let i be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i < len G holds
Int (cell G,i,(width G)) = { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) }
let G be Go-board; :: thesis: ( 1 <= i & i < len G implies Int (cell G,i,(width G)) = { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) } )
assume A1: ( 1 <= i & i < len G ) ; :: thesis: Int (cell G,i,(width G)) = { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) }
cell G,i,(width G) = (v_strip G,i) /\ (h_strip G,(width G)) by GOBOARD5:def 3;
then A2: Int (cell G,i,(width G)) = (Int (v_strip G,i)) /\ (Int (h_strip G,(width G))) by TOPS_1:46;
A3: Int (v_strip G,i) = { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 ) } by A1, Th17;
A4: Int (h_strip G,(width G)) = { |[r,s]| where r, s is Real : (G * 1,(width G)) `2 < s } by Th19;
thus Int (cell G,i,(width G)) c= { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) } c= Int (cell G,i,(width G))

proof

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