) - (x0 /* q1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((r ") (#) ((x0 /* (r + q1)) - (x0 /* q1))) is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
f + N is V30() real ext-real Element of REAL
[.f,(f + N).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
q1 is V30() real ext-real Element of REAL
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (N + g1)) - (x0 /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (x0 /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (N + g1)) + (- (x0 /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((x0 /* (N + g1)) - (x0 /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((x0 /* (N + g1)) - (x0 /* g1))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f + x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f + x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
(N,f) + (N,x0) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
min (r,g1) is V30() real ext-real Element of REAL
N - (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - g1 <= b₁ & b₁ <= N ) } is set
[.(N - (min (r,g1))),N.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - r <= b₁ & b₁ <= N ) } is set
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f + x0) is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f + x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f + x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f + x0) /* (q2 + h1)) - ((f + x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f + x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f + x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f + x0) /* (q2 + h1)) + (- ((f + x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f + x0) /* (q2 + h1)) - ((f + x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f + x0) /* (q2 + h1)) - ((f + x0) /* h1))) is V30() real ext-real Element of REAL
h2 is set
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
- (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) + (- (f /* h1))) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (- (f /* h1))) . h1) + (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(- (f /* h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1) is V30() real ext-real Element of REAL
- (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((x0 /* (q2 + h1)) + (- (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1)) + (((x0 /* (q2 + h1)) + (- (x0 /* h1))) . h1) is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(- (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1)) + (((x0 /* (q2 + h1)) . h1) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((- (f /* h1)) . h1) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + (((- (f /* h1)) . h1) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
- ((f /* h1) . h1) is V30() real ext-real Element of REAL
(- ((f /* h1) . h1)) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + ((- ((f /* h1) . h1)) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
- ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(- ((f /* h1) . h1)) + (- ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + ((- ((f /* h1) . h1)) + (- ((x0 /* h1) . h1))) is V30() real ext-real Element of REAL
((f /* h1) . h1) + ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) + ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) + (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1) - (((f /* h1) . h1) + ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) + (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h1) + (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1) - (((f /* h1) + (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) - ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) + (- ((f /* h1) + (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) - ((f /* h1) + (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f + x0) /* (q2 + h1)) - ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f + x0) /* (q2 + h1)) + (- ((f /* h1) + (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f + x0) /* (q2 + h1)) - ((f /* h1) + (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f + x0) /* (q2 + h1)) - ((f + x0) /* h1)) . h1 is V30() real ext-real Element of REAL
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) + ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q2 ") (#) (((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f - x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- x0 is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) x0 is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
f + (- x0) is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
N is V30() real ext-real Element of REAL
(N,(f - x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
(N,f) - (N,x0) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
min (r,g1) is V30() real ext-real Element of REAL
N - (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - g1 <= b₁ & b₁ <= N ) } is set
[.(N - (min (r,g1))),N.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - r <= b₁ & b₁ <= N ) } is set
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f - x0) is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f - x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f - x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f - x0) /* (q2 + h1)) - ((f - x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f - x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f - x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f - x0) /* (q2 + h1)) + (- ((f - x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f - x0) /* (q2 + h1)) - ((f - x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f - x0) /* (q2 + h1)) - ((f - x0) /* h1))) is V30() real ext-real Element of REAL
h2 is set
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) + (- ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (f /* h1)) . h1) - (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) - (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) - (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((f /* h1) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) - (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* (q2 + h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* (q2 + h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (x0 /* (q2 + h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1) - (((f /* h1) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h1) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* h1) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1) - (((f /* h1) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) - ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) + (- ((f /* h1) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) - ((f /* h1) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f - x0) /* (q2 + h1)) - ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f - x0) /* (q2 + h1)) + (- ((f /* h1) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f - x0) /* (q2 + h1)) - ((f /* h1) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f - x0) /* (q2 + h1)) - ((f - x0) /* h1)) . h1 is V30() real ext-real Element of REAL
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) - ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) + (- ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f (#) x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f (#) x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) + ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
min (r,g1) is V30() real ext-real Element of REAL
N - (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - g1 <= b₁ & b₁ <= N ) } is set
[.(N - (min (r,g1))),N.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - r <= b₁ & b₁ <= N ) } is set
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f (#) x0) is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f (#) x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f (#) x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f (#) x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f (#) x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f (#) x0) /* (q2 + h1)) + (- ((f (#) x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1))) is V30() real ext-real Element of REAL
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* h1) + ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((f /* h1) + ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((f /* h1) . h1) + (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
((f /* h1) . h1) + (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h1 is V30() real ext-real Element of REAL
h1 is set
h1 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* h1) . h2 is V30() real ext-real Element of REAL
((f /* h1) . h2) - (f . N) is V30() real ext-real Element of REAL
abs (((f /* h1) . h2) - (f . N)) is V30() real ext-real Element of REAL
h1 . h2 is V30() real ext-real Element of REAL
f . (h1 . h2) is V30() real ext-real Element of REAL
(f . (h1 . h2)) - (f . N) is V30() real ext-real Element of REAL
abs ((f . (h1 . h2)) - (f . N)) is V30() real ext-real Element of REAL
(f . N) - (f . N) is V30() real ext-real Element of REAL
abs ((f . N) - (f . N)) is V30() real ext-real Element of REAL
h1 is set
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(x0 /* h1) . h2 is V30() real ext-real Element of REAL
((x0 /* h1) . h2) - (x0 . N) is V30() real ext-real Element of REAL
abs (((x0 /* h1) . h2) - (x0 . N)) is V30() real ext-real Element of REAL
h1 . h2 is V30() real ext-real Element of REAL
x0 . (h1 . h2) is V30() real ext-real Element of REAL
(x0 . (h1 . h2)) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . (h1 . h2)) - (x0 . N)) is V30() real ext-real Element of REAL
(x0 . N) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . N) - (x0 . N)) is V30() real ext-real Element of REAL
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
lim q2 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim q2) * (lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1))) . h1 is V30() real ext-real Element of REAL
(q2 ") . h1 is V30() real ext-real Element of REAL
(((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) . h1) is V30() real ext-real Element of REAL
((f (#) x0) /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f (#) x0) /* h1) . h1 is V30() real ext-real Element of REAL
(((f (#) x0) /* (q2 + h1)) . h1) - (((f (#) x0) /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f (#) x0) /* (q2 + h1)) . h1) - (((f (#) x0) /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) (#) (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f (#) x0) /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f (#) x0) /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) (#) (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h1) (#) (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f /* h1) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f /* h1) (#) (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) (#) (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
((f /* h1) . h1) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) * ((x0 /* h1) . h1))) is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1))) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1))) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1)) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1)) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1)) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1)) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
(q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) . h1 is V30() real ext-real Element of REAL
q2 (#) (q2 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q2 (#) (q2 ")) (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 (#) (q2 ")) (#) ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(q2 (#) (q2 ")) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 (#) (q2 ")) . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(q2 ") . h1 is V30() real ext-real Element of REAL
(q2 . h1) * ((q2 ") . h1) is V30() real ext-real Element of REAL
((q2 . h1) * ((q2 ") . h1)) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(q2 . h1) " is V30() real ext-real Element of REAL
(q2 . h1) * ((q2 . h1) ") is V30() real ext-real Element of REAL
((q2 . h1) * ((q2 . h1) ")) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
1 * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
