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

now :: thesis: ( ( r = 1 & p in {((G * (1,(j + 1))) - |[1,0]|)} ) or ( r < 1 & p in Int (cell (G,0,j)) ) )

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

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

then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,(j + 1))) - |[1,0]|)) by A4, RLVECT_1:10
.= 1 * ((G * (1,(j + 1))) - |[1,0]|) by RLVECT_1:4
.= (G * (1,(j + 1))) - |[1,0]| by RLVECT_1:def 8 ;
hence p in {((G * (1,(j + 1))) - |[1,0]|)} by TARSKI:def 1; :: thesis: verum

end;

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

set r3 = (1 - r) * (1 / 2);
1 - r > 0 by A7, XREAL_1:50;
then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68;
set r2 = (G * (1,1)) `1 ;
set s1 = (G * (1,j)) `2 ;
set s2 = (G * (1,(j + 1))) `2 ;
A9: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + (r * ((G * (1,j)) `2)) = (G * (1,j)) `2 ;
A10: j + 1 <= width G by A2, NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then A11: 1 <= len G by NAT_1:14;
j < j + 1 by XREAL_1:29;
then A12: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A10, A11, GOBOARD5:4;
then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6;
then A13: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68;
r * ((G * (1,j)) `2) <= r * ((G * (1,(j + 1))) `2) by A5, A12, XREAL_1:64;
then A14: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) by A13, A9, XREAL_1:8;
A15: 1 <= j + 1 by A1, NAT_1:13;
A16: G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| by EUCLID:53
.= |[((G * (1,1)) `1),((G * (1,j)) `2)]| by A1, A2, A11, GOBOARD5:2 ;
(G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29;
then A17: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19;
A18: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) = (G * (1,(j + 1))) `2 ;
((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A12, XREAL_1:6;
then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68;
then A19: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) < (G * (1,(j + 1))) `2 by A18, XREAL_1:8;
A20: Int (cell (G,0,j)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th20;
A21: G * (1,(j + 1)) = |[((G * (1,(j + 1))) `1),((G * (1,(j + 1))) `2)]| by EUCLID:53
.= |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| by A15, A10, A11, GOBOARD5:2 ;
p = (((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by A4, RLVECT_1:34
.= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by RLVECT_1:def 7
.= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by EUCLID:58
.= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (r * (|[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| - |[1,0]|)) by A21, A16, EUCLID:56
.= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + ((r * |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]|) - (r * |[1,0]|)) by RLVECT_1:34
.= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,(j + 1))) `2))]| - (r * |[1,0]|)) by EUCLID:58
.= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,(j + 1))) `2))]| - |[(r * 1),(r * 0)]|) by EUCLID:58
.= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:62
.= (|[(((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:58
.= |[((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]| + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:62
.= |[(((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)) + ((r * ((G * (1,1)) `1)) - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)))]| by EUCLID:56 ;
hence p in Int (cell (G,0,j)) by A17, A14, A19, A20; :: thesis: verum

end;

end;

end;

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