let j be Nat; :: thesis: for G being Go-board st 1 <= j & j < width G holds
Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (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,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } )
cell (G,(len G),j) = (v_strip (G,(len G))) /\ (h_strip (G,j)) by GOBOARD5:def 3;
then A1: Int (cell (G,(len G),j)) = (Int (v_strip (G,(len G)))) /\ (Int (h_strip (G,j))) by TOPS_1:17;
assume ( 1 <= j & j < width G ) ; :: thesis: Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) }
then A2: Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by Th17;
A3: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by Th13;
thus Int (cell (G,(len G),j)) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (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 : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } c= Int (cell (G,(len G),j))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Int (cell (G,(len G),j)) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } )
assume A4: x in Int (cell (G,(len G),j)) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) }
then x in Int (v_strip (G,(len G))) by A1, XBOOLE_0:def 4;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: (G * ((len G),1)) `1 < r1 by A3;
x in Int (h_strip (G,j)) by A1, A4, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: ( (G * (1,j)) `2 < s2 & s2 < (G * (1,(j + 1))) `2 ) by A2;
s1 = s2 by A5, A7, SPPOL_2:1;
hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A5, A6, A8; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } or x in Int (cell (G,(len G),j)) )
assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ; :: thesis: x in Int (cell (G,(len G),j))
then A9: ex r, s being Real st
( x = |[r,s]| & (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) ;
then A10: x in Int (h_strip (G,j)) by A2;
x in Int (v_strip (G,(len G))) by A3, A9;
hence x in Int (cell (G,(len G),j)) by A1, A10, XBOOLE_0:def 4; :: thesis: verum