set j = 1;
let f be constant standard special_circular_sequence; :: thesis: ( f is clockwise_oriented iff (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f) )
set r = Rotate (f,(S-max (L~ f)));
A1: Rotate (f,(S-max (L~ f))) is_sequence_on GoB (Rotate (f,(S-max (L~ f)))) by GOBOARD5:def 5;
len (Rotate (f,(S-max (L~ f)))) > 2 by TOPREAL8:3;
then A2: 1 + 1 in dom (Rotate (f,(S-max (L~ f)))) by FINSEQ_3:25;
then consider i2, j2 being Nat such that
A3: [i2,j2] in Indices (GoB (Rotate (f,(S-max (L~ f))))) and
A4: (Rotate (f,(S-max (L~ f)))) /. (1 + 1) = (GoB (Rotate (f,(S-max (L~ f))))) * (i2,j2) by A1, GOBOARD1:def 9;
A5: i2 <= len (GoB (Rotate (f,(S-max (L~ f))))) by A3, MATRIX_0:32;
set i = i_e_s (Rotate (f,(S-max (L~ f))));
A6: S-max (L~ f) in rng f by SPRECT_2:42;
then A7: (Rotate (f,(S-max (L~ f)))) /. 1 = S-max (L~ f) by FINSEQ_6:92;
A8: 2 <= len f by TOPREAL8:3;
thus ( f is clockwise_oriented implies (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f) ) :: thesis: ( (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f) implies f is clockwise_oriented )

proof

set k = (S-max (L~ f)) .. f;
(S-max (L~ f)) .. f < len f by SPRECT_5:14;
then A9: ((S-max (L~ f)) .. f) + 1 <= len f by NAT_1:13;
1 <= ((S-max (L~ f)) .. f) + 1 by NAT_1:11;
then A10: ((S-max (L~ f)) .. f) + 1 in dom f by A9, FINSEQ_3:25;
then f /. (((S-max (L~ f)) .. f) + 1) = f . (((S-max (L~ f)) .. f) + 1) by PARTFUN1:def 6;
then A11: f /. (((S-max (L~ f)) .. f) + 1) in rng f by A10, FUNCT_1:3;
A12: rng f c= L~ f by A8, SPPOL_2:18;
A13: f /. ((S-max (L~ f)) .. f) = S-max (L~ f) by A6, FINSEQ_5:38;
(S-max (L~ f)) .. f <= ((S-max (L~ f)) .. f) + 1 by NAT_1:13;
then A14: f /. (((S-max (L~ f)) .. f) + 1) = (Rotate (f,(S-max (L~ f)))) /. (((((S-max (L~ f)) .. f) + 1) + 1) -' ((S-max (L~ f)) .. f)) by A6, A9, FINSEQ_6:175
.= (Rotate (f,(S-max (L~ f)))) /. ((((S-max (L~ f)) .. f) + (1 + 1)) -' ((S-max (L~ f)) .. f))
.= (Rotate (f,(S-max (L~ f)))) /. 2 by NAT_D:34 ;
f is_sequence_on GoB f by GOBOARD5:def 5;
then A15: f is_sequence_on GoB (Rotate (f,(S-max (L~ f)))) by REVROT_1:28;
assume f is clockwise_oriented ; :: thesis: (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f)
then consider i, j being Nat such that
A16: [(i + 1),j] in Indices (GoB (Rotate (f,(S-max (L~ f))))) and
A17: [i,j] in Indices (GoB (Rotate (f,(S-max (L~ f))))) and
A18: f /. ((S-max (L~ f)) .. f) = (GoB (Rotate (f,(S-max (L~ f))))) * ((i + 1),j) and
A19: f /. (((S-max (L~ f)) .. f) + 1) = (GoB (Rotate (f,(S-max (L~ f))))) * (i,j) by A6, A9, A13, A15, Th24, FINSEQ_4:21;
A20: ( 1 <= i + 1 & i + 1 <= len (GoB (Rotate (f,(S-max (L~ f))))) ) by A16, MATRIX_0:32;
A21: ( 1 <= j & j <= width (GoB (Rotate (f,(S-max (L~ f))))) ) by A16, MATRIX_0:32;
A22: ( 1 <= j & j <= width (GoB (Rotate (f,(S-max (L~ f))))) ) by A16, MATRIX_0:32;
( 1 <= i & i <= len (GoB (Rotate (f,(S-max (L~ f))))) ) by A17, MATRIX_0:32;
then (f /. (((S-max (L~ f)) .. f) + 1)) `2 = ((GoB (Rotate (f,(S-max (L~ f))))) * (1,j)) `2 by A19, A21, GOBOARD5:1
.= (f /. ((S-max (L~ f)) .. f)) `2 by A18, A20, A22, GOBOARD5:1
.= S-bound (L~ f) by A13, EUCLID:52 ;
hence (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f) by A14, A11, A12, SPRECT_2:11; :: thesis: verum

end;

assume A23: (Rotate (f,(S-max (L~ f)))) /. 2 in S-most (L~ f) ; :: thesis: f is clockwise_oriented
A24: [(i_e_s (Rotate (f,(S-max (L~ f))))),1] in Indices (GoB (Rotate (f,(S-max (L~ f))))) by JORDAN5D:def 6;
then A25: 1 <= i_e_s (Rotate (f,(S-max (L~ f)))) by MATRIX_0:32;
A26: L~ (Rotate (f,(S-max (L~ f)))) = L~ f by REVROT_1:33;
then A27: (GoB (Rotate (f,(S-max (L~ f))))) * ((i_e_s (Rotate (f,(S-max (L~ f))))),1) = (Rotate (f,(S-max (L~ f)))) /. 1 by A7, JORDAN5D:def 6;
then ((GoB (Rotate (f,(S-max (L~ f))))) * ((i_e_s (Rotate (f,(S-max (L~ f))))),1)) `2 = S-bound (L~ f) by A7, EUCLID:52
.= (S-min (L~ f)) `2 by EUCLID:52 ;
then ((GoB (Rotate (f,(S-max (L~ f))))) * (i2,j2)) `2 = ((GoB (Rotate (f,(S-max (L~ f))))) * ((i_e_s (Rotate (f,(S-max (L~ f))))),1)) `2 by A23, A4, PSCOMP_1:55;
then A28: j2 = 1 by A24, A3, JORDAN1G:6;
A29: 1 <= width (GoB (Rotate (f,(S-max (L~ f))))) by A24, MATRIX_0:32;
rng (Rotate (f,(S-max (L~ f)))) = rng f by FINSEQ_6:90, SPRECT_2:42;
then 1 in dom (Rotate (f,(S-max (L~ f)))) by FINSEQ_3:31, SPRECT_2:42;
then |.(1 - 1).| + |.((i_e_s (Rotate (f,(S-max (L~ f))))) - i2).| = 1 by A1, A24, A27, A2, A3, A4, A28, GOBOARD1:def 9;
then A30: 0 + |.((i_e_s (Rotate (f,(S-max (L~ f))))) - i2).| = 1 by ABSVALUE:2;
((GoB (Rotate (f,(S-max (L~ f))))) * (i2,j2)) `1 <= ((GoB (Rotate (f,(S-max (L~ f))))) * ((i_e_s (Rotate (f,(S-max (L~ f))))),1)) `1 by A7, A27, A23, A4, PSCOMP_1:55;
then (i_e_s (Rotate (f,(S-max (L~ f))))) - i2 >= 0 by A25, A29, A28, A5, GOBOARD5:3, XREAL_1:48;
then A31: (i_e_s (Rotate (f,(S-max (L~ f))))) - i2 = 1 by A30, ABSVALUE:def 1;
then i2 = (i_e_s (Rotate (f,(S-max (L~ f))))) - 1 ;
then A32: i2 = (i_e_s (Rotate (f,(S-max (L~ f))))) -' 1 by A25, XREAL_1:233;
then A33: 1 <= (i_e_s (Rotate (f,(S-max (L~ f))))) -' 1 by A3, MATRIX_0:32;
A34: i_e_s (Rotate (f,(S-max (L~ f)))) <= len (GoB (Rotate (f,(S-max (L~ f))))) by A24, MATRIX_0:32;
A35: 1 + 1 <= len (Rotate (f,(S-max (L~ f)))) by TOPREAL8:3;
then A36: Int (left_cell ((Rotate (f,(S-max (L~ f)))),1)) c= LeftComp (Rotate (f,(S-max (L~ f)))) by GOBOARD9:21;
Int (left_cell ((Rotate (f,(S-max (L~ f)))),1)) <> {} by A35, GOBOARD9:15;
then consider p being object such that
A37: p in Int (left_cell ((Rotate (f,(S-max (L~ f)))),1)) by XBOOLE_0:def 1;
[(((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1) + 1),1] in Indices (GoB (Rotate (f,(S-max (L~ f))))) by A24, A25, XREAL_1:235;
then ( 1 -' 1 = 1 - 1 & left_cell ((Rotate (f,(S-max (L~ f)))),1,(GoB (Rotate (f,(S-max (L~ f)))))) = cell ((GoB (Rotate (f,(S-max (L~ f))))),((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1),(1 -' 1)) ) by A35, A1, A27, A3, A4, A28, A31, A32, GOBRD13:25, XREAL_1:233;
then A38: left_cell ((Rotate (f,(S-max (L~ f)))),1) = cell ((GoB (Rotate (f,(S-max (L~ f))))),((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1),0) by A35, JORDAN1H:21;
(i_e_s (Rotate (f,(S-max (L~ f))))) - 1 < i_e_s (Rotate (f,(S-max (L~ f)))) by XREAL_1:146;
then (i_e_s (Rotate (f,(S-max (L~ f))))) -' 1 < len (GoB (Rotate (f,(S-max (L~ f))))) by A34, A31, A32, XXREAL_0:2;
then A39: Int (left_cell ((Rotate (f,(S-max (L~ f)))),1)) = { |[t,s]| where t, s is Real : ( ((GoB (Rotate (f,(S-max (L~ f))))) * (((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1),1)) `1 < t & t < ((GoB (Rotate (f,(S-max (L~ f))))) * ((((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1) + 1),1)) `1 & s < ((GoB (Rotate (f,(S-max (L~ f))))) * (1,1)) `2 ) } by A33, A38, GOBOARD6:24;
reconsider p = p as Point of (TOP-REAL 2) by A37;
A40: ( LeftComp (Rotate (f,(S-max (L~ f)))) is_a_component_of (L~ (Rotate (f,(S-max (L~ f))))) ` & UBD (L~ (Rotate (f,(S-max (L~ f))))) is_a_component_of (L~ (Rotate (f,(S-max (L~ f))))) ` ) by GOBOARD9:def 1, JORDAN2C:124;
consider t, s being Real such that
A41: p = |[t,s]| and
((GoB (Rotate (f,(S-max (L~ f))))) * (((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1),1)) `1 < t and
t < ((GoB (Rotate (f,(S-max (L~ f))))) * ((((i_e_s (Rotate (f,(S-max (L~ f))))) -' 1) + 1),1)) `1 and
A42: s < ((GoB (Rotate (f,(S-max (L~ f))))) * (1,1)) `2 by A39, A37;

now :: thesis: not south_halfline p meets L~ (Rotate (f,(S-max (L~ f))))

assume south_halfline p meets L~ (Rotate (f,(S-max (L~ f)))) ; :: thesis: contradiction
then (south_halfline p) /\ (L~ (Rotate (f,(S-max (L~ f))))) <> {} by XBOOLE_0:def 7;
then consider a being object such that
A43: a in (south_halfline p) /\ (L~ (Rotate (f,(S-max (L~ f))))) by XBOOLE_0:def 1;
A44: a in L~ (Rotate (f,(S-max (L~ f)))) by A43, XBOOLE_0:def 4;
A45: a in south_halfline p by A43, XBOOLE_0:def 4;
reconsider a = a as Point of (TOP-REAL 2) by A43;
a `2 <= p `2 by A45, TOPREAL1:def 12;
then a `2 <= s by A41, EUCLID:52;
then A46: a `2 < ((GoB (Rotate (f,(S-max (L~ f))))) * (1,1)) `2 by A42, XXREAL_0:2;
((GoB (Rotate (f,(S-max (L~ f))))) * ((i_e_s (Rotate (f,(S-max (L~ f))))),1)) `2 = ((GoB (Rotate (f,(S-max (L~ f))))) * (1,1)) `2 by A25, A34, A29, GOBOARD5:1;
then a `2 < S-bound (L~ (Rotate (f,(S-max (L~ f))))) by A26, A7, A27, A46, EUCLID:52;
hence contradiction by A44, PSCOMP_1:24; :: thesis: verum

end;

then A47: south_halfline p c= UBD (L~ (Rotate (f,(S-max (L~ f))))) by JORDAN2C:128;
p in south_halfline p by TOPREAL1:38;
then LeftComp (Rotate (f,(S-max (L~ f)))) meets UBD (L~ (Rotate (f,(S-max (L~ f))))) by A36, A37, A47, XBOOLE_0:3;
then Rotate (f,(S-max (L~ f))) is clockwise_oriented by A40, GOBOARD9:1, JORDAN1H:41;
hence f is clockwise_oriented by JORDAN1H:40; :: thesis: verum