let G be Go-board; for i1, j1, i2, j2 being Nat st 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds
( j1 = j2 & |.(i1 - i2).| <= 1 )
let i1, j1, i2, j2 be Nat; ( 1 <= i1 & i1 + 1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) implies ( j1 = j2 & |.(i1 - i2).| <= 1 ) )
assume that
A1:
1 <= i1
and
A2:
i1 + 1 <= len G
and
A3:
( 1 <= j1 & j1 <= width G )
and
A4:
1 <= i2
and
A5:
i2 + 1 <= len G
and
A6:
( 1 <= j2 & j2 <= width G )
; ( not LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) or ( j1 = j2 & |.(i1 - i2).| <= 1 ) )
A7:
1 <= i1 + 1
by A1, NAT_1:13;
A8:
i1 < len G
by A2, NAT_1:13;
assume
LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))
; ( j1 = j2 & |.(i1 - i2).| <= 1 )
then consider x being object such that
A9:
x in LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1)))
and
A10:
x in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2)))
by XBOOLE_0:3;
reconsider p = x as Point of (TOP-REAL 2) by A9;
consider r1 being Real such that
A11:
p = ((1 - r1) * (G * (i1,j1))) + (r1 * (G * ((i1 + 1),j1)))
and
A12:
r1 >= 0
and
A13:
r1 <= 1
by A9;
consider r2 being Real such that
A14:
p = ((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2)))
and
A15:
r2 >= 0
and
A16:
r2 <= 1
by A10;
A17:
1 <= i2 + 1
by A4, NAT_1:13;
A18:
i2 < len G
by A5, NAT_1:13;
assume A19:
( not j1 = j2 or not |.(i1 - i2).| <= 1 )
; contradiction
per cases
( j1 <> j2 or |.(i1 - i2).| > 1 )
by A19;
suppose
j1 <> j2
;
contradictionthen A20:
(
j1 < j2 or
j2 < j1 )
by XXREAL_0:1;
A21:
(G * (i2,j2)) `2 =
(G * (1,j2)) `2
by A4, A6, A18, GOBOARD5:1
.=
(G * ((i2 + 1),j2)) `2
by A5, A6, A17, GOBOARD5:1
;
(G * (i1,j1)) `2 =
(G * (1,j1)) `2
by A1, A3, A8, GOBOARD5:1
.=
(G * ((i1 + 1),j1)) `2
by A2, A3, A7, GOBOARD5:1
;
then 1
* ((G * (i1,j1)) `2) =
((1 - r1) * ((G * (i1,j1)) `2)) + (r1 * ((G * ((i1 + 1),j1)) `2))
.=
(((1 - r1) * (G * (i1,j1))) `2) + (r1 * ((G * ((i1 + 1),j1)) `2))
by TOPREAL3:4
.=
(((1 - r1) * (G * (i1,j1))) `2) + ((r1 * (G * ((i1 + 1),j1))) `2)
by TOPREAL3:4
.=
p `2
by A11, TOPREAL3:2
.=
(((1 - r2) * (G * (i2,j2))) `2) + ((r2 * (G * ((i2 + 1),j2))) `2)
by A14, TOPREAL3:2
.=
((1 - r2) * ((G * (i2,j2)) `2)) + ((r2 * (G * ((i2 + 1),j2))) `2)
by TOPREAL3:4
.=
((1 - r2) * ((G * (i2,j2)) `2)) + (r2 * ((G * ((i2 + 1),j2)) `2))
by TOPREAL3:4
.=
(G * (1,j2)) `2
by A5, A6, A17, A21, GOBOARD5:1
.=
(G * (i1,j2)) `2
by A1, A6, A8, GOBOARD5:1
;
hence
contradiction
by A1, A3, A6, A8, A20, GOBOARD5:4;
verum end; suppose A22:
|.(i1 - i2).| > 1
;
contradictionA23:
(G * ((i2 + 1),j2)) `1 =
(G * ((i2 + 1),1)) `1
by A5, A6, A17, GOBOARD5:2
.=
(G * ((i2 + 1),j1)) `1
by A3, A5, A17, GOBOARD5:2
;
A24:
(G * (i2,j2)) `1 =
(G * (i2,1)) `1
by A4, A6, A18, GOBOARD5:2
.=
(G * (i2,j1)) `1