lim (f /* h1) is V30() real ext-real Element of REAL
lim (f /* (q2 + h1)) is V30() real ext-real Element of REAL
(lim (f /* (q2 + h1))) - (f . N) is V30() real ext-real Element of REAL
lim ((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
lim (((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) is V30() real ext-real Element of REAL
lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) is V30() real ext-real Element of REAL
(lim (((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1)))) + (lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (lim (f /* (q2 + h1))) is V30() real ext-real Element of REAL
((lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (lim (f /* (q2 + h1)))) + (lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (f . N) is V30() real ext-real Element of REAL
lim (x0 /* h1) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) * (lim (x0 /* h1)) is V30() real ext-real Element of REAL
((lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (f . N)) + ((lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) * (lim (x0 /* h1))) is V30() real ext-real Element of REAL
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom x0 is V71() V72() V73() Element of K19(REAL)
f / x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f / x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) - ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
(x0 . N) ^2 is V30() real ext-real Element of REAL
K115((x0 . N),(x0 . N)) is V30() real ext-real set
(((N,f) * (x0 . N)) - ((N,x0) * (f . N))) / ((x0 . N) ^2) is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
q1 is V30() real ext-real Element of REAL
N - q1 is V30() real ext-real Element of REAL
[.(N - q1),N.] is V71() V72() V73() V102() Element of K19(REAL)
min (r,q1) is V30() real ext-real Element of REAL
N - (min (r,q1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - q1 <= b₁ & b₁ <= N ) } is set
[.(N - (min (r,q1))),N.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - r <= b₁ & b₁ <= N ) } is set
min (g1,(min (r,q1))) is V30() real ext-real Element of REAL
N - (min (g1,(min (r,q1)))) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - (min (r,q1)) <= b₁ & b₁ <= N ) } is set
[.(N - (min (g1,(min (r,q1))))),N.] is V71() V72() V73() V102() Element of K19(REAL)
x0 ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom (x0 ^) is V71() V72() V73() Element of K19(REAL)
h1 is set
h2 is V30() real ext-real Element of REAL
x0 " {0} is V71() V72() V73() Element of K19(REAL)
(dom x0) \ (x0 " {0}) is V71() V72() V73() Element of K19(REAL)
x0 . h2 is V30() real ext-real Element of REAL
x0 " {0} is V71() V72() V73() Element of K19(REAL)
(dom x0) \ (x0 " {0}) is V71() V72() V73() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N - g1 <= b₁ & b₁ <= N ) } is set
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f / x0) is V71() V72() V73() Element of K19(REAL)
h1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
h1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h2 is V71() V72() V73() Element of K19(REAL)
h1 + h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (h1 + h2) is V71() V72() V73() Element of K19(REAL)
(f / x0) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f / x0) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f / x0) /* (h1 + h2)) - ((f / x0) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f / x0) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f / x0) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f / x0) /* (h1 + h2)) + (- ((f / x0) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f / x0) /* (h1 + h2)) - ((f / x0) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h1 ") (#) (((f / x0) /* (h1 + h2)) - ((f / x0) /* h2))) is V30() real ext-real Element of REAL
(dom f) /\ ((dom x0) \ (x0 " {0})) is V71() V72() V73() Element of K19(REAL)
h2 is set
f /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) - (f /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + h2)) + (- (f /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) is V30() real ext-real Element of REAL
h2 is set
(dom f) /\ (dom (x0 ^)) is V71() V72() V73() Element of K19(REAL)
x0 /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) (#) (x0 /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) (#) (x0 /* h2)) + (- ((f /* h2) (#) (x0 /* (h1 + h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) . h2 is V30() real ext-real Element of REAL
(h1 ") . h2 is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) . h2) is V30() real ext-real Element of REAL
((f /* (h1 + h2)) (#) (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((f /* h2) (#) (x0 /* (h1 + h2))) . h2 is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2)) is V30() real ext-real Element of REAL
(f /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2)) is V30() real ext-real Element of REAL
(f /* h2) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h2)) . h2) - ((f /* h2) . h2) is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((f /* h2) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2) is V30() real ext-real Element of REAL
(((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2))) - (((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2)) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2))) - (((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2))) is V30() real ext-real Element of REAL
((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) is V30() real ext-real Element of REAL
(((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2))) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2))) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2))) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)))) is V30() real ext-real Element of REAL
((f /* (h1 + h2)) - (f /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2) is V30() real ext-real Element of REAL
(((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)))) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) - (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (h1 + h2)) + (- (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2)) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2))) is V30() real ext-real Element of REAL
((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2 is V30() real ext-real Element of REAL
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2))) is V30() real ext-real Element of REAL
(h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2) is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2)) is V30() real ext-real Element of REAL
((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2) - (((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2)) is V30() real ext-real Element of REAL
(f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2 is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2) is V30() real ext-real Element of REAL
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) + (- ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) + ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((x0 /* h2) + ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((x0 /* h2) . h2) + (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((x0 /* h2) . h2) + (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f (#) (x0 ^) is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
(f (#) (x0 ^)) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) - ((f / x0) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) + (- ((f / x0) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f (#) (x0 ^)) /* (h1 + h2)) - ((f / x0) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f (#) (x0 ^)) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f (#) (x0 ^)) /* (h1 + h2)) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f (#) (x0 ^)) /* (h1 + h2)) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 ^) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) /" (x0 /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (h1 + h2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + h2)) (#) ((x0 /* (h1 + h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 ^) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) (#) ((x0 ^) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f /* h2) (#) ((x0 ^) /* h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) (#) ((x0 ^) /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) /" (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* h2) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* h2) (#) ((x0 /* h2) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f /* h2) /" (x0 /* h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) /" (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((x0 /* (h1 + h2)) (#) (x0 /* h2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
lim ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h2 is V30() real ext-real Element of REAL
h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2 is V30() real ext-real Element of REAL
h1 (#) (h1 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h1 (#) (h1 ")) (#) ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 (#) (h1 ")) (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
(h1 (#) (h1 ")) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 (#) (h1 ")) . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(h1 ") . h2 is V30() real ext-real Element of REAL
(h1 . h2) * ((h1 ") . h2) is V30() real ext-real Element of REAL
((h1 . h2) * ((h1 ") . h2)) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(h1 . h2) " is V30() real ext-real Element of REAL
(h1 . h2) * ((h1 . h2) ") is V30() real ext-real Element of REAL
((h1 . h2) * ((h1 . h2) ")) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
1 * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 . h2 is V30() real ext-real Element of REAL
h2 is V30() real ext-real set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* h2) . c2 is V30() real ext-real Element of REAL
((f /* h2) . c2) - (f . N) is V30() real ext-real Element of REAL
abs (((f /* h2) . c2) - (f . N)) is V30() real ext-real Element of REAL
h2 . c2 is V30() real ext-real Element of REAL
f . (h2 . c2) is V30() real ext-real Element of REAL
(f . (h2 . c2)) - (f . N) is V30() real ext-real Element of REAL
abs ((f . (h2 . c2)) - (f . N)) is V30() real ext-real Element of REAL
(f . N) - (f . N) is V30() real ext-real Element of REAL
abs ((f . N) - (f . N)) is V30() real ext-real Element of REAL
lim (f /* h2) is V30() real ext-real Element of REAL
h2 is V30() real ext-real set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(x0 /* h2) . c2 is V30() real ext-real Element of REAL
((x0 /* h2) . c2) - (x0 . N) is V30() real ext-real Element of REAL
abs (((x0 /* h2) . c2) - (x0 . N)) is V30() real ext-real Element of REAL
h2 . c2 is V30() real ext-real Element of REAL
x0 . (h2 . c2) is V30() real ext-real Element of REAL
(x0 . (h2 . c2)) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . (h2 . c2)) - (x0 . N)) is V30() real ext-real Element of REAL
(x0 . N) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . N) - (x0 . N)) is V30() real ext-real Element of REAL
lim (x0 /* h2) is V30() real ext-real Element of REAL
lim (h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
lim h1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim h1) * (lim ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
lim (x0 /* (h1 + h2)) is V30() real ext-real Element of REAL
(lim (x0 /* (h1 + h2))) - (x0 . N) is V30() real ext-real Element of REAL
lim ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V30() real ext-real Element of REAL
lim ((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
(lim ((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) is V30() real ext-real Element of REAL
lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
(lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
((lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
(lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2)) is V30() real ext-real Element of REAL
((lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
(((lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
(dom f) /\ ((dom x0) \ (x0 " {0})) is V71() V72() V73() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f / x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,(f / x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) - ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
