let i, j be Element of NAT ; for G being Go-board st 1 <= i & i < len G & 1 <= j & j + 1 < width G holds
LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) c= ((Int (cell G,i,j)) \/ (Int (cell G,i,(j + 1)))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}
let G be Go-board; ( 1 <= i & i < len G & 1 <= j & j + 1 < width G implies LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) c= ((Int (cell G,i,j)) \/ (Int (cell G,i,(j + 1)))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))} )
assume that
A1:
1 <= i
and
A2:
i < len G
and
A3:
1 <= j
and
A4:
j + 1 < width G
; LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) c= ((Int (cell G,i,j)) \/ (Int (cell G,i,(j + 1)))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}
set p1 = G * i,j;
set p2 = G * i,(j + 1);
set q2 = G * (i + 1),(j + 1);
set q3 = G * (i + 1),(j + 2);
set r = (((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ));
A5:
j + 1 >= 1
by NAT_1:11;
set I1 = Int (cell G,i,j);
set I2 = Int (cell G,i,(j + 1));
j <= j + 1
by NAT_1:11;
then A6:
j < width G
by A4, XXREAL_0:2;
then A7:
LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1)))) c= (Int (cell G,i,j)) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}
by A1, A2, A3, Th44;
j < j + 1
by XREAL_1:31;
then
(G * i,j) `2 < (G * i,(j + 1)) `2
by A1, A2, A3, A4, GOBOARD5:5;
then A8:
((G * i,(j + 1)) `2 ) - ((G * i,j) `2 ) > 0
by XREAL_1:52;
A9:
(j + 1) + 1 = j + (1 + 1)
;
then A10:
j + 2 >= 1
by NAT_1:11;
A11:
j + (1 + 1) <= width G
by A4, A9, NAT_1:13;
A12:
( i + 1 >= 1 & i + 1 <= len G )
by A2, NAT_1:11, NAT_1:13;
then A13: (G * (i + 1),(j + 1)) `1 =
(G * (i + 1),1) `1
by A4, A5, GOBOARD5:3
.=
(G * (i + 1),(j + 2)) `1
by A11, A10, A12, GOBOARD5:3
;
A14: (G * (i + 1),(j + 1)) `2 =
(G * 1,(j + 1)) `2
by A4, A5, A12, GOBOARD5:2
.=
(G * i,(j + 1)) `2
by A1, A2, A4, A5, GOBOARD5:2
;
j + 1 < j + 2
by XREAL_1:8;
then
(G * (i + 1),(j + 1)) `2 < (G * (i + 1),(j + 2)) `2
by A5, A11, A12, GOBOARD5:5;
then A15:
((G * i,(j + 1)) `2 ) - ((G * i,j) `2 ) < ((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )
by A14, XREAL_1:11;
then A16:
((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )) = ((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )
by A8, XCMPLX_1:88;
(G * i,j) `1 =
(G * i,1) `1
by A1, A2, A3, A6, GOBOARD5:3
.=
(G * i,(j + 1)) `1
by A1, A2, A4, A5, GOBOARD5:3
;
then A17: ((G * i,(j + 1)) + (G * (i + 1),(j + 1))) `1 =
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) `1 ) + ((G * (i + 1),(j + 1)) `1 ))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) `1 ) + ((G * (i + 1),(j + 2)) `1 )))
by A13, Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) + (G * (i + 1),(j + 1))) `1 )) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) `1 ) + ((G * (i + 1),(j + 2)) `1 )))
by Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) + (G * (i + 1),(j + 1))) `1 )) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) + (G * (i + 1),(j + 2))) `1 ))
by Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) + (G * (i + 1),(j + 1))) `1 )) + ((((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) `1 )
by Lm3
.=
(((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1)))) `1 ) + ((((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) `1 )
by Lm3
.=
(((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1)))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2))))) `1
by Lm1
;
((G * i,(j + 1)) + (G * (i + 1),(j + 1))) `2 =
((G * i,(j + 1)) `2 ) + ((((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) + (1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))))) * ((G * (i + 1),(j + 1)) `2 ))
by Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) `2 ) + ((G * (i + 1),(j + 1)) `2 ))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) `2 ) + ((G * (i + 1),(j + 2)) `2 )))
by A14, A16
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) `2 ) + ((G * (i + 1),(j + 1)) `2 ))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) + (G * (i + 1),(j + 2))) `2 ))
by Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) + (G * (i + 1),(j + 1))) `2 )) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * (((G * i,(j + 1)) + (G * (i + 1),(j + 2))) `2 ))
by Lm1
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * (((G * i,j) + (G * (i + 1),(j + 1))) `2 )) + ((((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) `2 )
by Lm3
.=
(((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1)))) `2 ) + ((((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) `2 )
by Lm3
.=
(((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1)))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2))))) `2
by Lm1
;
then ((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1)))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) =
|[(((G * i,(j + 1)) + (G * (i + 1),(j + 1))) `1 ),(((G * i,(j + 1)) + (G * (i + 1),(j + 1))) `2 )]|
by A17, EUCLID:57
.=
(G * i,(j + 1)) + (G * (i + 1),(j + 1))
by EUCLID:57
;
then A18: (1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))) =
((1 / 2) * ((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,j) + (G * (i + 1),(j + 1))))) + ((1 / 2) * (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))))
by EUCLID:36
.=
(((1 / 2) * (1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))))) * ((G * i,j) + (G * (i + 1),(j + 1)))) + ((1 / 2) * (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))))
by EUCLID:34
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))))) + ((1 / 2) * (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))))
by EUCLID:34
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))))) + (((1 / 2) * ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2))))
by EUCLID:34
.=
((1 - ((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )))) * ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))))) + (((((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 ))) * ((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))))
by EUCLID:34
;
(((G * i,(j + 1)) `2 ) - ((G * i,j) `2 )) / (((G * (i + 1),(j + 2)) `2 ) - ((G * i,j) `2 )) < 1
by A15, A8, XREAL_1:191;
then
(1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))) in LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2))))
by A15, A8, A18;
then A19:
LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) = (LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))) \/ (LSeg ((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))))
by TOPREAL1:11;
A20: ((Int (cell G,i,j)) \/ (Int (cell G,i,(j + 1)))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))} =
(Int (cell G,i,j)) \/ ((Int (cell G,i,(j + 1))) \/ ({((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))} \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}))
by XBOOLE_1:4
.=
(Int (cell G,i,j)) \/ (((Int (cell G,i,(j + 1))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))})
by XBOOLE_1:4
.=
((Int (cell G,i,j)) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}) \/ ((Int (cell G,i,(j + 1))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))})
by XBOOLE_1:4
;
LSeg ((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) c= (Int (cell G,i,(j + 1))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}
by A1, A2, A4, A5, A9, Th46;
hence
LSeg ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))),((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 2)))) c= ((Int (cell G,i,j)) \/ (Int (cell G,i,(j + 1)))) \/ {((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),(j + 1))))}
by A19, A7, A20, XBOOLE_1:13; verum