let p be Point of (TOP-REAL 2); :: thesis:
let f be non constant standard clockwise_oriented special_circular_sequence; :: thesis:
set g = Rotate f,(N-min (L~ f));
A1: L~ f = L~ (Rotate f,(N-min (L~ f))) by REVROT_1:33;
N-min (L~ f) in rng f by SPRECT_2:43;
then A2: (Rotate f,(N-min (L~ f))) /. 1 = N-min (L~ (Rotate f,(N-min (L~ f)))) by A1, FINSEQ_6:98;
reconsider u = p as Point of (Euclid 2) by XX;
assume p in RightComp f ; :: thesis:
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_1:55;
then consider r being real number such that
A3: r > 0 and
A4: Ball u,r c= RightComp (Rotate f,(N-min (L~ f))) by GOBOARD6:8;
reconsider r = r as Real by XREAL_0:def 1;
A5: W-bound (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f)))))) = W-bound (L~ (Rotate f,(N-min (L~ f)))) by SPRECT_1:66;
A6: W-bound (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f)))))) = lower_bound (proj1 .: (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f))))))) by SPRECT_1:48;
A7: proj1 .: (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f)))))) is bounded_below by SPRECT_1:46;
set x = |[((p `1 ) - (r / 2)),(N-bound (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f)))))))]|;
reconsider k = |[((p `1 ) - (r / 2)),(p `2 )]| as Point of (Euclid 2) by XX;
A8: r / 2 > 0 by A3, XREAL_1:141;
dist u,k = (Pitag_dist 2) . 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 TOPREAL3:12
.= sqrt ((((p `1 ) - ((p `1 ) - (r / 2))) ^2 ) + (((p `2 ) - (|[((p `1 ) - (r / 2)),(p `2 )]| `2 )) ^2 )) by EUCLID:56
.= sqrt ((((p `1 ) - ((p `1 ) - (r / 2))) ^2 ) + (((p `2 ) - (p `2 )) ^2 )) by EUCLID:56
.= r / 2 by A8, SQUARE_1:89 ;
then dist u,k < r / 1 by A3, XREAL_1:78;
then A9: k in Ball u,r by METRIC_1:12;
RightComp (Rotate f,(N-min (L~ f))) misses LeftComp (Rotate f,(N-min (L~ f))) by Th24;
then A10: not |[((p `1 ) - (r / 2)),(p `2 )]| in LeftComp (Rotate f,(N-min (L~ f))) by A4, A9, XBOOLE_0:3;
then |[((p `1 ) - (r / 2)),(p `2 )]| `1 <= E-bound (L~ (Rotate f,(N-min (L~ f)))) by A2, JORDAN2C:119;
then (p `1 ) - (r / 2) <= E-bound (L~ (Rotate f,(N-min (L~ f)))) by EUCLID:56;
then A11: |[((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:56;
|[((p `1 ) - (r / 2)),(p `2 )]| `1 >= W-bound (L~ (Rotate f,(N-min (L~ f)))) by A2, A10, JORDAN2C:118;
then (p `1 ) - (r / 2) >= W-bound (L~ (Rotate f,(N-min (L~ f)))) by EUCLID:56;
then A12: |[((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:56;
|[((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:56;
then A13: |[((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_1:68;
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 (TOP-REAL 2) : ( 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_1:27;
then A14: |[((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 A11, A12, A13;
A15: 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_3:26;
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 28
.= (p `1 ) - (r / 2) by EUCLID:56 ;
then (p `1 ) - (r / 2) in proj1 .: (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f)))))) by A14, A15, FUNCT_2:43;
then A16: lower_bound (proj1 .: (L~ (SpStSeq (L~ (Rotate f,(N-min (L~ f))))))) <= (p `1 ) - (r / 2) by A7, SEQ_4:def 5;
(p `1 ) - (r / 2) < (p `1 ) - 0 by A8, XREAL_1:17;
hence W-bound (L~ f) < p `1 by A1, A5, A6, A16, XXREAL_0:2; :: thesis: