let i be 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]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),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]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} )
assume that
A1: 1 <= i and
A2: i < len G ; :: thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) or x in (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} )
assume A3: x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) ; :: thesis: x in (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),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 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) and
A5: 0 <= r and
A6: r <= 1 by A3;

now :: thesis: ( ( r = 1 & p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} ) or ( r < 1 & p in Int (cell (G,i,0)) ) )

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

case r = 1 ; :: thesis: p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}

then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by A4, RLVECT_1:10
.= 1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) by RLVECT_1:4
.= (1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))) by RLVECT_1:def 8 ;
hence p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),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);
set s3 = r * (1 / 2);
set s2 = (G * (1,1)) `2 ;
set r1 = (G * (i,1)) `1 ;
set r2 = (G * ((i + 1),1)) `1 ;
A8: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ;
A9: i + 1 <= len G by A2, NAT_1:13;
0 <> width G by MATRIX_0:def 10;
then A10: 1 <= width G by NAT_1:14;
i < i + 1 by XREAL_1:29;
then A11: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A9, A10, GOBOARD5:3;
then A12: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6;
then A13: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) 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 * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A12, XREAL_1:68;
then A16: (G * (i,1)) `1 < (((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 A13, A8, XREAL_1:8;
(G * (1,1)) `2 < ((G * (1,1)) `2) + (1 - r) by A14, XREAL_1:29;
then A17: ((G * (1,1)) `2) - (1 - r) < (G * (1,1)) `2 by XREAL_1:19;
A18: 1 <= i + 1 by A1, NAT_1:13;
A19: G * ((i + 1),1) = |[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]| by EUCLID:53
.= |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| by A18, A9, A10, GOBOARD5:1 ;
A20: ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A11, XREAL_1:6;
then A21: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A5, XREAL_1:64;
A22: 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, Th24;
A23: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ;
((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 A15, A20, XREAL_1:68;
then A24: (((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))) < (G * ((i + 1),1)) `1 by A21, A23, XREAL_1:8;
A25: G * (i,1) = |[((G * (i,1)) `1),((G * (i,1)) `2)]| by EUCLID:53
.= |[((G * (i,1)) `1),((G * (1,1)) `2)]| by A1, A2, A10, GOBOARD5:1 ;
p = (((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by A4, RLVECT_1:34
.= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by RLVECT_1:def 7
.= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by EUCLID:58
.= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,(1 - r)]|) + ((r * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) by RLVECT_1:def 7
.= ((((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 * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) by A19, A25, EUCLID:56
.= ((((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 * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) by A19, A25, EUCLID:56
.= (|[(((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 * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) by EUCLID:58
.= (|[(((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 * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| by EUCLID:58
.= |[((((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 * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| by EUCLID:62
.= |[((((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)))),(((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r)) + ((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))))]| by EUCLID:56 ;
hence p in Int (cell (G,i,0)) by A17, A16, A24, A22; :: thesis: verum

end;

end;

end;

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