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 and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: 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 ) )
A5: ( i < len G & j < width G ) by A2, A4, NAT_1:13;
then 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, A3, GOBOARD5:6;
A7: v_strip G,i = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 ) } by A1, A3, A5, GOBOARD5:9;

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 A8: 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 p in v_strip G,i 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 A7;
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 )
p in h_strip G,j by A8, XBOOLE_0:def 4;
then ex r, s being Real st
( |[r,s]| = p & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) by A6;
hence ( (G * i,j) `2 <= p `2 & p `2 <= (G * i,(j + 1)) `2 ) by EUCLID:56; :: thesis: verum

end;

assume that
A9: ( (G * i,j) `1 <= p `1 & p `1 <= (G * (i + 1),j) `1 ) and
A10: ( (G * i,j) `2 <= p `2 & p `2 <= (G * i,(j + 1)) `2 ) ; :: thesis: p in cell G,i,j
A11: p = |[(p `1 ),(p `2 )]| by EUCLID:57;
then A12: p in h_strip G,j by A6, A10;
p in v_strip G,i by A7, A9, A11;
hence p in cell G,i,j by A12, XBOOLE_0:def 4; :: thesis: verum