let p be Point of (); :: thesis: for f being V8() standard clockwise_oriented special_circular_sequence st p in RightComp f holds
W-bound (L~ f) < p `1

let f be V8() standard clockwise_oriented special_circular_sequence; :: thesis: ( p in RightComp f implies W-bound (L~ f) < p `1 )
set g = Rotate (f,(N-min (L~ f)));
A1: L~ f = L~ (Rotate (f,(N-min (L~ f)))) by REVROT_1:33;
reconsider u = p as Point of () by EUCLID:22;
assume p in RightComp f ; :: thesis: W-bound (L~ f) < p `1
then p in RightComp (Rotate (f,(N-min (L~ f)))) by REVROT_1:37;
then u in Int (RightComp (Rotate (f,(N-min (L~ f))))) by TOPS_1:23;
then consider r being Real such that
A2: r > 0 and
A3: Ball (u,r) c= RightComp (Rotate (f,(N-min (L~ f)))) by GOBOARD6:5;
reconsider r = r as Real ;
reconsider k = |[((p `1) - (r / 2)),(p `2)]| as Point of () by EUCLID:22;
dist (u,k) = () . (u,k) by METRIC_1:def 1
.= sqrt ((((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2)) by TOPREAL3:7
.= sqrt ((((p `1) - ((p `1) - (r / 2))) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2)) by EUCLID:52
.= sqrt ((((p `1) - ((p `1) - (r / 2))) ^2) + (((p `2) - (p `2)) ^2)) by EUCLID:52
.= r / 2 by ;
then dist (u,k) < r / 1 by ;
then A4: k in Ball (u,r) by METRIC_1:11;
RightComp (Rotate (f,(N-min (L~ f)))) misses LeftComp (Rotate (f,(N-min (L~ f)))) by Th14;
then A5: not |[((p `1) - (r / 2)),(p `2)]| in LeftComp (Rotate (f,(N-min (L~ f)))) by ;
set x = |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]|;
A6: LSeg ((NW-corner (L~ (Rotate (f,(N-min (L~ f)))))),(NE-corner (L~ (Rotate (f,(N-min (L~ f))))))) c= L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f)))))) by SPRECT_3:14;
A7: proj1 . |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| = |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| `1 by PSCOMP_1:def 5
.= (p `1) - (r / 2) by EUCLID:52 ;
N-min (L~ f) in rng f by SPRECT_2:39;
then A8: (Rotate (f,(N-min (L~ f)))) /. 1 = N-min (L~ (Rotate (f,(N-min (L~ f))))) by ;
then |[((p `1) - (r / 2)),(p `2)]| `1 <= E-bound (L~ (Rotate (f,(N-min (L~ f))))) by ;
then (p `1) - (r / 2) <= E-bound (L~ (Rotate (f,(N-min (L~ f))))) by EUCLID:52;
then A9: |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| `1 <= E-bound (L~ (Rotate (f,(N-min (L~ f))))) by EUCLID:52;
|[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| `2 = N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))) by EUCLID:52;
then A10: |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| `2 = N-bound (L~ (Rotate (f,(N-min (L~ f))))) by SPRECT_1:60;
|[((p `1) - (r / 2)),(p `2)]| `1 >= W-bound (L~ (Rotate (f,(N-min (L~ f))))) by ;
then (p `1) - (r / 2) >= W-bound (L~ (Rotate (f,(N-min (L~ f))))) by EUCLID:52;
then A11: |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| `1 >= W-bound (L~ (Rotate (f,(N-min (L~ f))))) by EUCLID:52;
LSeg ((NW-corner (L~ (Rotate (f,(N-min (L~ f)))))),(NE-corner (L~ (Rotate (f,(N-min (L~ f))))))) = { q where q is Point of () : ( q `1 <= E-bound (L~ (Rotate (f,(N-min (L~ f))))) & q `1 >= W-bound (L~ (Rotate (f,(N-min (L~ f))))) & q `2 = N-bound (L~ (Rotate (f,(N-min (L~ f))))) ) } by SPRECT_1:25;
then |[((p `1) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))))]| in LSeg ((NW-corner (L~ (Rotate (f,(N-min (L~ f)))))),(NE-corner (L~ (Rotate (f,(N-min (L~ f))))))) by A9, A11, A10;
then ( proj1 .: (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))) is bounded_below & (p `1) - (r / 2) in proj1 .: (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))) ) by ;
then A12: lower_bound (proj1 .: (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f)))))))) <= (p `1) - (r / 2) by SEQ_4:def 2;
r / 2 > 0 by ;
then A13: (p `1) - (r / 2) < (p `1) - 0 by XREAL_1:15;
( W-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))) = W-bound (L~ (Rotate (f,(N-min (L~ f))))) & W-bound (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f))))))) = lower_bound (proj1 .: (L~ (SpStSeq (L~ (Rotate (f,(N-min (L~ f)))))))) ) by ;
hence W-bound (L~ f) < p `1 by ; :: thesis: verum