(x0 . N) ^2 is V30() real ext-real Element of REAL
K115((x0 . N),(x0 . N)) is V30() real ext-real set
(((N,f) * (x0 . N)) - ((N,x0) * (f . N))) / ((x0 . N) ^2) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N - g1 is V30() real ext-real Element of REAL
[.(N - g1),N.] is V71() V72() V73() V102() Element of K19(REAL)
r is V30() real ext-real Element of REAL
q1 is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
x0 . q1 is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
N - r is V30() real ext-real Element of REAL
[.(N - r),N.] is V71() V72() V73() V102() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
f ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
(x0,(f ^)) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
(f . x0) ^2 is V30() real ext-real Element of REAL
K115((f . x0),(f . x0)) is V30() real ext-real set
(x0,f) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- ((x0,f) / ((f . x0) ^2)) is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
x0 - N is V30() real ext-real Element of REAL
[.(x0 - N),x0.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
min (g1,N) is V30() real ext-real Element of REAL
x0 - (min (g1,N)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - N <= b₁ & b₁ <= x0 ) } is set
[.(x0 - (min (g1,N))),x0.] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - g1 <= b₁ & b₁ <= x0 ) } is set
dom (f ^) is V71() V72() V73() Element of K19(REAL)
q1 is set
q2 is V30() real ext-real Element of REAL
f " {0} is V71() V72() V73() Element of K19(REAL)
(dom f) \ (f " {0}) is V71() V72() V73() Element of K19(REAL)
f . q2 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
q1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng q2 is V71() V72() V73() Element of K19(REAL)
q1 + q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q1 + q2) is V71() V72() V73() Element of K19(REAL)
(f ^) /* (q1 + q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f ^) /* q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f ^) /* (q1 + q2)) - ((f ^) /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f ^) /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f ^) /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f ^) /* (q1 + q2)) + (- ((f ^) /* q2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2))) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
f " {0} is V71() V72() V73() Element of K19(REAL)
(dom f) \ (f " {0}) is V71() V72() V73() Element of K19(REAL)
f /* (q1 + q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q1 + q2)) - (f /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q1 + q2)) + (- (f /* q2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V30() real ext-real Element of REAL
- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
K116(1) (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . h1 is V30() real ext-real Element of REAL
q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) . h1 is V30() real ext-real Element of REAL
q1 (#) (q1 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q1 (#) (q1 ")) (#) ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q1 (#) (q1 ")) (#) ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(q1 (#) (q1 ")) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
((q1 (#) (q1 ")) . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(q1 ") . h1 is V30() real ext-real Element of REAL
(q1 . h1) * ((q1 ") . h1) is V30() real ext-real Element of REAL
((q1 . h1) * ((q1 ") . h1)) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(q1 . h1) " is V30() real ext-real Element of REAL
(q1 . h1) * ((q1 . h1) ") is V30() real ext-real Element of REAL
((q1 . h1) * ((q1 . h1) ")) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
1 * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) (#) (f /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* q2) + ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((f /* q2) + ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(f /* q2) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
((f /* q2) . h1) + (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) - ((f /* q2) . h1) is V30() real ext-real Element of REAL
((f /* q2) . h1) + (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1)) is V30() real ext-real Element of REAL
h1 is V30() real ext-real set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* q2) . h2 is V30() real ext-real Element of REAL
((f /* q2) . h2) - (f . x0) is V30() real ext-real Element of REAL
abs (((f /* q2) . h2) - (f . x0)) is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
f . (q2 . h2) is V30() real ext-real Element of REAL
(f . (q2 . h2)) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (q2 . h2)) - (f . x0)) is V30() real ext-real Element of REAL
(f . x0) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . x0) - (f . x0)) is V30() real ext-real Element of REAL
lim (f /* q2) is V30() real ext-real Element of REAL
lim (q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
lim q1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim q1) * (lim ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
lim (f /* (q1 + q2)) is V30() real ext-real Element of REAL
(lim (f /* (q1 + q2))) - (f . x0) is V30() real ext-real Element of REAL
lim ((f /* (q1 + q2)) (#) (f /* q2)) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
(f /* q2) . h1 is V30() real ext-real Element of REAL
((q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2))) . h1 is V30() real ext-real Element of REAL
(q1 ") . h1 is V30() real ext-real Element of REAL
(((f ^) /* (q1 + q2)) - ((f ^) /* q2)) . h1 is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f ^) /* (q1 + q2)) - ((f ^) /* q2)) . h1) is V30() real ext-real Element of REAL
((f ^) /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
((f ^) /* q2) . h1 is V30() real ext-real Element of REAL
(((f ^) /* (q1 + q2)) . h1) - (((f ^) /* q2) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f ^) /* (q1 + q2)) . h1) - (((f ^) /* q2) . h1)) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q1 + q2)) ") . h1 is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) ") . h1) - (((f ^) /* q2) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) ") . h1) - (((f ^) /* q2) . h1)) is V30() real ext-real Element of REAL
(f /* q2) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* q2) ") . h1 is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) ") . h1) - (((f /* q2) ") . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) ") . h1) - (((f /* q2) ") . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) " is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) . h1) ") - (((f /* q2) ") . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) . h1) ") - (((f /* q2) ") . h1)) is V30() real ext-real Element of REAL
((f /* q2) . h1) " is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) . h1) ") - (((f /* q2) . h1) ") is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) . h1) ") - (((f /* q2) . h1) ")) is V30() real ext-real Element of REAL
1 / ((f /* (q1 + q2)) . h1) is V30() real ext-real Element of REAL
(1 / ((f /* (q1 + q2)) . h1)) - (((f /* q2) . h1) ") is V30() real ext-real Element of REAL
((q1 ") . h1) * ((1 / ((f /* (q1 + q2)) . h1)) - (((f /* q2) . h1) ")) is V30() real ext-real Element of REAL
1 / ((f /* q2) . h1) is V30() real ext-real Element of REAL
(1 / ((f /* (q1 + q2)) . h1)) - (1 / ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((1 / ((f /* (q1 + q2)) . h1)) - (1 / ((f /* q2) . h1))) is V30() real ext-real Element of REAL
1 * ((f /* q2) . h1) is V30() real ext-real Element of REAL
1 * ((f /* (q1 + q2)) . h1) is V30() real ext-real Element of REAL
(1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) * ((f /* q2) . h1) is V30() real ext-real Element of REAL
((1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1))) / (((f /* (q1 + q2)) . h1) * ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * (((1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1))) / (((f /* (q1 + q2)) . h1) * ((f /* q2) . h1))) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) - ((f /* q2) . h1) is V30() real ext-real Element of REAL
- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) (#) (f /* q2)) . h1 is V30() real ext-real Element of REAL
(- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1))) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1))) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
- (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(- (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((- (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * (- ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1))) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
- (((q1 ") . h1) * ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1))) is V30() real ext-real Element of REAL
((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) (#) (f /* q2)) . h1) " is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) . h1) ") is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) . h1) ")) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) (#) (f /* q2)) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((f /* (q1 + q2)) (#) (f /* q2)) ") . h1 is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) ") . h1) is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) ") . h1)) is V30() real ext-real Element of REAL
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q1 + q2)) (#) (f /* q2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) (#) (((f /* (q1 + q2)) (#) (f /* q2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1 is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1) is V30() real ext-real Element of REAL
- (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
K116(1) (#) (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(- (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2)))) . h1 is V30() real ext-real Element of REAL
(- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) /" ((f /* (q1 + q2)) (#) (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) (#) (((f /* (q1 + q2)) (#) (f /* q2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1 is V30() real ext-real Element of REAL
lim (- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
(lim (- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))))) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- (x0,f) is V30() real ext-real Element of REAL
(- (x0,f)) / ((f . x0) ^2) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
(x0,(f ^)) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
(f . x0) ^2 is V30() real ext-real Element of REAL
K115((f . x0),(f . x0)) is V30() real ext-real set
(x0,f) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- ((x0,f) / ((f . x0) ^2)) is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 - N is V30() real ext-real Element of REAL
[.(x0 - N),x0.] is V71() V72() V73() V102() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 - N is V30() real ext-real Element of REAL
[.(x0 - N),x0.] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
f . r is V30() real ext-real Element of REAL
g1 is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
x0 is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
[.x0,(x0 + g1).] is V71() V72() V73() V102() Element of K19(REAL)
r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng r is V71() V72() V73() Element of K19(REAL)
g1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
g1 + r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (g1 + r) is V71() V72() V73() Element of K19(REAL)
g1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 + r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (g1 + r)) - (f /* r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (g1 + r)) + (- (f /* r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(g1 ") (#) ((f /* (g1 + r)) - (f /* r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((g1 ") (#) ((f /* (g1 + r)) - (f /* r))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f + x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f + x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
(N,f) + (N,x0) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
N + 0 is V30() real ext-real Element of REAL
min (r,g1) is V30() real ext-real Element of REAL
N + (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + g1 ) } is set
