let G be Go-board; :: thesis: for p being Point of (TOP-REAL 2)
for i, j being Element of 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); :: thesis: for i, j being Element of 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 Element of NAT ; :: thesis: ( 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 ; :: thesis: ( 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, GOBOARD5:3;
A7: i < len G by A2, NAT_1:13;
then A8: (G * 1,j) `2 = (G * i,j) `2 by A1, A3, A5, GOBOARD5:2;
j + 1 >= 1 by NAT_1:11;
then A9: (G * 1,(j + 1)) `2 = (G * i,(j + 1)) `2 by A1, A4, A7, GOBOARD5:2;
A10: (G * i,1) `1 = (G * i,j) `1 by A1, A3, A7, A5, GOBOARD5:3;
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 ) ) :: thesis: ( (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) ; :: thesis: ( (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, GOBOARD6:29;
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:56; :: thesis: 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 ; :: thesis: p in Int (cell G,i,j)
p = |[(p `1 ),(p `2 )]| by EUCLID:57;
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, GOBOARD6:29; :: thesis: verum