let i be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i < len G holds
LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((G * (i + 1),1) - |[0 ,1]|) c= (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)}
let G be Go-board; :: thesis: ( 1 <= i & i < len G implies LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((G * (i + 1),1) - |[0 ,1]|) c= (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)} )
assume that
A1: 1 <= i and
A2: i < len G ; :: thesis: LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((G * (i + 1),1) - |[0 ,1]|) c= (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((G * (i + 1),1) - |[0 ,1]|) or x in (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)} )
assume A3: x in LSeg (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|),((G * (i + 1),1) - |[0 ,1]|) ; :: thesis: x in (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)}
then reconsider p = x as Point of (TOP-REAL 2) ;
consider r being Real such that
A4: p = ((1 - r) * (((1 / 2) * ((G * i,1) + (G * (i + 1),1))) - |[0 ,1]|)) + (r * ((G * (i + 1),1) - |[0 ,1]|)) and
A5: 0 <= r and
A6: r <= 1 by A3;

now

per cases ( r = 1 or r < 1 ) by A6, XXREAL_0:1;

case r = 1 ; :: thesis: p in {((G * (i + 1),1) - |[0 ,1]|)}

then p = (0. (TOP-REAL 2)) + (1 * ((G * (i + 1),1) - |[0 ,1]|)) by A4, EUCLID:33
.= 1 * ((G * (i + 1),1) - |[0 ,1]|) by EUCLID:31
.= (G * (i + 1),1) - |[0 ,1]| by EUCLID:33 ;
hence p in {((G * (i + 1),1) - |[0 ,1]|)} by TARSKI:def 1; :: thesis: verum

end;

case A7: r < 1 ; :: thesis: p in Int (cell G,i,0 )

set r3 = (1 - r) * (1 / 2);
1 - r > 0 by A7, XREAL_1:52;
then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:70;
set s1 = (G * 1,1) `2 ;
set r1 = (G * i,1) `1 ;
set r2 = (G * (i + 1),1) `1 ;
A9: (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 ))) + (r * ((G * i,1) `1 )) = (G * i,1) `1 ;
A10: i + 1 <= len G by A2, NAT_1:13;
0 <> width G by GOBOARD1:def 5;
then A11: 1 <= width G by NAT_1:14;
i < i + 1 by XREAL_1:31;
then A12: (G * i,1) `1 < (G * (i + 1),1) `1 by A1, A10, A11, GOBOARD5:4;
then ((G * i,1) `1 ) + ((G * i,1) `1 ) < ((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) by XREAL_1:8;
then A13: ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * i,1) `1 )) < ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 )) by A8, XREAL_1:70;
r * ((G * i,1) `1 ) <= r * ((G * (i + 1),1) `1 ) by A5, A12, XREAL_1:66;
then A14: (G * i,1) `1 < (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + (r * ((G * (i + 1),1) `1 )) by A13, A9, XREAL_1:10;
A15: 1 <= i + 1 by A1, NAT_1:13;
A16: G * i,1 = |[((G * i,1) `1 ),((G * i,1) `2 )]| by EUCLID:57
.= |[((G * i,1) `1 ),((G * 1,1) `2 )]| by A1, A2, A11, GOBOARD5:2 ;
(G * 1,1) `2 < ((G * 1,1) `2 ) + 1 by XREAL_1:31;
then A17: ((G * 1,1) `2 ) - 1 < (G * 1,1) `2 by XREAL_1:21;
A18: (((1 - r) * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ))) + (r * ((G * (i + 1),1) `1 )) = (G * (i + 1),1) `1 ;
((G * i,1) `1 ) + ((G * (i + 1),1) `1 ) < ((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 ) by A12, XREAL_1:8;
then ((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 )) < ((1 - r) * (1 / 2)) * (((G * (i + 1),1) `1 ) + ((G * (i + 1),1) `1 )) by A8, XREAL_1:70;
then A19: (((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + (r * ((G * (i + 1),1) `1 )) < (G * (i + 1),1) `1 by A18, XREAL_1:10;
A20: Int (cell G,i,0 ) = { |[r9,s9]| where r9, s9 is Real : ( (G * i,1) `1 < r9 & r9 < (G * (i + 1),1) `1 & s9 < (G * 1,1) `2 ) } by A1, A2, Th27;
A21: G * (i + 1),1 = |[((G * (i + 1),1) `1 ),((G * (i + 1),1) `2 )]| by EUCLID:57
.= |[((G * (i + 1),1) `1 ),((G * 1,1) `2 )]| by A15, A10, A11, GOBOARD5:2 ;
p = (((1 - r) * ((1 / 2) * ((G * i,1) + (G * (i + 1),1)))) - ((1 - r) * |[0 ,1]|)) + (r * ((G * (i + 1),1) - |[0 ,1]|)) by A4, EUCLID:53
.= ((((1 - r) * (1 / 2)) * ((G * i,1) + (G * (i + 1),1))) - ((1 - r) * |[0 ,1]|)) + (r * ((G * (i + 1),1) - |[0 ,1]|)) by EUCLID:34
.= ((((1 - r) * (1 / 2)) * ((G * i,1) + (G * (i + 1),1))) - |[((1 - r) * 0 ),((1 - r) * 1)]|) + (r * ((G * (i + 1),1) - |[0 ,1]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|) - |[0 ,(1 - r)]|) + (r * (|[((G * (i + 1),1) `1 ),((G * 1,1) `2 )]| - |[0 ,1]|)) by A21, A16, EUCLID:60
.= ((((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|) - |[0 ,(1 - r)]|) + ((r * |[((G * (i + 1),1) `1 ),((G * 1,1) `2 )]|) - (r * |[0 ,1]|)) by EUCLID:53
.= ((((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|) - |[0 ,(1 - r)]|) + (|[(r * ((G * (i + 1),1) `1 )),(r * ((G * 1,1) `2 ))]| - (r * |[0 ,1]|)) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|) - |[0 ,(1 - r)]|) + (|[(r * ((G * (i + 1),1) `1 )),(r * ((G * 1,1) `2 ))]| - |[(r * 0 ),(r * 1)]|) by EUCLID:62
.= ((((1 - r) * (1 / 2)) * |[(((G * i,1) `1 ) + ((G * (i + 1),1) `1 )),(((G * 1,1) `2 ) + ((G * 1,1) `2 ))]|) - |[0 ,(1 - r)]|) + |[((r * ((G * (i + 1),1) `1 )) - 0 ),((r * ((G * 1,1) `2 )) - r)]| by EUCLID:66
.= (|[(((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))),(((1 - r) * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 )))]| - |[0 ,(1 - r)]|) + |[((r * ((G * (i + 1),1) `1 )) - 0 ),((r * ((G * 1,1) `2 )) - r)]| by EUCLID:62
.= |[((((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) - 0 ),((((1 - r) * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 ))) - (1 - r))]| + |[((r * ((G * (i + 1),1) `1 )) - 0 ),((r * ((G * 1,1) `2 )) - r)]| by EUCLID:66
.= |[((((1 - r) * (1 / 2)) * (((G * i,1) `1 ) + ((G * (i + 1),1) `1 ))) + (r * ((G * (i + 1),1) `1 ))),(((((1 - r) * (1 / 2)) * (((G * 1,1) `2 ) + ((G * 1,1) `2 ))) - (1 - r)) + ((r * ((G * 1,1) `2 )) - r))]| by EUCLID:60 ;
hence p in Int (cell G,i,0 ) by A17, A14, A19, A20; :: thesis: verum

end;

end;

end;

hence x in (Int (cell G,i,0 )) \/ {((G * (i + 1),1) - |[0 ,1]|)} by XBOOLE_0:def 3; :: thesis: verum