Article JORDAN5D, MML version 4.99.1005
:: JORDAN5D:th 3
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
st 2 <= len b2
holds b2 /. len b2 in LSeg(b2,(len b2) -' 1);
:: JORDAN5D:th 4
theorem
for b1 being Element of NAT
st 3 <= b1
holds b1 mod (b1 -' 1) = 1;
:: JORDAN5D:th 5
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b1 in proj2 b2
holds ex b3 being Element of NAT st
1 <= b3 & b3 + 1 <= len b2 & b2 . b3 = b1;
:: JORDAN5D:th 6
theorem
for b1 being Element of REAL
for b2 being FinSequence of REAL
st b1 in proj2 b2
holds (Incr b2) . 1 <= b1 & b1 <= (Incr b2) . len Incr b2;
:: JORDAN5D:th 7
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= width GoB b1
holds ((GoB b1) *(1,b3)) `1 <= (b1 /. b2) `1 &
(b1 /. b2) `1 <= ((GoB b1) *(len GoB b1,b3)) `1;
:: JORDAN5D:th 8
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= len GoB b1
holds ((GoB b1) *(b3,1)) `2 <= (b1 /. b2) `2 &
(b1 /. b2) `2 <= ((GoB b1) *(b3,width GoB b1)) `2;
:: JORDAN5D:th 9
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & b2 <= len GoB b1
holds ex b3, b4 being Element of NAT st
b3 in dom b1 & [b2,b4] in Indices GoB b1 & b1 /. b3 = (GoB b1) *(b2,b4);
:: JORDAN5D:th 10
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & b2 <= width GoB b1
holds ex b3, b4 being Element of NAT st
b3 in dom b1 & [b4,b2] in Indices GoB b1 & b1 /. b3 = (GoB b1) *(b4,b2);
:: JORDAN5D:th 11
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len GoB b1 & 1 <= b3 & b3 <= width GoB b1
holds ex b4 being Element of NAT st
b4 in dom b1 &
[b2,b3] in Indices GoB b1 &
(b1 /. b4) `1 = ((GoB b1) *(b2,b3)) `1;
:: JORDAN5D:th 12
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len GoB b1 & 1 <= b3 & b3 <= width GoB b1
holds ex b4 being Element of NAT st
b4 in dom b1 &
[b2,b3] in Indices GoB b1 &
(b1 /. b4) `2 = ((GoB b1) *(b2,b3)) `2;
:: JORDAN5D:th 13
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & b2 <= len b1
holds S-bound L~ b1 <= (b1 /. b2) `2 & (b1 /. b2) `2 <= N-bound L~ b1;
:: JORDAN5D:th 14
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & b2 <= len b1
holds W-bound L~ b1 <= (b1 /. b2) `1 & (b1 /. b2) `1 <= E-bound L~ b1;
:: JORDAN5D:th 15
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = W-bound L~ b1 & b3 in L~ b1}
holds b2 = (proj2 || W-most L~ b1) .: the carrier of (TOP-REAL 2) | W-most L~ b1;
:: JORDAN5D:th 16
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = E-bound L~ b1 & b3 in L~ b1}
holds b2 = (proj2 || E-most L~ b1) .: the carrier of (TOP-REAL 2) | E-most L~ b1;
:: JORDAN5D:th 17
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = N-bound L~ b1 & b3 in L~ b1}
holds b2 = (proj1 || N-most L~ b1) .: the carrier of (TOP-REAL 2) | N-most L~ b1;
:: JORDAN5D:th 18
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = S-bound L~ b1 & b3 in L~ b1}
holds b2 = (proj1 || S-most L~ b1) .: the carrier of (TOP-REAL 2) | S-most L~ b1;
:: JORDAN5D:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds b2 = (proj1 || L~ b1) .: the carrier of (TOP-REAL 2) | L~ b1;
:: JORDAN5D:th 20
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds b2 = (proj2 || L~ b1) .: the carrier of (TOP-REAL 2) | L~ b1;
:: JORDAN5D:th 21
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = W-bound L~ b1 & b3 in L~ b1}
holds lower_bound b2 = inf (proj2 || W-most L~ b1);
:: JORDAN5D:th 22
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = W-bound L~ b1 & b3 in L~ b1}
holds upper_bound b2 = sup (proj2 || W-most L~ b1);
:: JORDAN5D:th 23
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = E-bound L~ b1 & b3 in L~ b1}
holds lower_bound b2 = inf (proj2 || E-most L~ b1);
:: JORDAN5D:th 24