[.N,(N + (min (r,g1))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + r ) } is set
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
dom (f + x0) is V71() V72() V73() Element of K19(REAL)
{N} is V51() V71() V72() V73() V99() V100() V101() set
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f + x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f + x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f + x0) /* (q2 + h1)) - ((f + x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f + x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f + x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f + x0) /* (q2 + h1)) + (- ((f + x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f + x0) /* (q2 + h1)) - ((f + x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f + x0) /* (q2 + h1)) - ((f + x0) /* h1))) is V30() real ext-real Element of REAL
h2 is set
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
- (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) + (- (f /* h1))) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (- (f /* h1))) . h1) + (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(- (f /* h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1) is V30() real ext-real Element of REAL
- (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((x0 /* (q2 + h1)) + (- (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1)) + (((x0 /* (q2 + h1)) + (- (x0 /* h1))) . h1) is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(- (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((- (f /* h1)) . h1)) + (((x0 /* (q2 + h1)) . h1) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((- (f /* h1)) . h1) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + (((- (f /* h1)) . h1) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
- ((f /* h1) . h1) is V30() real ext-real Element of REAL
(- ((f /* h1) . h1)) + ((- (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + ((- ((f /* h1) . h1)) + ((- (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
- ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(- ((f /* h1) . h1)) + (- ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) + ((- ((f /* h1) . h1)) + (- ((x0 /* h1) . h1))) is V30() real ext-real Element of REAL
((f /* h1) . h1) + ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) + ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) + ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) + (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1) - (((f /* h1) . h1) + ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) + (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h1) + (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) . h1) - (((f /* h1) + (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) - ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) + (x0 /* (q2 + h1))) + (- ((f /* h1) + (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (q2 + h1)) + (x0 /* (q2 + h1))) - ((f /* h1) + (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f + x0) /* (q2 + h1)) - ((f /* h1) + (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f + x0) /* (q2 + h1)) + (- ((f /* h1) + (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f + x0) /* (q2 + h1)) - ((f /* h1) + (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f + x0) /* (q2 + h1)) - ((f + x0) /* h1)) . h1 is V30() real ext-real Element of REAL
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) + ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q2 ") (#) (((f /* (q2 + h1)) - (f /* h1)) + ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f - x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
- x0 is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) x0 is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
f + (- x0) is V1() V4( REAL ) V6() complex-valued ext-real-valued real-valued set
N is V30() real ext-real Element of REAL
(N,(f - x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
(N,f) - (N,x0) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
min (r,g1) is V30() real ext-real Element of REAL
N + 0 is V30() real ext-real Element of REAL
N + (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + g1 ) } is set
[.N,(N + (min (r,g1))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + r ) } is set
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f - x0) is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f - x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f - x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f - x0) /* (q2 + h1)) - ((f - x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f - x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f - x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f - x0) /* (q2 + h1)) + (- ((f - x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f - x0) /* (q2 + h1)) - ((f - x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f - x0) /* (q2 + h1)) - ((f - x0) /* h1))) is V30() real ext-real Element of REAL
h2 is set
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (f /* h1)) + (- ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (f /* h1)) . h1) - (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) - (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) - (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((f /* h1) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) - ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) - (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* (q2 + h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* (q2 + h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (x0 /* (q2 + h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1) - (((f /* h1) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h1) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* h1) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) . h1) - (((f /* h1) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) - ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (q2 + h1)) - (x0 /* (q2 + h1))) + (- ((f /* h1) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (q2 + h1)) - (x0 /* (q2 + h1))) - ((f /* h1) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
((f - x0) /* (q2 + h1)) - ((f /* h1) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f - x0) /* (q2 + h1)) + (- ((f /* h1) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f - x0) /* (q2 + h1)) - ((f /* h1) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((f - x0) /* (q2 + h1)) - ((f - x0) /* h1)) . h1 is V30() real ext-real Element of REAL
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) - ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) + (- ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f /* (q2 + h1)) - (f /* h1)) - ((x0 /* (q2 + h1)) - (x0 /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f (#) x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f (#) x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) + ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
N + 0 is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
min (r,g1) is V30() real ext-real Element of REAL
N + (min (r,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + g1 ) } is set
[.N,(N + (min (r,g1))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + r ) } is set
{N} is V51() V71() V72() V73() V99() V100() V101() set
dom (f (#) x0) is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
(f (#) x0) /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f (#) x0) /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f (#) x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f (#) x0) /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f (#) x0) /* (q2 + h1)) + (- ((f (#) x0) /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1))) is V30() real ext-real Element of REAL
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h1 is V30() real ext-real Element of REAL
x0 /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(x0 /* h1) . h2 is V30() real ext-real Element of REAL
((x0 /* h1) . h2) - (x0 . N) is V30() real ext-real Element of REAL
abs (((x0 /* h1) . h2) - (x0 . N)) is V30() real ext-real Element of REAL
h1 . h2 is V30() real ext-real Element of REAL
x0 . (h1 . h2) is V30() real ext-real Element of REAL
(x0 . (h1 . h2)) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . (h1 . h2)) - (x0 . N)) is V30() real ext-real Element of REAL
(x0 . N) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . N) - (x0 . N)) is V30() real ext-real Element of REAL
x0 /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (q2 + h1)) - (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (q2 + h1)) + (- (x0 /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) . h1 is V30() real ext-real Element of REAL
q2 (#) (q2 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q2 (#) (q2 ")) (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q2 (#) (q2 ")) (#) ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(q2 (#) (q2 ")) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 (#) (q2 ")) . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(q2 ") . h1 is V30() real ext-real Element of REAL
(q2 . h1) * ((q2 ") . h1) is V30() real ext-real Element of REAL
((q2 . h1) * ((q2 ") . h1)) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(q2 . h1) " is V30() real ext-real Element of REAL
(q2 . h1) * ((q2 . h1) ") is V30() real ext-real Element of REAL
((q2 . h1) * ((q2 . h1) ")) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
1 * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
h1 is set
h1 is V30() real ext-real set
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* h1) . h2 is V30() real ext-real Element of REAL
((f /* h1) . h2) - (f . N) is V30() real ext-real Element of REAL
abs (((f /* h1) . h2) - (f . N)) is V30() real ext-real Element of REAL
h1 . h2 is V30() real ext-real Element of REAL
f . (h1 . h2) is V30() real ext-real Element of REAL
(f . (h1 . h2)) - (f . N) is V30() real ext-real Element of REAL
abs ((f . (h1 . h2)) - (f . N)) is V30() real ext-real Element of REAL
(f . N) - (f . N) is V30() real ext-real Element of REAL
abs ((f . N) - (f . N)) is V30() real ext-real Element of REAL
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((q2 ") (#) (((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1))) . h1 is V30() real ext-real Element of REAL
(q2 ") . h1 is V30() real ext-real Element of REAL
(((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f (#) x0) /* (q2 + h1)) - ((f (#) x0) /* h1)) . h1) is V30() real ext-real Element of REAL
((f (#) x0) /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f (#) x0) /* h1) . h1 is V30() real ext-real Element of REAL
(((f (#) x0) /* (q2 + h1)) . h1) - (((f (#) x0) /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f (#) x0) /* (q2 + h1)) . h1) - (((f (#) x0) /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) (#) (x0 /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f (#) x0) /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f (#) x0) /* h1) . h1)) is V30() real ext-real Element of REAL
(f /* h1) (#) (x0 /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h1) (#) (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f /* h1) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) (#) (x0 /* (q2 + h1))) . h1) - (((f /* h1) (#) (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
(x0 /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) (#) (x0 /* h1)) . h1)) is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
(x0 /* h1) . h1 is V30() real ext-real Element of REAL
((f /* h1) . h1) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
(((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") . h1) * ((((f /* (q2 + h1)) . h1) * ((x0 /* (q2 + h1)) . h1)) - (((f /* h1) . h1) * ((x0 /* h1) . h1))) is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1))) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") . h1) * (((x0 /* (q2 + h1)) . h1) - ((x0 /* h1) . h1))) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((x0 /* (q2 + h1)) - (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1)) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((((q2 ") . h1) * (((x0 /* (q2 + h1)) - (x0 /* h1)) . h1)) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1))) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1)) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") . h1) * (((f /* (q2 + h1)) - (f /* h1)) . h1)) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1) is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) . h1) * ((f /* (q2 + h1)) . h1)) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1 is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) . h1) * ((x0 /* h1) . h1)) is V30() real ext-real Element of REAL
