set i = 1;
let f be non constant standard special_circular_sequence; :: thesis: ( f is clockwise_oriented iff (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f) )
set r = Rotate f,(W-min (L~ f));
A1: Rotate f,(W-min (L~ f)) is_sequence_on GoB (Rotate f,(W-min (L~ f))) by GOBOARD5:def 5;
A2: 1 + 1 <= len (Rotate f,(W-min (L~ f))) by TOPREAL8:3;
then A3: Int (left_cell (Rotate f,(W-min (L~ f))),1) c= LeftComp (Rotate f,(W-min (L~ f))) by GOBOARD9:24;
set j = i_s_w (Rotate f,(W-min (L~ f)));
A4: W-min (L~ f) in rng f by SPRECT_2:47;
then A5: (Rotate f,(W-min (L~ f))) /. 1 = W-min (L~ f) by FINSEQ_6:98;
A6: 2 <= len f by TOPREAL8:3;
thus ( f is clockwise_oriented implies (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f) ) :: thesis: ( (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f) implies f is clockwise_oriented )

proof

set k = (W-min (L~ f)) .. f;
(W-min (L~ f)) .. f < len f by SPRECT_5:21;
then A7: ((W-min (L~ f)) .. f) + 1 <= len f by NAT_1:13;
1 <= ((W-min (L~ f)) .. f) + 1 by NAT_1:11;
then A8: ((W-min (L~ f)) .. f) + 1 in dom f by A7, FINSEQ_3:27;
then f /. (((W-min (L~ f)) .. f) + 1) = f . (((W-min (L~ f)) .. f) + 1) by PARTFUN1:def 8;
then A9: f /. (((W-min (L~ f)) .. f) + 1) in rng f by A8, FUNCT_1:12;
A10: rng f c= L~ f by A6, SPPOL_2:18;
A11: f /. ((W-min (L~ f)) .. f) = W-min (L~ f) by A4, FINSEQ_5:41;
(W-min (L~ f)) .. f <= ((W-min (L~ f)) .. f) + 1 by NAT_1:13;
then A12: f /. (((W-min (L~ f)) .. f) + 1) = (Rotate f,(W-min (L~ f))) /. (((((W-min (L~ f)) .. f) + 1) + 1) -' ((W-min (L~ f)) .. f)) by A4, A7, REVROT_1:10
.= (Rotate f,(W-min (L~ f))) /. ((((W-min (L~ f)) .. f) + (1 + 1)) -' ((W-min (L~ f)) .. f))
.= (Rotate f,(W-min (L~ f))) /. 2 by NAT_D:34 ;
f is_sequence_on GoB f by GOBOARD5:def 5;
then A13: f is_sequence_on GoB (Rotate f,(W-min (L~ f))) by REVROT_1:28;
assume f is clockwise_oriented ; :: thesis: (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f)
then consider i, j being Element of NAT such that
A14: [i,j] in Indices (GoB (Rotate f,(W-min (L~ f)))) and
A15: [i,(j + 1)] in Indices (GoB (Rotate f,(W-min (L~ f)))) and
A16: f /. ((W-min (L~ f)) .. f) = (GoB (Rotate f,(W-min (L~ f)))) * i,j and
A17: f /. (((W-min (L~ f)) .. f) + 1) = (GoB (Rotate f,(W-min (L~ f)))) * i,(j + 1) by A4, A7, A11, A13, Th23, FINSEQ_4:31;
A18: ( 1 <= i & i <= len (GoB (Rotate f,(W-min (L~ f)))) ) by A14, MATRIX_1:39;
A19: ( 1 <= j + 1 & j + 1 <= width (GoB (Rotate f,(W-min (L~ f)))) ) by A15, MATRIX_1:39;
A20: ( 1 <= j & j <= width (GoB (Rotate f,(W-min (L~ f)))) ) by A14, MATRIX_1:39;
( 1 <= i & i <= len (GoB (Rotate f,(W-min (L~ f)))) ) by A14, MATRIX_1:39;
then (f /. (((W-min (L~ f)) .. f) + 1)) `1 = ((GoB (Rotate f,(W-min (L~ f)))) * i,1) `1 by A17, A19, GOBOARD5:3
.= (f /. ((W-min (L~ f)) .. f)) `1 by A16, A18, A20, GOBOARD5:3
.= W-bound (L~ f) by A11, EUCLID:56 ;
hence (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f) by A12, A9, A10, SPRECT_2:16; :: thesis: verum

end;