by A3, A4, A18, GOBOARD5:2
;
A25:
((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) =
(((1 - r1) * (G * (i1,j1))) `1) + (r1 * ((G * ((i1 + 1),j1)) `1))
by TOPREAL3:4
.=
(((1 - r1) * (G * (i1,j1))) `1) + ((r1 * (G * ((i1 + 1),j1))) `1)
by TOPREAL3:4
.=
p `1
by A11, TOPREAL3:2
.=
(((1 - r2) * (G * (i2,j2))) `1) + ((r2 * (G * ((i2 + 1),j2))) `1)
by A14, TOPREAL3:2
.=
((1 - r2) * ((G * (i2,j2)) `1)) + ((r2 * (G * ((i2 + 1),j2))) `1)
by TOPREAL3:4
.=
((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1))
by A23, A24, TOPREAL3:4
;
now contradictionper cases
( i1 + 1 < i2 or i2 + 1 < i1 )
by A22, Th1;
suppose A26:
i1 + 1
< i2
;
contradiction
i2 < i2 + 1
by XREAL_1:29;
then
(G * (i2,j1)) `1 < (G * ((i2 + 1),j1)) `1
by A3, A4, A5, GOBOARD5:3;
then
(
((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * (i2,j1)) `1)) = 1
* ((G * (i2,j1)) `1) &
r2 * ((G * (i2,j1)) `1) <= r2 * ((G * ((i2 + 1),j1)) `1) )
by A15, XREAL_1:64;
then A27:
(G * (i2,j1)) `1 <= ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1))
by XREAL_1:6;
i1 < i1 + 1
by XREAL_1:29;
then A28:
(G * (i1,j1)) `1 <= (G * ((i1 + 1),j1)) `1
by A1, A2, A3, GOBOARD5:3;
1
- r1 >= 0
by A13, XREAL_1:48;
then
(
((1 - r1) * ((G * ((i1 + 1),j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) = 1
* ((G * ((i1 + 1),j1)) `1) &
(1 - r1) * ((G * (i1,j1)) `1) <= (1 - r1) * ((G * ((i1 + 1),j1)) `1) )
by A28, XREAL_1:64;
then A29:
((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1)) <= (G * ((i1 + 1),j1)) `1
by XREAL_1:6;
(G * ((i1 + 1),j1)) `1 < (G * (i2,j1)) `1
by A3, A7, A18, A26, GOBOARD5:3;
hence
contradiction
by A25, A29, A27, XXREAL_0:2;
verum end; suppose A30:
i2 + 1
< i1
;
contradiction
i1 < i1 + 1
by XREAL_1:29;
then
(G * (i1,j1)) `1 < (G * ((i1 + 1),j1)) `1
by A1, A2, A3, GOBOARD5:3;
then
(
((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * (i1,j1)) `1)) = 1
* ((G * (i1,j1)) `1) &
r1 * ((G * (i1,j1)) `1) <= r1 * ((G * ((i1 + 1),j1)) `1) )
by A12, XREAL_1:64;
then A31:
(G * (i1,j1)) `1 <= ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * ((i1 + 1),j1)) `1))
by XREAL_1:6;
i2 < i2 + 1
by XREAL_1:29;
then A32:
(G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1
by A3, A4, A5, GOBOARD5:3;
1
- r2 >= 0
by A16, XREAL_1:48;
then
(
((1 - r2) * ((G * ((i2 + 1),j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) = 1
* ((G * ((i2 + 1),j1)) `1) &
(1 - r2) * ((G * (i2,j1)) `1) <= (1 - r2) * ((G * ((i2 + 1),j1)) `1) )
by A32, XREAL_1:64;
then A33:
((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) <= (G * ((i2 + 1),j1)) `1
by XREAL_1:6;
(G * ((i2 + 1),j1)) `1 < (G * (i1,j1)) `1
by A3, A8, A17, A30, GOBOARD5:3;
hence
contradiction
by A25, A33, A31, XXREAL_0:2;
verum end; end; end; hence
contradiction
;
verum end; end;