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = E-bound L~ b1 & b3 in L~ b1}
holds upper_bound b2 = sup (proj2 || E-most L~ b1);
:: JORDAN5D:th 25
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds lower_bound b2 = inf (proj1 || L~ b1);
:: JORDAN5D:th 26
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = S-bound L~ b1 & b3 in L~ b1}
holds lower_bound b2 = inf (proj1 || S-most L~ b1);
:: JORDAN5D:th 27
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = S-bound L~ b1 & b3 in L~ b1}
holds upper_bound b2 = sup (proj1 || S-most L~ b1);
:: JORDAN5D:th 28
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = N-bound L~ b1 & b3 in L~ b1}
holds lower_bound b2 = inf (proj1 || N-most L~ b1);
:: JORDAN5D:th 29
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = N-bound L~ b1 & b3 in L~ b1}
holds upper_bound b2 = sup (proj1 || N-most L~ b1);
:: JORDAN5D:th 30
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds lower_bound b2 = inf (proj2 || L~ b1);
:: JORDAN5D:th 31
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `1 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds upper_bound b2 = sup (proj1 || L~ b1);
:: JORDAN5D:th 32
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool REAL
st b2 = {b3 `2 where b3 is Element of the carrier of TOP-REAL 2: b3 in L~ b1}
holds upper_bound b2 = sup (proj2 || L~ b1);
:: JORDAN5D:th 33
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 in L~ b2 & 1 <= b3 & b3 <= width GoB b2
holds ((GoB b2) *(1,b3)) `1 <= b1 `1;
:: JORDAN5D:th 34
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 in L~ b2 & 1 <= b3 & b3 <= width GoB b2
holds b1 `1 <= ((GoB b2) *(len GoB b2,b3)) `1;
:: JORDAN5D:th 35
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 in L~ b2 & 1 <= b3 & b3 <= len GoB b2
holds ((GoB b2) *(b3,1)) `2 <= b1 `2;
:: JORDAN5D:th 36
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 in L~ b2 & 1 <= b3 & b3 <= len GoB b2
holds b1 `2 <= ((GoB b2) *(b3,width GoB b2)) `2;
:: JORDAN5D:th 37
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len GoB b1 & 1 <= b3 & b3 <= width GoB b1
holds ex b4 being Element of the carrier of TOP-REAL 2 st
b4 `1 = ((GoB b1) *(b2,b3)) `1 & b4 in L~ b1;
:: JORDAN5D:th 38
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 1 <= b2 & b2 <= len GoB b1 & 1 <= b3 & b3 <= width GoB b1
holds ex b4 being Element of the carrier of TOP-REAL 2 st
b4 `2 = ((GoB b1) *(b2,b3)) `2 & b4 in L~ b1;
:: JORDAN5D:th 39
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
W-bound L~ b1 = ((GoB b1) *(1,1)) `1;
:: JORDAN5D:th 40
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
S-bound L~ b1 = ((GoB b1) *(1,1)) `2;
:: JORDAN5D:th 41
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
E-bound L~ b1 = ((GoB b1) *(len GoB b1,1)) `1;
:: JORDAN5D:th 42
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
N-bound L~ b1 = ((GoB b1) *(1,width GoB b1)) `2;
:: JORDAN5D:th 43
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
for b5 being non empty finite Element of bool NAT
st 1 <= b2 &
b2 <= len b1 &
1 <= b3 &
b3 <= len GoB b1 &
b5 = {b6 where b6 is Element of NAT: [b3,b6] in Indices GoB b1 &
(ex b7 being Element of NAT st
b7 in dom b1 & b1 /. b7 = (GoB b1) *(b3,b6))} &
(b1 /. b2) `1 = ((GoB b1) *(b3,1)) `1 &
b4 = min b5
holds ((GoB b1) *(b3,b4)) `2 <= (b1 /. b2) `2;
:: JORDAN5D:th 44
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
for b5 being non empty finite Element of bool NAT
st 1 <= b2 &
b2 <= len b1 &
1 <= b3 &
b3 <= width GoB b1 &
b5 = {b6 where b6 is Element of NAT: [b6,b3] in Indices GoB b1 &
(ex b7 being Element of NAT st
b7 in dom b1 & b1 /. b7 = (GoB b1) *(b6,b3))} &
