let j be Element of NAT ; :: thesis: for G being Go-board st 1 <= j & j < width G holds
Int (cell G,0 ,j) = { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) }
let G be Go-board; :: thesis: ( 1 <= j & j < width G implies Int (cell G,0 ,j) = { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } )
assume A1: ( 1 <= j & j < width G ) ; :: thesis: Int (cell G,0 ,j) = { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) }
cell G,0 ,j = (v_strip G,0 ) /\ (h_strip G,j) by GOBOARD5:def 3;
then A2: Int (cell G,0 ,j) = (Int (v_strip G,0 )) /\ (Int (h_strip G,j)) by TOPS_1:46;
A3: Int (v_strip G,0 ) = { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } by Th15;
A4: Int (h_strip G,j) = { |[r,s]| where r, s is Real : ( (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } by A1, Th20;
thus Int (cell G,0 ,j) c= { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } c= Int (cell G,0 ,j)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Int (cell G,0 ,j) or x in { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } )
assume A5: x in Int (cell G,0 ,j) ; :: thesis: x in { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) }
then x in Int (v_strip G,0 ) by A2, XBOOLE_0:def 4;
then consider r1, s1 being Real such that
A6: x = |[r1,s1]| and
A7: r1 < (G * 1,1) `1 by A3;
x in Int (h_strip G,j) by A2, A5, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A8: x = |[r2,s2]| and
A9: ( (G * 1,j) `2 < s2 & s2 < (G * 1,(j + 1)) `2 ) by A4;
( r1 = r2 & s1 = s2 ) by A6, A8, SPPOL_2:1;
hence x in { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } by A6, A7, A9; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } or x in Int (cell G,0 ,j) )
assume x in { |[r,s]| where r, s is Real : ( r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } ; :: thesis: x in Int (cell G,0 ,j)
then ex r, s being Real st
( x = |[r,s]| & r < (G * 1,1) `1 & (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) ;
then ( x in Int (v_strip G,0 ) & x in Int (h_strip G,j) ) by A3, A4;
hence x in Int (cell G,0 ,j) by A2, XBOOLE_0:def 4; :: thesis: verum