A21: [1,(i_s_w (Rotate f,(W-min (L~ f))))] in Indices (GoB (Rotate f,(W-min (L~ f)))) by JORDAN5D:def 1;
then A22: ( 1 <= len (GoB (Rotate f,(W-min (L~ f)))) & i_s_w (Rotate f,(W-min (L~ f))) <= width (GoB (Rotate f,(W-min (L~ f)))) ) by MATRIX_1:39;
len (Rotate f,(W-min (L~ f))) > 2 by TOPREAL8:3;
then A23: 1 + 1 in dom (Rotate f,(W-min (L~ f))) by FINSEQ_3:27;
then consider i2, j2 being Element of NAT such that
A24: [i2,j2] in Indices (GoB (Rotate f,(W-min (L~ f)))) and
A25: (Rotate f,(W-min (L~ f))) /. (1 + 1) = (GoB (Rotate f,(W-min (L~ f)))) * i2,j2 by A1, GOBOARD1:def 11;
A26: 1 <= j2 by A24, MATRIX_1:39;
A27: L~ (Rotate f,(W-min (L~ f))) = L~ f by REVROT_1:33;
then A28: (GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f)))) = (Rotate f,(W-min (L~ f))) /. 1 by A5, JORDAN5D:def 1;
assume A29: (Rotate f,(W-min (L~ f))) /. 2 in W-most (L~ f) ; :: thesis: f is clockwise_oriented
then ((GoB (Rotate f,(W-min (L~ f)))) * i2,j2) `1 = ((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `1 by A5, A28, A25, PSCOMP_1:88;
then A30: i2 = 1 by A21, A24, JORDAN1G:7;
rng (Rotate f,(W-min (L~ f))) = rng f by FINSEQ_6:96, SPRECT_2:47;
then 1 in dom (Rotate f,(W-min (L~ f))) by FINSEQ_3:33, SPRECT_2:47;
then (abs (1 - 1)) + (abs ((i_s_w (Rotate f,(W-min (L~ f)))) - j2)) = 1 by A1, A21, A28, A23, A24, A25, A30, GOBOARD1:def 11;
then 0 + (abs ((i_s_w (Rotate f,(W-min (L~ f)))) - j2)) = 1 by ABSVALUE:7;
then A31: abs (j2 - (i_s_w (Rotate f,(W-min (L~ f))))) = 1 by UNIFORM1:13;
((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `2 <= ((GoB (Rotate f,(W-min (L~ f)))) * i2,j2) `2 by A5, A28, A29, A25, PSCOMP_1:88;
then j2 - (i_s_w (Rotate f,(W-min (L~ f)))) >= 0 by A22, A30, A26, GOBOARD5:5, XREAL_1:50;
then A32: j2 - (i_s_w (Rotate f,(W-min (L~ f)))) = 1 by A31, ABSVALUE:def 1;
then j2 = (i_s_w (Rotate f,(W-min (L~ f)))) + 1 ;
then ( 1 -' 1 = 1 - 1 & left_cell (Rotate f,(W-min (L~ f))),1,(GoB (Rotate f,(W-min (L~ f)))) = cell (GoB (Rotate f,(W-min (L~ f)))),(1 -' 1),(i_s_w (Rotate f,(W-min (L~ f)))) ) by A2, A1, A21, A28, A24, A25, A30, GOBRD13:22, XREAL_1:235;
then A33: left_cell (Rotate f,(W-min (L~ f))),1 = cell (GoB (Rotate f,(W-min (L~ f)))),0 ,(i_s_w (Rotate f,(W-min (L~ f)))) by A2, JORDAN1H:27;
Int (left_cell (Rotate f,(W-min (L~ f))),1) <> {} by A2, GOBOARD9:18;
then consider p being set such that
A34: p in Int (left_cell (Rotate f,(W-min (L~ f))),1) by XBOOLE_0:def 1;
reconsider p = p as Point of (TOP-REAL 2) by A34;
A35: ( LeftComp (Rotate f,(W-min (L~ f))) is_a_component_of (L~ (Rotate f,(W-min (L~ f)))) ` & UBD (L~ (Rotate f,(W-min (L~ f)))) is_a_component_of (L~ (Rotate f,(W-min (L~ f)))) ` ) by GOBOARD9:def 1, JORDAN2C:132;
A36: 1 <= i_s_w (Rotate f,(W-min (L~ f))) by A21, MATRIX_1:39;
(i_s_w (Rotate f,(W-min (L~ f)))) + 1 <= width (GoB (Rotate f,(W-min (L~ f)))) by A24, A32, MATRIX_1:39;
then i_s_w (Rotate f,(W-min (L~ f))) < width (GoB (Rotate f,(W-min (L~ f)))) by NAT_1:13;
then Int (left_cell (Rotate f,(W-min (L~ f))),1) = { |[t,s]| where t, s is Real : ( t < ((GoB (Rotate f,(W-min (L~ f)))) * 1,1) `1 & ((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `2 < s & s < ((GoB (Rotate f,(W-min (L~ f)))) * 1,((i_s_w (Rotate f,(W-min (L~ f)))) + 1)) `2 ) } by A36, A33, GOBOARD6:23;
then consider t, s being Real such that
A37: p = |[t,s]| and
A38: t < ((GoB (Rotate f,(W-min (L~ f)))) * 1,1) `1 and
((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `2 < s and
s < ((GoB (Rotate f,(W-min (L~ f)))) * 1,((i_s_w (Rotate f,(W-min (L~ f)))) + 1)) `2 by A34;