(((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) . h1 is V30() real ext-real Element of REAL
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) . h1) + ((((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) . h1) is V30() real ext-real Element of REAL
(((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) . h1 is V30() real ext-real Element of REAL
(f /* h1) + ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((f /* h1) + ((f /* (q2 + h1)) - (f /* h1))) . h1 is V30() real ext-real Element of REAL
(f /* h1) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) - (f /* h1)) . h1 is V30() real ext-real Element of REAL
((f /* h1) . h1) + (((f /* (q2 + h1)) - (f /* h1)) . h1) is V30() real ext-real Element of REAL
(f /* (q2 + h1)) . h1 is V30() real ext-real Element of REAL
((f /* (q2 + h1)) . h1) - ((f /* h1) . h1) is V30() real ext-real Element of REAL
((f /* h1) . h1) + (((f /* (q2 + h1)) . h1) - ((f /* h1) . h1)) is V30() real ext-real Element of REAL
lim (q2 (#) ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
lim q2 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim q2) * (lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) is V30() real ext-real Element of REAL
lim (f /* (q2 + h1)) is V30() real ext-real Element of REAL
lim (f /* h1) is V30() real ext-real Element of REAL
(lim (f /* (q2 + h1))) - (lim (f /* h1)) is V30() real ext-real Element of REAL
(lim (f /* (q2 + h1))) - (f . N) is V30() real ext-real Element of REAL
lim ((((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) + (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
lim (((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1))) is V30() real ext-real Element of REAL
lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1)) is V30() real ext-real Element of REAL
(lim (((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1))) (#) (f /* (q2 + h1)))) + (lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (lim (f /* (q2 + h1))) is V30() real ext-real Element of REAL
((lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (lim (f /* (q2 + h1)))) + (lim (((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) (#) (x0 /* h1))) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (f . N) is V30() real ext-real Element of REAL
lim (x0 /* h1) is V30() real ext-real Element of REAL
(lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) * (lim (x0 /* h1)) is V30() real ext-real Element of REAL
((lim ((q2 ") (#) ((x0 /* (q2 + h1)) - (x0 /* h1)))) * (f . N)) + ((lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)))) * (lim (x0 /* h1))) is V30() real ext-real Element of REAL
(dom f) /\ (dom x0) is V71() V72() V73() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom x0 is V71() V72() V73() Element of K19(REAL)
f / x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
(N,(f / x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) - ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
(x0 . N) ^2 is V30() real ext-real Element of REAL
K115((x0 . N),(x0 . N)) is V30() real ext-real set
(((N,f) * (x0 . N)) - ((N,x0) * (f . N))) / ((x0 . N) ^2) is V30() real ext-real Element of REAL
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
dom f is V71() V72() V73() Element of K19(REAL)
r is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
q1 is V30() real ext-real Element of REAL
N + q1 is V30() real ext-real Element of REAL
[.N,(N + q1).] is V71() V72() V73() V102() Element of K19(REAL)
0 + N is V30() real ext-real Element of REAL
min (q1,g1) is V30() real ext-real Element of REAL
N + (min (q1,g1)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + g1 ) } is set
[.N,(N + (min (q1,g1))).] is V71() V72() V73() V102() Element of K19(REAL)
N + 0 is V30() real ext-real Element of REAL
min (r,(min (q1,g1))) is V30() real ext-real Element of REAL
N + (min (r,(min (q1,g1)))) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + q1 ) } is set
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + (min (q1,g1)) ) } is set
[.N,(N + (min (r,(min (q1,g1))))).] is V71() V72() V73() V102() Element of K19(REAL)
x0 ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom (x0 ^) is V71() V72() V73() Element of K19(REAL)
h1 is set
h2 is V30() real ext-real Element of REAL
x0 " {0} is V71() V72() V73() Element of K19(REAL)
(dom x0) \ (x0 " {0}) is V71() V72() V73() Element of K19(REAL)
x0 . h2 is V30() real ext-real Element of REAL
x0 " {0} is V71() V72() V73() Element of K19(REAL)
(dom x0) \ (x0 " {0}) is V71() V72() V73() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( N <= b₁ & b₁ <= N + r ) } is set
(dom f) /\ ((dom x0) \ (x0 " {0})) is V71() V72() V73() Element of K19(REAL)
dom (f / x0) is V71() V72() V73() Element of K19(REAL)
{N} is V51() V71() V72() V73() V99() V100() V101() set
h1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
h1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h2 is V71() V72() V73() Element of K19(REAL)
h1 + h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (h1 + h2) is V71() V72() V73() Element of K19(REAL)
(f / x0) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f / x0) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f / x0) /* (h1 + h2)) - ((f / x0) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f / x0) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f / x0) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f / x0) /* (h1 + h2)) + (- ((f / x0) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f / x0) /* (h1 + h2)) - ((f / x0) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h1 ") (#) (((f / x0) /* (h1 + h2)) - ((f / x0) /* h2))) is V30() real ext-real Element of REAL
h2 is set
f /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) - (f /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + h2)) + (- (f /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) is V30() real ext-real Element of REAL
h2 is set
(dom f) /\ (dom (x0 ^)) is V71() V72() V73() Element of K19(REAL)
x0 /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
x0 /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) (#) (x0 /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) (x0 /* (h1 + h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) (#) (x0 /* h2)) + (- ((f /* h2) (#) (x0 /* (h1 + h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) . h2 is V30() real ext-real Element of REAL
(h1 ") . h2 is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) . h2) is V30() real ext-real Element of REAL
((f /* (h1 + h2)) (#) (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((f /* h2) (#) (x0 /* (h1 + h2))) . h2 is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2)) is V30() real ext-real Element of REAL
(f /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((f /* (h1 + h2)) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) (#) (x0 /* (h1 + h2))) . h2)) is V30() real ext-real Element of REAL
(f /* h2) . h2 is V30() real ext-real Element of REAL
((f /* (h1 + h2)) . h2) - ((f /* h2) . h2) is V30() real ext-real Element of REAL
(((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((f /* h2) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2) is V30() real ext-real Element of REAL
(((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2))) - (((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2)) is V30() real ext-real Element of REAL
((h1 ") . h2) * ((((((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) * ((x0 /* h2) . h2)) + (((f /* h2) . h2) * ((x0 /* h2) . h2))) - (((f /* h2) . h2) * ((x0 /* (h1 + h2)) . h2))) is V30() real ext-real Element of REAL
((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2)) is V30() real ext-real Element of REAL
(((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2))) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2))) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) . h2) - ((f /* h2) . h2))) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)))) is V30() real ext-real Element of REAL
((f /* (h1 + h2)) - (f /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2) is V30() real ext-real Element of REAL
(((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)))) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) - (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (x0 /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (x0 /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(x0 /* (h1 + h2)) + (- (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2)) is V30() real ext-real Element of REAL
((((h1 ") . h2) * (((f /* (h1 + h2)) - (f /* h2)) . h2)) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2))) is V30() real ext-real Element of REAL
((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2 is V30() real ext-real Element of REAL
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2))) is V30() real ext-real Element of REAL
(h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2) is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) . h2) * ((x0 /* h2) . h2)) - (((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2)) is V30() real ext-real Element of REAL
((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2) - (((f /* h2) . h2) * (((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2)) is V30() real ext-real Element of REAL
(f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2 is V30() real ext-real Element of REAL
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) . h2) - (((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2) is V30() real ext-real Element of REAL
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) + (- ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) + ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((x0 /* h2) + ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
(x0 /* h2) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((x0 /* h2) . h2) + (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2) is V30() real ext-real Element of REAL
((x0 /* h2) . h2) + (((x0 /* (h1 + h2)) . h2) - ((x0 /* h2) . h2)) is V30() real ext-real Element of REAL
(x0 /* (h1 + h2)) (#) (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f (#) (x0 ^) is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
(f (#) (x0 ^)) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) - ((f / x0) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) + (- ((f / x0) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f (#) (x0 ^)) /* (h1 + h2)) - ((f / x0) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f (#) (x0 ^)) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f (#) (x0 ^)) /* (h1 + h2)) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f (#) (x0 ^)) /* (h1 + h2)) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f (#) (x0 ^)) /* (h1 + h2)) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 ^) /* (h1 + h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) (#) ((x0 ^) /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + h2)) /" (x0 /* (h1 + h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* (h1 + h2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + h2)) (#) ((x0 /* (h1 + h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f (#) (x0 ^)) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 ^) /* h2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) (#) ((x0 ^) /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) (#) ((x0 ^) /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f /* h2) (#) ((x0 ^) /* h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) (#) ((x0 ^) /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* h2) /" (x0 /* h2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(x0 /* h2) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* h2) (#) ((x0 /* h2) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* h2) /" (x0 /* h2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) + (- ((f /* h2) /" (x0 /* h2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (h1 + h2)) /" (x0 /* (h1 + h2))) - ((f /* h2) /" (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((x0 /* (h1 + h2)) (#) (x0 /* h2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2)))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 ") (#) (((f /* (h1 + h2)) (#) (x0 /* h2)) - ((f /* h2) (#) (x0 /* (h1 + h2))))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) /" ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) (#) (((x0 /* (h1 + h2)) (#) (x0 /* h2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
lim ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 . h2 is V30() real ext-real Element of REAL
h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) . h2 is V30() real ext-real Element of REAL
h1 (#) (h1 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(h1 (#) (h1 ")) (#) ((x0 /* (h1 + h2)) - (x0 /* h2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((h1 (#) (h1 ")) (#) ((x0 /* (h1 + h2)) - (x0 /* h2))) . h2 is V30() real ext-real Element of REAL
(h1 (#) (h1 ")) . h2 is V30() real ext-real Element of REAL
((x0 /* (h1 + h2)) - (x0 /* h2)) . h2 is V30() real ext-real Element of REAL
((h1 (#) (h1 ")) . h2) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(h1 ") . h2 is V30() real ext-real Element of REAL
(h1 . h2) * ((h1 ") . h2) is V30() real ext-real Element of REAL
((h1 . h2) * ((h1 ") . h2)) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
(h1 . h2) " is V30() real ext-real Element of REAL
(h1 . h2) * ((h1 . h2) ") is V30() real ext-real Element of REAL
((h1 . h2) * ((h1 . h2) ")) * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
1 * (((x0 /* (h1 + h2)) - (x0 /* h2)) . h2) is V30() real ext-real Element of REAL
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 . h2 is V30() real ext-real Element of REAL
h2 is V30() real ext-real set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* h2) . c2 is V30() real ext-real Element of REAL
((f /* h2) . c2) - (f . N) is V30() real ext-real Element of REAL
abs (((f /* h2) . c2) - (f . N)) is V30() real ext-real Element of REAL
h2 . c2 is V30() real ext-real Element of REAL
f . (h2 . c2) is V30() real ext-real Element of REAL
(f . (h2 . c2)) - (f . N) is V30() real ext-real Element of REAL
abs ((f . (h2 . c2)) - (f . N)) is V30() real ext-real Element of REAL
(f . N) - (f . N) is V30() real ext-real Element of REAL
abs ((f . N) - (f . N)) is V30() real ext-real Element of REAL
lim (f /* h2) is V30() real ext-real Element of REAL
h2 is V30() real ext-real set
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
c2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(x0 /* h2) . c2 is V30() real ext-real Element of REAL
((x0 /* h2) . c2) - (x0 . N) is V30() real ext-real Element of REAL
abs (((x0 /* h2) . c2) - (x0 . N)) is V30() real ext-real Element of REAL
h2 . c2 is V30() real ext-real Element of REAL
x0 . (h2 . c2) is V30() real ext-real Element of REAL
(x0 . (h2 . c2)) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . (h2 . c2)) - (x0 . N)) is V30() real ext-real Element of REAL
(x0 . N) - (x0 . N) is V30() real ext-real Element of REAL
abs ((x0 . N) - (x0 . N)) is V30() real ext-real Element of REAL
lim (x0 /* h2) is V30() real ext-real Element of REAL
lim (h1 (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
lim h1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim h1) * (lim ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
lim (x0 /* (h1 + h2)) is V30() real ext-real Element of REAL
(lim (x0 /* (h1 + h2))) - (x0 . N) is V30() real ext-real Element of REAL
lim ((x0 /* (h1 + h2)) (#) (x0 /* h2)) is V30() real ext-real Element of REAL
lim ((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
(lim ((((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) - ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2)) is V30() real ext-real Element of REAL
lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))) is V30() real ext-real Element of REAL
(lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
((lim (((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2))) (#) (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
(lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2)) is V30() real ext-real Element of REAL
((lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2))))) is V30() real ext-real Element of REAL
(((lim ((h1 ") (#) ((f /* (h1 + h2)) - (f /* h2)))) * (lim (x0 /* h2))) - (lim ((f /* h2) (#) ((h1 ") (#) ((x0 /* (h1 + h2)) - (x0 /* h2)))))) / (lim ((x0 /* (h1 + h2)) (#) (x0 /* h2))) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f / x0 is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
N is V30() real ext-real Element of REAL
x0 . N is V30() real ext-real Element of REAL
(N,(f / x0)) is V30() real ext-real Element of REAL
(N,f) is V30() real ext-real Element of REAL
(N,f) * (x0 . N) is V30() real ext-real Element of REAL
(N,x0) is V30() real ext-real Element of REAL
f . N is V30() real ext-real Element of REAL
(N,x0) * (f . N) is V30() real ext-real Element of REAL
((N,f) * (x0 . N)) - ((N,x0) * (f . N)) is V30() real ext-real Element of REAL
(x0 . N) ^2 is V30() real ext-real Element of REAL
K115((x0 . N),(x0 . N)) is V30() real ext-real set
(((N,f) * (x0 . N)) - ((N,x0) * (f . N))) / ((x0 . N) ^2) is V30() real ext-real Element of REAL
dom x0 is V71() V72() V73() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
N + g1 is V30() real ext-real Element of REAL
[.N,(N + g1).] is V71() V72() V73() V102() Element of K19(REAL)
r is V30() real ext-real Element of REAL
q1 is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
x0 . q1 is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
N + r is V30() real ext-real Element of REAL
[.N,(N + r).] is V71() V72() V73() V102() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
dom f is V71() V72() V73() Element of K19(REAL)
f ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
(x0,(f ^)) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
(f . x0) ^2 is V30() real ext-real Element of REAL
K115((f . x0),(f . x0)) is V30() real ext-real set
(x0,f) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- ((x0,f) / ((f . x0) ^2)) is V30() real ext-real Element of REAL
0 + x0 is V30() real ext-real Element of REAL
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
[.x0,(x0 + g1).] is V71() V72() V73() V102() Element of K19(REAL)
min (g1,N) is V30() real ext-real Element of REAL
x0 + (min (g1,N)) is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + N ) } is set
[.x0,(x0 + (min (g1,N))).] is V71() V72() V73() V102() Element of K19(REAL)
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + g1 ) } is set
dom (f ^) is V71() V72() V73() Element of K19(REAL)
q1 is set
q2 is V30() real ext-real Element of REAL
f " {0} is V71() V72() V73() Element of K19(REAL)
(dom f) \ (f " {0}) is V71() V72() V73() Element of K19(REAL)
f . q2 is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
q1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng q2 is V71() V72() V73() Element of K19(REAL)
q1 + q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q1 + q2) is V71() V72() V73() Element of K19(REAL)
(f ^) /* (q1 + q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f ^) /* q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f ^) /* (q1 + q2)) - ((f ^) /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f ^) /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) ((f ^) /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f ^) /* (q1 + q2)) + (- ((f ^) /* q2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2))) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . h1 is V30() real ext-real Element of REAL
f " {0} is V71() V72() V73() Element of K19(REAL)
(dom f) \ (f " {0}) is V71() V72() V73() Element of K19(REAL)
f /* (q1 + q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q1 + q2)) - (f /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q1 + q2)) + (- (f /* q2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V30() real ext-real Element of REAL
- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
K116(1) (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
h1 is set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . h1 is V30() real ext-real Element of REAL
q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) . h1 is V30() real ext-real Element of REAL
q1 (#) (q1 ") is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(q1 (#) (q1 ")) (#) ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((q1 (#) (q1 ")) (#) ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(q1 (#) (q1 ")) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
((q1 (#) (q1 ")) . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(q1 ") . h1 is V30() real ext-real Element of REAL
(q1 . h1) * ((q1 ") . h1) is V30() real ext-real Element of REAL
((q1 . h1) * ((q1 ") . h1)) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(q1 . h1) " is V30() real ext-real Element of REAL
(q1 . h1) * ((q1 . h1) ") is V30() real ext-real Element of REAL
((q1 . h1) * ((q1 . h1) ")) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
1 * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) (#) (f /* q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* q2) + ((f /* (q1 + q2)) - (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((f /* q2) + ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(f /* q2) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
((f /* q2) . h1) + (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) - ((f /* q2) . h1) is V30() real ext-real Element of REAL
((f /* q2) . h1) + (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1)) is V30() real ext-real Element of REAL
h1 is V30() real ext-real set
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* q2) . h2 is V30() real ext-real Element of REAL
((f /* q2) . h2) - (f . x0) is V30() real ext-real Element of REAL
abs (((f /* q2) . h2) - (f . x0)) is V30() real ext-real Element of REAL
q2 . h2 is V30() real ext-real Element of REAL
f . (q2 . h2) is V30() real ext-real Element of REAL
(f . (q2 . h2)) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . (q2 . h2)) - (f . x0)) is V30() real ext-real Element of REAL
(f . x0) - (f . x0) is V30() real ext-real Element of REAL
abs ((f . x0) - (f . x0)) is V30() real ext-real Element of REAL
lim (f /* q2) is V30() real ext-real Element of REAL
lim (q1 (#) ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
lim q1 is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
(lim q1) * (lim ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
lim (f /* (q1 + q2)) is V30() real ext-real Element of REAL
(lim (f /* (q1 + q2))) - (f . x0) is V30() real ext-real Element of REAL
lim ((f /* (q1 + q2)) (#) (f /* q2)) is V30() real ext-real Element of REAL
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(f /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
(f /* q2) . h1 is V30() real ext-real Element of REAL
((q1 ") (#) (((f ^) /* (q1 + q2)) - ((f ^) /* q2))) . h1 is V30() real ext-real Element of REAL
(q1 ") . h1 is V30() real ext-real Element of REAL
(((f ^) /* (q1 + q2)) - ((f ^) /* q2)) . h1 is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f ^) /* (q1 + q2)) - ((f ^) /* q2)) . h1) is V30() real ext-real Element of REAL
((f ^) /* (q1 + q2)) . h1 is V30() real ext-real Element of REAL
((f ^) /* q2) . h1 is V30() real ext-real Element of REAL
(((f ^) /* (q1 + q2)) . h1) - (((f ^) /* q2) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f ^) /* (q1 + q2)) . h1) - (((f ^) /* q2) . h1)) is V30() real ext-real Element of REAL
(f /* (q1 + q2)) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q1 + q2)) ") . h1 is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) ") . h1) - (((f ^) /* q2) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) ") . h1) - (((f ^) /* q2) . h1)) is V30() real ext-real Element of REAL
(f /* q2) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* q2) ") . h1 is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) ") . h1) - (((f /* q2) ") . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) ") . h1) - (((f /* q2) ") . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) " is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) . h1) ") - (((f /* q2) ") . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) . h1) ") - (((f /* q2) ") . h1)) is V30() real ext-real Element of REAL
((f /* q2) . h1) " is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) . h1) ") - (((f /* q2) . h1) ") is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) . h1) ") - (((f /* q2) . h1) ")) is V30() real ext-real Element of REAL
1 / ((f /* (q1 + q2)) . h1) is V30() real ext-real Element of REAL
(1 / ((f /* (q1 + q2)) . h1)) - (((f /* q2) . h1) ") is V30() real ext-real Element of REAL
((q1 ") . h1) * ((1 / ((f /* (q1 + q2)) . h1)) - (((f /* q2) . h1) ")) is V30() real ext-real Element of REAL
1 / ((f /* q2) . h1) is V30() real ext-real Element of REAL
(1 / ((f /* (q1 + q2)) . h1)) - (1 / ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((1 / ((f /* (q1 + q2)) . h1)) - (1 / ((f /* q2) . h1))) is V30() real ext-real Element of REAL
1 * ((f /* q2) . h1) is V30() real ext-real Element of REAL
1 * ((f /* (q1 + q2)) . h1) is V30() real ext-real Element of REAL
(1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) * ((f /* q2) . h1) is V30() real ext-real Element of REAL
((1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1))) / (((f /* (q1 + q2)) . h1) * ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * (((1 * ((f /* q2) . h1)) - (1 * ((f /* (q1 + q2)) . h1))) / (((f /* (q1 + q2)) . h1) * ((f /* q2) . h1))) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) . h1) - ((f /* q2) . h1) is V30() real ext-real Element of REAL
- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) (#) (f /* q2)) . h1 is V30() real ext-real Element of REAL
(- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1))) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((- (((f /* (q1 + q2)) . h1) - ((f /* q2) . h1))) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) - (f /* q2)) . h1 is V30() real ext-real Element of REAL
- (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(- (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((- (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((q1 ") . h1) * (- ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1))) is V30() real ext-real Element of REAL
((q1 ") . h1) * ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
- (((q1 ") . h1) * ((((f /* (q1 + q2)) - (f /* q2)) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1))) is V30() real ext-real Element of REAL
((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1) is V30() real ext-real Element of REAL
(((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((q1 ") . h1) * (((f /* (q1 + q2)) - (f /* q2)) . h1)) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1 is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1) is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) / (((f /* (q1 + q2)) (#) (f /* q2)) . h1)) is V30() real ext-real Element of REAL
(((f /* (q1 + q2)) (#) (f /* q2)) . h1) " is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) . h1) ") is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) . h1) ")) is V30() real ext-real Element of REAL
((f /* (q1 + q2)) (#) (f /* q2)) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(((f /* (q1 + q2)) (#) (f /* q2)) ") . h1 is V30() real ext-real Element of REAL
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) ") . h1) is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) . h1) * ((((f /* (q1 + q2)) (#) (f /* q2)) ") . h1)) is V30() real ext-real Element of REAL
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (q1 + q2)) (#) (f /* q2)) " is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) (#) (((f /* (q1 + q2)) (#) (f /* q2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1 is V30() real ext-real Element of REAL
- ((((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1) is V30() real ext-real Element of REAL
- (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
K116(1) (#) (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(- (((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))) /" ((f /* (q1 + q2)) (#) (f /* q2)))) . h1 is V30() real ext-real Element of REAL
(- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) /" ((f /* (q1 + q2)) (#) (f /* q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) (#) (((f /* (q1 + q2)) (#) (f /* q2)) ") is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) /" ((f /* (q1 + q2)) (#) (f /* q2))) . h1 is V30() real ext-real Element of REAL
lim (- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2)))) is V30() real ext-real Element of REAL
(lim (- ((q1 ") (#) ((f /* (q1 + q2)) - (f /* q2))))) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- (x0,f) is V30() real ext-real Element of REAL
(- (x0,f)) / ((f . x0) ^2) is V30() real ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
f ^ is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
f . x0 is V30() real ext-real Element of REAL
(x0,(f ^)) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
(f . x0) ^2 is V30() real ext-real Element of REAL
K115((f . x0),(f . x0)) is V30() real ext-real set
(x0,f) / ((f . x0) ^2) is V30() real ext-real Element of REAL
- ((x0,f) / ((f . x0) ^2)) is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
[.x0,(x0 + g1).] is V71() V72() V73() V102() Element of K19(REAL)
f . r is V30() real ext-real Element of REAL
g1 is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
[.x0,(x0 + g1).] is V71() V72() V73() V102() Element of K19(REAL)
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
diff (f,x0) is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
N is V30() real ext-real Element of REAL
x0 + N is V30() real ext-real Element of REAL
[.x0,(x0 + N).] is V71() V72() V73() V102() Element of K19(REAL)
g1 is V30() real ext-real Element of REAL
x0 - g1 is V30() real ext-real Element of REAL
[.(x0 - g1),x0.] is V71() V72() V73() V102() Element of K19(REAL)
min (g1,N) is V30() real ext-real Element of REAL
x0 - (min (g1,N)) is V30() real ext-real Element of REAL
x0 + (min (g1,N)) is V30() real ext-real Element of REAL
].(x0 - (min (g1,N))),(x0 + (min (g1,N))).[ is V71() V72() V73() V97() V98() V102() Element of K19(REAL)
q1 is V71() V72() V73() Neighbourhood of x0
q2 is set
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( not b₁ <= x0 - (min (g1,N)) & not x0 + (min (g1,N)) <= b₁ ) } is set
h1 is V30() real ext-real Element of REAL
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 - g1 <= b₁ & b₁ <= x0 ) } is set
{ b₁ where b₁ is V30() real ext-real Element of REAL : ( x0 <= b₁ & b₁ <= x0 + N ) } is set
{x0} is V51() V71() V72() V73() V99() V100() V101() set
N is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
N " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
g1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng g1 is V71() V72() V73() Element of K19(REAL)
N + g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (N + g1) is V71() V72() V73() Element of K19(REAL)
f /* (N + g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* g1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + g1)) - (f /* g1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* g1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (N + g1)) + (- (f /* g1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(N ") (#) ((f /* (N + g1)) - (f /* g1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((N ") (#) ((f /* (N + g1)) - (f /* g1))) is V30() real ext-real Element of REAL
r is V71() V72() V73() Neighbourhood of x0
r is V71() V72() V73() Neighbourhood of x0
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N ^\ r is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent subsequence of N
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(N ^\ r) . q2 is V30() real ext-real Element of REAL
q2 + r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . (q2 + r) is V30() real ext-real Element of REAL
r + 0 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 ^\ r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent subsequence of g1
rng (g1 ^\ r) is V71() V72() V73() Element of K19(REAL)
h1 is set
g1 . r is V30() real ext-real Element of REAL
(g1 ^\ r) . 0 is V30() real ext-real Element of REAL
0 + r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (0 + r) is V30() real ext-real Element of REAL
(N ^\ r) + (g1 ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N + g1) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N + g1
(N ^\ r) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* ((N ^\ r) + (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* (g1 ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* (g1 ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* ((N ^\ r) + (g1 ^\ r))) + (- (f /* (g1 ^\ r))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) ((f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + g1)) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* (N + g1)
((f /* (N + g1)) ^\ r) - (f /* (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) ^\ r) + (- (f /* (g1 ^\ r))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) (((f /* (N + g1)) ^\ r) - (f /* (g1 ^\ r))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* g1) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* g1
((f /* (N + g1)) ^\ r) - ((f /* g1) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* g1) ^\ r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* g1) ^\ r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (N + g1)) ^\ r) + (- ((f /* g1) ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) (((f /* (N + g1)) ^\ r) - ((f /* g1) ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) - (f /* g1)) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (f /* (N + g1)) - (f /* g1)
((N ^\ r) ") (#) (((f /* (N + g1)) - (f /* g1)) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N ") ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N "
((N ") ^\ r) (#) (((f /* (N + g1)) - (f /* g1)) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((N ") (#) ((f /* (N + g1)) - (f /* g1))) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (N ") (#) ((f /* (N + g1)) - (f /* g1))
rng ((N ^\ r) + (g1 ^\ r)) is V71() V72() V73() Element of K19(REAL)
lim (((N ^\ r) ") (#) ((f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r)))) is V30() real ext-real Element of REAL
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N ^\ r is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent subsequence of N
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
(N ^\ r) . q2 is V30() real ext-real Element of REAL
q2 + r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . (q2 + r) is V30() real ext-real Element of REAL
r + 0 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 ^\ r is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent subsequence of g1
rng (g1 ^\ r) is V71() V72() V73() Element of K19(REAL)
h1 is set
g1 . r is V30() real ext-real Element of REAL
(g1 ^\ r) . 0 is V30() real ext-real Element of REAL
0 + r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
g1 . (0 + r) is V30() real ext-real Element of REAL
(N ^\ r) + (g1 ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N + g1) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N + g1
(N ^\ r) " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* ((N ^\ r) + (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (g1 ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* (g1 ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* (g1 ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* ((N ^\ r) + (g1 ^\ r))) + (- (f /* (g1 ^\ r))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) ((f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + g1)) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* (N + g1)
((f /* (N + g1)) ^\ r) - (f /* (g1 ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) ^\ r) + (- (f /* (g1 ^\ r))) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) (((f /* (N + g1)) ^\ r) - (f /* (g1 ^\ r))) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* g1) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* g1
((f /* (N + g1)) ^\ r) - ((f /* g1) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* g1) ^\ r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* g1) ^\ r) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (N + g1)) ^\ r) + (- ((f /* g1) ^\ r)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ^\ r) ") (#) (((f /* (N + g1)) ^\ r) - ((f /* g1) ^\ r)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) - (f /* g1)) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (f /* (N + g1)) - (f /* g1)
((N ^\ r) ") (#) (((f /* (N + g1)) - (f /* g1)) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N ") ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N "
((N ") ^\ r) (#) (((f /* (N + g1)) - (f /* g1)) ^\ r) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((N ") (#) ((f /* (N + g1)) - (f /* g1))) ^\ r is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (N ") (#) ((f /* (N + g1)) - (f /* g1))
rng ((N ^\ r) + (g1 ^\ r)) is V71() V72() V73() Element of K19(REAL)
lim (((N ^\ r) ") (#) ((f /* ((N ^\ r) + (g1 ^\ r))) - (f /* (g1 ^\ r)))) is V30() real ext-real Element of REAL
q1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . q2 is V30() real ext-real Element of REAL
q1 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
N * q1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N
q2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
h1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N . h1 is V30() real ext-real Element of REAL
q2 is V1() V4( NAT ) V5( NAT ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K19(K20(NAT,NAT))
N * q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N
h1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N
lim N is V11() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V51() V55() V71() V72() V73() V74() V75() V76() V77() V99() V100() V101() V102() Element of REAL
lim h1 is V30() real ext-real Element of REAL
lim (N * q2) is V30() real ext-real Element of REAL
g1 * q2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent subsequence of g1
rng (g1 * q2) is V71() V72() V73() Element of K19(REAL)
(N + g1) * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N + g1
rng ((N + g1) * q2) is V71() V72() V73() Element of K19(REAL)
h2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
c2 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
h2 + c2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (h2 + c2) is V71() V72() V73() Element of K19(REAL)
h2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (h2 + c2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* c2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h2 + c2)) - (f /* c2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* c2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* c2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h2 + c2)) + (- (f /* c2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h2 ") (#) ((f /* (h2 + c2)) - (f /* c2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h2 ") (#) ((f /* (h2 + c2)) - (f /* c2))) is V30() real ext-real Element of REAL
f /* ((N + g1) * q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N + g1) * q2)) - (f /* c2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N + g1) * q2)) + (- (f /* c2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h2 ") (#) ((f /* ((N + g1) * q2)) - (f /* c2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + g1)) * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* (N + g1)
((f /* (N + g1)) * q2) - (f /* c2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) * q2) + (- (f /* c2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h2 ") (#) (((f /* (N + g1)) * q2) - (f /* c2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N ") * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N "
((N ") * q2) (#) (((f /* (N + g1)) * q2) - (f /* c2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* g1) * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* g1
((f /* (N + g1)) * q2) - ((f /* g1) * q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* g1) * q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* g1) * q2) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (N + g1)) * q2) + (- ((f /* g1) * q2)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ") * q2) (#) (((f /* (N + g1)) * q2) - ((f /* g1) * q2)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) - (f /* g1)) * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (f /* (N + g1)) - (f /* g1)
((N ") * q2) (#) (((f /* (N + g1)) - (f /* g1)) * q2) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q2 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (N ") (#) ((f /* (N + g1)) - (f /* g1))
g1 * q1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent subsequence of g1
rng (g1 * q1) is V71() V72() V73() Element of K19(REAL)
(N + g1) * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N + g1
rng ((N + g1) * q1) is V71() V72() V73() Element of K19(REAL)
h1 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
c1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
h1 + c1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (h1 + c1) is V71() V72() V73() Element of K19(REAL)
h1 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (h1 + c1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* c1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (h1 + c1)) - (f /* c1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* c1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* c1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (h1 + c1)) + (- (f /* c1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((f /* (h1 + c1)) - (f /* c1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((h1 ") (#) ((f /* (h1 + c1)) - (f /* c1))) is V30() real ext-real Element of REAL
f /* ((N + g1) * q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N + g1) * q1)) - (f /* c1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* ((N + g1) * q1)) + (- (f /* c1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) ((f /* ((N + g1) * q1)) - (f /* c1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (N + g1)) * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* (N + g1)
((f /* (N + g1)) * q1) - (f /* c1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) * q1) + (- (f /* c1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(h1 ") (#) (((f /* (N + g1)) * q1) - (f /* c1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(N ") * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of N "
((N ") * q1) (#) (((f /* (N + g1)) * q1) - (f /* c1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* g1) * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of f /* g1
((f /* (N + g1)) * q1) - ((f /* g1) * q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- ((f /* g1) * q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) ((f /* g1) * q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((f /* (N + g1)) * q1) + (- ((f /* g1) * q1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
((N ") * q1) (#) (((f /* (N + g1)) * q1) - ((f /* g1) * q1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((f /* (N + g1)) - (f /* g1)) * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (f /* (N + g1)) - (f /* g1)
((N ") * q1) (#) (((f /* (N + g1)) - (f /* g1)) * q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued subsequence of (N ") (#) ((f /* (N + g1)) - (f /* g1))
g1 is V30() real ext-real set
n1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
n2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . n1 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . n2 is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
max ((q1 . n1),(q2 . n2)) is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
((N ") (#) ((f /* (N + g1)) - (f /* g1))) . m is V30() real ext-real Element of REAL
(((N ") (#) ((f /* (N + g1)) - (f /* g1))) . m) - (x0,f) is V30() real ext-real Element of REAL
abs ((((N ") (#) ((f /* (N + g1)) - (f /* g1))) . m) - (x0,f)) is V30() real ext-real Element of REAL
N . m is V30() real ext-real Element of REAL
k is V30() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q2 . k is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
dom q2 is V71() V72() V73() V74() V75() V76() V99() Element of K19(NAT)
(((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q2) . k is V30() real ext-real Element of REAL
((((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q2) . k) - (x0,f) is V30() real ext-real Element of REAL
abs (((((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q2) . k) - (x0,f)) is V30() real ext-real Element of REAL
N . m is V30() real ext-real Element of REAL
k is V30() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
q1 . k is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
dom q1 is V71() V72() V73() V74() V75() V76() V99() Element of K19(NAT)
(((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q1) . k is V30() real ext-real Element of REAL
((((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q1) . k) - (x0,f) is V30() real ext-real Element of REAL
abs (((((N ") (#) ((f /* (N + g1)) - (f /* g1))) * q1) . k) - (x0,f)) is V30() real ext-real Element of REAL
N . m is V30() real ext-real Element of REAL
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() Element of NAT
N is V71() V72() V73() Neighbourhood of x0
f is V1() V4( REAL ) V5( REAL ) V6() complex-valued ext-real-valued real-valued Element of K19(K20(REAL,REAL))
x0 is V30() real ext-real Element of REAL
diff (f,x0) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
(x0,f) is V30() real ext-real Element of REAL
dom f is V71() V72() V73() Element of K19(REAL)
N is V71() V72() V73() Neighbourhood of x0
g1 is V30() real ext-real set
x0 - g1 is V30() real ext-real Element of REAL
x0 + g1 is V30() real ext-real Element of REAL
].(x0 - g1),(x0 + g1).[ is V71() V72() V73() V97() V98() V102() Element of K19(REAL)
g1 / 2 is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 + r is V30() real ext-real Element of REAL
[.x0,(x0 + r).] is V71() V72() V73() V102() Element of K19(REAL)
x0 + (g1 / 2) is V30() real ext-real Element of REAL
(x0 + (g1 / 2)) - x0 is V30() real ext-real Element of REAL
abs ((x0 + (g1 / 2)) - x0) is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 - r is V30() real ext-real Element of REAL
[.(x0 - r),x0.] is V71() V72() V73() V102() Element of K19(REAL)
x0 - (g1 / 2) is V30() real ext-real Element of REAL
(x0 - (g1 / 2)) - x0 is V30() real ext-real Element of REAL
abs ((x0 - (g1 / 2)) - x0) is V30() real ext-real Element of REAL
- (g1 / 2) is V30() real ext-real Element of REAL
(- (g1 / 2)) + 0 is V30() real ext-real Element of REAL
abs ((- (g1 / 2)) + 0) is V30() real ext-real Element of REAL
abs (g1 / 2) is V30() real ext-real Element of REAL
{x0} is V51() V71() V72() V73() V99() V100() V101() set
q1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng q1 is V71() V72() V73() Element of K19(REAL)
r is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
r + q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (r + q1) is V71() V72() V73() Element of K19(REAL)
r " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (r + q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* q1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (r + q1)) - (f /* q1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) is V30() real ext-real non positive set
K116(1) (#) (f /* q1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (r + q1)) + (- (f /* q1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(r ") (#) ((f /* (r + q1)) - (f /* q1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((r ") (#) ((f /* (r + q1)) - (f /* q1))) is V30() real ext-real Element of REAL
h1 is V1() V4( NAT ) V5( REAL ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent Element of K19(K20(NAT,REAL))
rng h1 is V71() V72() V73() Element of K19(REAL)
q2 is V1() non-zero V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K19(K20(NAT,REAL))
q2 + h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
rng (q2 + h1) is V71() V72() V73() Element of K19(REAL)
q2 " is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* (q2 + h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
f /* h1 is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(f /* (q2 + h1)) - (f /* h1) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
- (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
K116(1) (#) (f /* h1) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(f /* (q2 + h1)) + (- (f /* h1)) is V1() V4( NAT ) V6() total complex-valued ext-real-valued real-valued set
(q2 ") (#) ((f /* (q2 + h1)) - (f /* h1)) is V1() V4( NAT ) V5( REAL ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim ((q2 ") (#) ((f /* (q2 + h1)) - (f /* h1))) is V30() real ext-real Element of REAL
r is V30() real ext-real Element of REAL
x0 + r is V30() real ext-real Element of REAL
[.x0,(x0 + r).] is V71() V72() V73() V102() Element of K19(REAL)
q1 is V30() real ext-real Element of REAL
x0 - q1 is V30() real ext-real Element of REAL
[.(x0 - q1),x0.] is V71() V72() V73() V102() Element of K19(REAL)