(b1 /. b2) `2 = ((GoB b1) *(1,b3)) `2 &
b4 = min b5
holds ((GoB b1) *(b4,b3)) `1 <= (b1 /. b2) `1;
:: JORDAN5D:th 45
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
for b5 being non empty finite Element of bool NAT
st 1 <= b2 &
b2 <= len b1 &
1 <= b3 &
b3 <= width GoB b1 &
b5 = {b6 where b6 is Element of NAT: [b6,b3] in Indices GoB b1 &
(ex b7 being Element of NAT st
b7 in dom b1 & b1 /. b7 = (GoB b1) *(b6,b3))} &
(b1 /. b2) `2 = ((GoB b1) *(1,b3)) `2 &
b4 = max b5
holds (b1 /. b2) `1 <= ((GoB b1) *(b4,b3)) `1;
:: JORDAN5D:th 46
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
for b5 being non empty finite Element of bool NAT
st 1 <= b2 &
b2 <= len b1 &
1 <= b3 &
b3 <= len GoB b1 &
b5 = {b6 where b6 is Element of NAT: [b3,b6] in Indices GoB b1 &
(ex b7 being Element of NAT st
b7 in dom b1 & b1 /. b7 = (GoB b1) *(b3,b6))} &
(b1 /. b2) `1 = ((GoB b1) *(b3,1)) `1 &
b4 = max b5
holds (b1 /. b2) `2 <= ((GoB b1) *(b3,b4)) `2;
:: JORDAN5D:funcnot 1 => JORDAN5D:func 1
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_s_w A1 -> Element of NAT means
[1,it] in Indices GoB a1 &
(GoB a1) *(1,it) = W-min L~ a1;
end;
:: JORDAN5D:def 1
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_s_w b1
iff
[1,b2] in Indices GoB b1 &
(GoB b1) *(1,b2) = W-min L~ b1;
:: JORDAN5D:funcnot 2 => JORDAN5D:func 2
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_n_w A1 -> Element of NAT means
[1,it] in Indices GoB a1 &
(GoB a1) *(1,it) = W-max L~ a1;
end;
:: JORDAN5D:def 2
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_n_w b1
iff
[1,b2] in Indices GoB b1 &
(GoB b1) *(1,b2) = W-max L~ b1;
:: JORDAN5D:funcnot 3 => JORDAN5D:func 3
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_s_e A1 -> Element of NAT means
[len GoB a1,it] in Indices GoB a1 &
(GoB a1) *(len GoB a1,it) = E-min L~ a1;
end;
:: JORDAN5D:def 3
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_s_e b1
iff
[len GoB b1,b2] in Indices GoB b1 &
(GoB b1) *(len GoB b1,b2) = E-min L~ b1;
:: JORDAN5D:funcnot 4 => JORDAN5D:func 4
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_n_e A1 -> Element of NAT means
[len GoB a1,it] in Indices GoB a1 &
(GoB a1) *(len GoB a1,it) = E-max L~ a1;
end;
:: JORDAN5D:def 4
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_n_e b1
iff
[len GoB b1,b2] in Indices GoB b1 &
(GoB b1) *(len GoB b1,b2) = E-max L~ b1;
:: JORDAN5D:funcnot 5 => JORDAN5D:func 5
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_w_s A1 -> Element of NAT means
[it,1] in Indices GoB a1 &
(GoB a1) *(it,1) = S-min L~ a1;
end;
:: JORDAN5D:def 5
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_w_s b1
iff
[b2,1] in Indices GoB b1 &
(GoB b1) *(b2,1) = S-min L~ b1;
:: JORDAN5D:funcnot 6 => JORDAN5D:func 6
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_e_s A1 -> Element of NAT means
[it,1] in Indices GoB a1 &
(GoB a1) *(it,1) = S-max L~ a1;
end;
:: JORDAN5D:def 6
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_e_s b1
iff
[b2,1] in Indices GoB b1 &
(GoB b1) *(b2,1) = S-max L~ b1;
:: JORDAN5D:funcnot 7 => JORDAN5D:func 7
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_w_n A1 -> Element of NAT means
[it,width GoB a1] in Indices GoB a1 &
(GoB a1) *(it,width GoB a1) = N-min L~ a1;
end;
:: JORDAN5D:def 7
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_w_n b1
iff
[b2,width GoB b1] in Indices GoB b1 &
(GoB b1) *(b2,width GoB b1) = N-min L~ b1;
:: JORDAN5D:funcnot 8 => JORDAN5D:func 8
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func i_e_n A1 -> Element of NAT means
[it,width GoB a1] in Indices GoB a1 &
(GoB a1) *(it,width GoB a1) = N-max L~ a1;
end;
:: JORDAN5D:def 8
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = i_e_n b1
iff
[b2,width GoB b1] in Indices GoB b1 &
(GoB b1) *(b2,width GoB b1) = N-max L~ b1;
:: JORDAN5D:th 47
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
1 <= i_w_n b1 & i_w_n b1 <= len GoB b1 & 1 <= i_e_n b1 & i_e_n b1 <= len GoB b1 & 1 <= i_w_s b1 & i_w_s b1 <= len GoB b1 & 1 <= i_e_s b1 & i_e_s b1 <= len GoB b1;
:: JORDAN5D:th 48
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
1 <= i_n_e b1 & i_n_e b1 <= width GoB b1 & 1 <= i_s_e b1 & i_s_e b1 <= width GoB b1 & 1 <= i_n_w b1 & i_n_w b1 <= width GoB b1 & 1 <= i_s_w b1 & i_s_w b1 <= width GoB b1;
:: JORDAN5D:funcnot 9 => JORDAN5D:func 9
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_s_w A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = W-min L~ a1;
end;
:: JORDAN5D:def 9
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_s_w b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = W-min L~ b1;
:: JORDAN5D:funcnot 10 => JORDAN5D:func 10
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_n_w A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = W-max L~ a1;
end;
:: JORDAN5D:def 10
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_n_w b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = W-max L~ b1;
:: JORDAN5D:funcnot 11 => JORDAN5D:func 11
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_s_e A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = E-min L~ a1;
end;
:: JORDAN5D:def 11
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_s_e b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = E-min L~ b1;
:: JORDAN5D:funcnot 12 => JORDAN5D:func 12
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_n_e A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = E-max L~ a1;
end;
:: JORDAN5D:def 12
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_n_e b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = E-max L~ b1;
:: JORDAN5D:funcnot 13 => JORDAN5D:func 13
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_w_s A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = S-min L~ a1;
end;
:: JORDAN5D:def 13
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_w_s b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = S-min L~ b1;
:: JORDAN5D:funcnot 14 => JORDAN5D:func 14
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_e_s A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = S-max L~ a1;
end;
:: JORDAN5D:def 14
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_e_s b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = S-max L~ b1;
:: JORDAN5D:funcnot 15 => JORDAN5D:func 15
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_w_n A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = N-min L~ a1;
end;
:: JORDAN5D:def 15
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_w_n b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = N-min L~ b1;
:: JORDAN5D:funcnot 16 => JORDAN5D:func 16
definition
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
func n_e_n A1 -> Element of NAT means
1 <= it & it + 1 <= len a1 & a1 . it = N-max L~ a1;
end;
:: JORDAN5D:def 16
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
b2 = n_e_n b1
iff
1 <= b2 & b2 + 1 <= len b1 & b1 . b2 = N-max L~ b1;
:: JORDAN5D:th 49
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
n_w_n b1 <> n_w_s b1;
:: JORDAN5D:th 50
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
n_s_w b1 <> n_s_e b1;
:: JORDAN5D:th 51
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
n_e_n b1 <> n_e_s b1;
:: JORDAN5D:th 52
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
n_n_w b1 <> n_n_e b1;