let G be Go-board; :: thesis: Int (cell G,(len G),(width G)) = { |[r,s]| where r, s is Real : ( (G * (len G),1) `1 < r & (G * 1,(width G)) `2 < s ) }
cell G,(len G),(width G) = (v_strip G,(len G)) /\ (h_strip G,(width G)) by GOBOARD5:def 3;
then A1: Int (cell G,(len G),(width G)) = (Int (v_strip G,(len G))) /\ (Int (h_strip G,(width G))) by TOPS_1:46;
A2: Int (v_strip G,(len G)) = { |[r,s]| where r, s is Real : (G * (len G),1) `1 < r } by Th16;
A3: 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,(len G),(width G)) c= { |[r,s]| where r, s is Real : ( (G * (len G),1) `1 < r & (G * 1,(width G)) `2 < s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( (G * (len G),1) `1 < r & (G * 1,(width G)) `2 < s ) } c= Int (cell G,(len G),(width G))

proof

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

end;

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