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 & i + 1 <= len G ) and
A2: ( 1 <= j & 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 ) )
A3: ( 1 <= i & i < len G ) by A1, NAT_1:13;
A4: ( 1 <= j & j < width G ) by A2, NAT_1:13;
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: (G * i,1) `1 = (G * i,j) `1 by A3, A4, GOBOARD5:3;
i + 1 >= 1 by NAT_1:11;
then A6: (G * (i + 1),1) `1 = (G * (i + 1),j) `1 by A1, A4, GOBOARD5:3;
A7: (G * 1,j) `2 = (G * i,j) `2 by A3, A4, GOBOARD5:2;
j + 1 >= 1 by NAT_1:11;
then A8: (G * 1,(j + 1)) `2 = (G * i,(j + 1)) `2 by A2, A3, 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 ) ) :: 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 A3, A4, GOBOARD6:29;
then consider r, s being Real such that
A9: p = |[r,s]| and
A10: ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 ) and
A11: ( (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) ;
thus ( (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 A5, A6, A7, A8, A9, A10, A11, EUCLID:56; :: thesis: verum

end;

assume A12: ( (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: 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 A5, A6, A7, A8, A12;
hence p in Int (cell G,i,j) by A3, A4, GOBOARD6:29; :: thesis: verum