now

A39: ((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `1 = ((GoB (Rotate f,(W-min (L~ f)))) * 1,1) `1 by A36, A22, GOBOARD5:3;
assume west_halfline p meets L~ (Rotate f,(W-min (L~ f))) ; :: thesis: contradiction
then (west_halfline p) /\ (L~ (Rotate f,(W-min (L~ f)))) <> {} by XBOOLE_0:def 7;
then consider a being set such that
A40: a in (west_halfline p) /\ (L~ (Rotate f,(W-min (L~ f)))) by XBOOLE_0:def 1;
A41: a in L~ (Rotate f,(W-min (L~ f))) by A40, XBOOLE_0:def 4;
A42: a in west_halfline p by A40, XBOOLE_0:def 4;
reconsider a = a as Point of (TOP-REAL 2) by A40;
a `1 <= p `1 by A42, TOPREAL1:def 15;
then a `1 <= t by A37, EUCLID:56;
then a `1 < ((GoB (Rotate f,(W-min (L~ f)))) * 1,(i_s_w (Rotate f,(W-min (L~ f))))) `1 by A38, A39, XXREAL_0:2;
then a `1 < W-bound (L~ (Rotate f,(W-min (L~ f)))) by A27, A5, A28, EUCLID:56;
hence contradiction by A41, PSCOMP_1:71; :: thesis: verum

end;

then A43: west_halfline p c= UBD (L~ (Rotate f,(W-min (L~ f)))) by JORDAN2C:134;
p in west_halfline p by TOPREAL1:45;
then LeftComp (Rotate f,(W-min (L~ f))) meets UBD (L~ (Rotate f,(W-min (L~ f)))) by A3, A34, A43, XBOOLE_0:3;
then Rotate f,(W-min (L~ f)) is clockwise_oriented by A35, GOBOARD9:3, JORDAN1H:49;
hence f is clockwise_oriented by JORDAN1H:48; :: thesis: verum