let f be non constant standard special_circular_sequence; :: thesis: LSeg (((GoB f) * 1,1) - |[1,1]|),(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|) misses L~ f
A1:
1 <= len (GoB f)
by GOBOARD7:34;
now let p be
Point of
(TOP-REAL 2);
:: thesis: ( p in LSeg (((GoB f) * 1,1) - |[1,1]|),(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|) implies p `1 < ((GoB f) * 1,1) `1 )assume A2:
p in LSeg (((GoB f) * 1,1) - |[1,1]|),
(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|)
;
:: thesis: p `1 < ((GoB f) * 1,1) `1
1
< width (GoB f)
by GOBOARD7:35;
then
1
+ 1
<= width (GoB f)
by NAT_1:13;
then A3:
((GoB f) * 1,2) `1 = ((GoB f) * 1,1) `1
by A1, GOBOARD5:3;
(((GoB f) * 1,1) - |[1,1]|) `1 =
(((GoB f) * 1,1) `1 ) - (|[1,1]| `1 )
by TOPREAL3:8
.=
(((GoB f) * 1,1) `1 ) - 1
by EUCLID:56
;
then A4:
((GoB f) * 1,1) - |[1,1]| = |[((((GoB f) * 1,1) `1 ) - 1),((((GoB f) * 1,1) - |[1,1]|) `2 )]|
by EUCLID:57;
(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|) `1 =
(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) `1 ) - (|[1,0 ]| `1 )
by TOPREAL3:8
.=
((1 / 2) * ((((GoB f) * 1,1) + ((GoB f) * 1,2)) `1 )) - (|[1,0 ]| `1 )
by TOPREAL3:9
.=
((1 / 2) * ((((GoB f) * 1,1) `1 ) + (((GoB f) * 1,1) `1 ))) - (|[1,0 ]| `1 )
by A3, TOPREAL3:7
.=
(1 * (((GoB f) * 1,1) `1 )) - 1
by EUCLID:56
;
then
((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]| = |[((((GoB f) * 1,1) `1 ) - 1),((((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|) `2 )]|
by EUCLID:57;
then
p `1 = (((GoB f) * 1,1) `1 ) - 1
by A2, A4, TOPREAL3:17;
hence
p `1 < ((GoB f) * 1,1) `1
by XREAL_1:46;
:: thesis: verum end;
hence
LSeg (((GoB f) * 1,1) - |[1,1]|),(((1 / 2) * (((GoB f) * 1,1) + ((GoB f) * 1,2))) - |[1,0 ]|) misses L~ f
by Th21; :: thesis: verum