let G be Go-board; for p being Point of (TOP-REAL 2)
for i, j being Nat st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( p in Int (cell (G,i,j)) iff ( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 ) )
let p be Point of (TOP-REAL 2); for i, j being Nat st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( p in Int (cell (G,i,j)) iff ( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 ) )
let i, j be Nat; ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( p in Int (cell (G,i,j)) iff ( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 ) ) )
assume that
A1:
1 <= i
and
A2:
i + 1 <= len G
and
A3:
1 <= j
and
A4:
j + 1 <= width G
; ( p in Int (cell (G,i,j)) iff ( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 ) )
set Z = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ;
A5:
j < width G
by A4, NAT_1:13;
i + 1 >= 1
by NAT_1:11;
then A6:
(G * ((i + 1),1)) `1 = (G * ((i + 1),j)) `1
by A2, A3, A5, GOBOARD5:2;
A7:
i < len G
by A2, NAT_1:13;
then A8:
(G * (1,j)) `2 = (G * (i,j)) `2
by A1, A3, A5, GOBOARD5:1;
j + 1 >= 1
by NAT_1:11;
then A9:
(G * (1,(j + 1))) `2 = (G * (i,(j + 1))) `2
by A1, A4, A7, GOBOARD5:1;
A10:
(G * (i,1)) `1 = (G * (i,j)) `1
by A1, A3, A7, A5, GOBOARD5:2;
thus
( p in Int (cell (G,i,j)) implies ( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 ) )
( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 implies p in Int (cell (G,i,j)) )proof
assume
p in Int (cell (G,i,j))
;
( (G * (i,j)) `1 < p `1 & p `1 < (G * ((i + 1),j)) `1 & (G * (i,j)) `2 < p `2 & p `2 < (G * (i,(j + 1))) `2 )
then
p in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) }
by A1, A3, A7, A5, GOBOARD6:26;
then
ex
r,
s being
Real st
(
p = |[r,s]| &
(G * (i,1)) `1 < r &
r < (G * ((i + 1),1)) `1 &
(G * (1,j)) `2 < s &
s < (G * (1,(j + 1))) `2 )
;
hence
(
(G * (i,j)) `1 < p `1 &
p `1 < (G * ((i + 1),j)) `1 &
(G * (i,j)) `2 < p `2 &
p `2 < (G * (i,(j + 1))) `2 )
by A10, A6, A8, A9, EUCLID:52;
verum
end;
assume that
A11:
(G * (i,j)) `1 < p `1
and
A12:
p `1 < (G * ((i + 1),j)) `1
and
A13:
(G * (i,j)) `2 < p `2
and
A14:
p `2 < (G * (i,(j + 1))) `2
; p in Int (cell (G,i,j))
p = |[(p `1),(p `2)]|
by EUCLID:53;
then
p in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) }
by A10, A6, A8, A9, A11, A12, A13, A14;
hence
p in Int (cell (G,i,j))
by A1, A3, A7, A5, GOBOARD6:26; verum