let G be Go-board; :: thesis: 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 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 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 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, GOBOARD5:5;
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, GOBOARD5:8;

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:52; :: 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:52; :: 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:53;
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