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 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 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 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 & i + 1 <= len G ) and
A2: ( 1 <= j & j + 1 <= width G ) ; :: thesis: ( p in 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 ) )
A3: i < len G by A1, NAT_1:13;
A4: j < width G by A2, NAT_1:13;
then A5: v_strip G,i = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 ) } by A1, A2, A3, GOBOARD5:9;
A6: h_strip G,j = { |[r,s]| where r, s is Real : ( (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } by A1, A2, A3, A4, GOBOARD5:6;

hereby :: 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 cell G,i,j )

assume p in 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 A7: ( p in v_strip G,i & p in h_strip G,j ) by XBOOLE_0:def 4;
then ex r, s being Real st
( |[r,s]| = p & (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 ) by A5;
hence ( (G * i,j) `1 <= p `1 & p `1 <= (G * (i + 1),j) `1 ) by EUCLID:56; :: thesis: ( (G * i,j) `2 <= p `2 & p `2 <= (G * i,(j + 1)) `2 )
ex r, s being Real st
( |[r,s]| = p & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) by A6, A7;
hence ( (G * i,j) `2 <= p `2 & p `2 <= (G * i,(j + 1)) `2 ) by EUCLID:56; :: thesis: verum

end;

assume that
A8: ( (G * i,j) `1 <= p `1 & p `1 <= (G * (i + 1),j) `1 ) and
A9: ( (G * i,j) `2 <= p `2 & p `2 <= (G * i,(j + 1)) `2 ) ; :: thesis: p in cell G,i,j
A10: p = |[(p `1 ),(p `2 )]| by EUCLID:57;
then A11: p in v_strip G,i by A5, A8;
p in h_strip G,j by A6, A9, A10;
hence p in cell G,i,j by A11, XBOOLE_0:def 4; :: thesis: verum