let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Nat st
( 1 <= i & i < len (Cage (C,n)) & S-min C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ex i being Nat st
( 1 <= i & i < len (Cage (C,n)) & S-min C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
consider p being Point of (TOP-REAL 2) such that
A1:
(south_halfline (S-min C)) /\ (L~ (Cage (C,n))) = {p}
by JORDAN1A:88, PSCOMP_1:58;
A2:
p in (south_halfline (S-min C)) /\ (L~ (Cage (C,n)))
by A1, TARSKI:def 1;
then A3:
p in south_halfline (S-min C)
by XBOOLE_0:def 4;
A4:
S-min C = |[((S-min C) `1),((S-min C) `2)]|
by EUCLID:53;
A5:
len (Gauge (C,n)) = width (Gauge (C,n))
by JORDAN8:def 1;
A6:
len (Gauge (C,n)) >= 4
by JORDAN8:10;
then A7:
1 < len (Gauge (C,n))
by XXREAL_0:2;
p in L~ (Cage (C,n))
by A2, XBOOLE_0:def 4;
then consider i being Nat such that
A8:
1 <= i
and
A9:
i + 1 <= len (Cage (C,n))
and
A10:
p in LSeg ((Cage (C,n)),i)
by SPPOL_2:13;
take
i
; ( 1 <= i & i < len (Cage (C,n)) & S-min C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
A11:
LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1)))
by A8, A9, TOPREAL1:def 3;
A12: (S-min C) `2 =
S-bound C
by EUCLID:52
.=
((Gauge (C,n)) * (1,2)) `2
by A7, JORDAN8:13
;
A13:
S-min C in S-most C
by PSCOMP_1:58;
thus A14:
( 1 <= i & i < len (Cage (C,n)) )
by A8, A9, NAT_1:13; S-min C in right_cell ((Cage (C,n)),i,(Gauge (C,n)))
then A15:
((Cage (C,n)) /. i) `2 = p `2
by A3, A10, A13, A11, JORDAN1A:80, SPPOL_1:40;
A16:
Cage (C,n) is_sequence_on Gauge (C,n)
by JORDAN9:def 1;
then consider i1, j1, i2, j2 being Nat such that
A17:
[i1,j1] in Indices (Gauge (C,n))
and
A18:
(Cage (C,n)) /. i = (Gauge (C,n)) * (i1,j1)
and
A19:
[i2,j2] in Indices (Gauge (C,n))
and
A20:
(Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i2,j2)
and
A21:
( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) )
by A8, A9, JORDAN8:3;
A22:
1 <= i1
by A17, MATRIX_0:32;
A23:
1 <= j2
by A19, MATRIX_0:32;
A24:
i2 <= i2 + 1
by NAT_1:11;
A25:
j2 <= width (Gauge (C,n))
by A19, MATRIX_0:32;
A26:
i1 <= len (Gauge (C,n))
by A17, MATRIX_0:32;
A27:
j1 <= width (Gauge (C,n))
by A17, MATRIX_0:32;
p `2 = S-bound (L~ (Cage (C,n)))
by A2, JORDAN1A:84, PSCOMP_1:58;
then
((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (i1,1)) `2
by A18, A15, A22, A26, JORDAN1A:72;
then A28:
1 >= j1
by A22, A26, A27, GOBOARD5:4;
A29:
1 <= j1
by A17, MATRIX_0:32;
then A30:
j1 = 1
by A28, XXREAL_0:1;
A31:
((Cage (C,n)) /. (i + 1)) `2 = p `2
by A3, A10, A14, A13, A11, JORDAN1A:80, SPPOL_1:40;
A32:
j1 = j2
proof
assume
j1 <> j2
;
contradiction
then
(
j1 < j2 or
j2 < j1 )
by XXREAL_0:1;
hence
contradiction
by A18, A20, A21, A15, A31, A22, A26, A29, A25, A23, A27, GOBOARD5:4;
verum
end;
then A33:
i2 < len (Gauge (C,n))
by A8, A9, A17, A18, A19, A20, A21, A26, A28, JORDAN10:3, NAT_1:13;
1 <= i2
by A19, MATRIX_0:32;
then A34:
((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1
by A8, A9, A17, A18, A19, A20, A21, A26, A29, A25, A23, A27, A28, A24, JORDAN10:3, JORDAN1A:18;
then
p `1 <= ((Cage (C,n)) /. i) `1
by A10, A11, TOPREAL1:3;
then A35:
(S-min C) `1 <= ((Gauge (C,n)) * ((i2 + 1),1)) `1
by A3, A8, A9, A17, A18, A19, A20, A21, A32, A30, JORDAN10:3, TOPREAL1:def 12;
((Cage (C,n)) /. (i + 1)) `1 <= p `1
by A10, A11, A34, TOPREAL1:3;
then A36:
((Gauge (C,n)) * (i2,1)) `1 <= (S-min C) `1
by A3, A20, A32, A30, TOPREAL1:def 12;
A37:
1 <= i2
by A19, MATRIX_0:32;
1 + 1 <= len (Gauge (C,n))
by A6, XXREAL_0:2;
then
((Gauge (C,n)) * (1,j1)) `2 <= (S-min C) `2
by A5, A30, A7, A12, SPRECT_3:12;
then
S-min C in { |[r,s]| where r, s is Real : ( ((Gauge (C,n)) * (i2,1)) `1 <= r & r <= ((Gauge (C,n)) * ((i2 + 1),1)) `1 & ((Gauge (C,n)) * (1,j1)) `2 <= s & s <= ((Gauge (C,n)) * (1,(j1 + 1))) `2 ) }
by A30, A12, A36, A35, A4;
then
S-min C in cell ((Gauge (C,n)),i2,j1)
by A5, A30, A37, A33, A7, GOBRD11:32;
hence
S-min C in right_cell ((Cage (C,n)),i,(Gauge (C,n)))
by A8, A9, A16, A17, A18, A19, A20, A21, A32, A28, GOBRD13:26, JORDAN10:3; verum