let i, j be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) }
let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } )
set A = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } ;
assume A1: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G ) ; :: thesis: cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) }

now

let p be set ; :: thesis: ( p in cell G,i,j implies p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } )
assume A2: p in cell G,i,j ; :: thesis: p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) }
then reconsider q = p as Point of (TOP-REAL 2) ;
A3: ( (G * i,j) `2 <= q `2 & q `2 <= (G * i,(j + 1)) `2 ) by A1, A2, Th19;
A4: p = |[(q `1 ),(q `2 )]| by EUCLID:57;
( (G * i,j) `1 <= q `1 & q `1 <= (G * (i + 1),j) `1 ) by A1, A2, Th19;
hence p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } by A4, A3; :: thesis: verum

end;

hence cell G,i,j c= { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } by TARSKI:def 3; :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } c= cell G,i,j

now

let p be set ; :: thesis: ( p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } implies p in cell G,i,j )
assume p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } ; :: thesis: p in cell G,i,j
then consider r, s being Real such that
A5: |[r,s]| = p and
A6: ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) ;
reconsider q = p as Point of (TOP-REAL 2) by A5;
( r = q `1 & s = q `2 ) by A5, EUCLID:56;
hence p in cell G,i,j by A1, A6, Th19; :: thesis: verum

end;

hence { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 & (G * i,j) `2 <= s & s <= (G * i,(j + 1)) `2 ) } c= cell G,i,j by TARSKI:def 3; :: thesis: verum