let n be Nat; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
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 ; :: thesis:
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; :: thesis:
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 ; :: thesis:
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; :: thesis:

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; :: thesis: