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]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}
let G be Go-board; :: thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} )
assume that
A1: 1 <= j and
A2: j < width G ; :: thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}
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]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) or x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} )
assume A3: x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) ; :: thesis: x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}
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 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) and
A5: 0 <= r and
A6: r <= 1 by A3;

now :: thesis: ( ( r = 1 & p in {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} ) 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 {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}

then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by A4, RLVECT_1:10
.= 1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) by RLVECT_1:4
.= (1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) by RLVECT_1:def 8 ;
hence p in {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} 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);
set s3 = r * (1 / 2);
set r2 = (G * (1,1)) `1 ;
set s1 = (G * (1,j)) `2 ;
set s2 = (G * (1,(j + 1))) `2 ;
A8: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ;
A9: j + 1 <= width G by A2, NAT_1:13;
0 <> len G by MATRIX_0:def 10;
then A10: 1 <= len G by NAT_1:14;
j < j + 1 by XREAL_1:29;
then A11: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A9, A10, GOBOARD5:4;
then A12: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6;
then A13: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64;
A14: 1 - r > 0 by A7, XREAL_1:50;
then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68;
then ((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 A12, XREAL_1:68;
then A16: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) by A13, A8, XREAL_1:8;
(G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r) by A14, XREAL_1:29;
then A17: ((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 by XREAL_1:19;
A18: 1 <= j + 1 by A1, NAT_1:13;
A19: 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 A18, A9, A10, GOBOARD5:2 ;
A20: ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A11, XREAL_1:6;
then A21: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64;
A22: 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;
A23: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ;
((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 A15, A20, XREAL_1:68;
then A24: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A21, A23, XREAL_1:8;
A25: 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, A10, GOBOARD5:2 ;
p = (((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by A4, RLVECT_1:34
.= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by RLVECT_1:def 7
.= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by EUCLID:58
.= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[(1 - r),0]|) + ((r * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) by RLVECT_1:def 7
.= ((((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 * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) by A19, A25, 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 * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by A19, A25, EUCLID:56
.= (|[(((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 * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58
.= (|[(((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 * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| 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 * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:62
.= |[(((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)) + ((r * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ;
hence p in Int (cell (G,0,j)) by A17, A16, A24, A22; :: thesis: verum

end;

end;

end;

hence x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by XBOOLE_0:def 3; :: thesis: verum