let G be V9 X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: cell G,0 ,0 = { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) }
A1: cell G,0 ,0 = (v_strip G,0 ) /\ (h_strip G,0 ) by GOBOARD5:def 3;
A2: v_strip G,0 = { |[r,s]| where r, s is Real : r <= (G * 1,1) `1 } by Th18;
A3: h_strip G,0 = { |[r,s]| where r, s is Real : s <= (G * 1,1) `2 } by Th21;
thus cell G,0 ,0 c= { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } c= cell G,0 ,0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in cell G,0 ,0 or x in { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } )
assume A4: x in cell G,0 ,0 ; :: thesis: x in { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) }
then x in v_strip G,0 by A1, XBOOLE_0:def 4;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: r1 <= (G * 1,1) `1 by A2;
x in h_strip G,0 by A1, A4, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: s2 <= (G * 1,1) `2 by A3;
( r1 = r2 & s1 = s2 ) by A5, A7, SPPOL_2:1;
hence x in { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } 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 : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } or x in cell G,0 ,0 )
assume x in { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } ; :: thesis: x in cell G,0 ,0
then ex r, s being Real st
( x = |[r,s]| & r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) ;
then ( x in v_strip G,0 & x in h_strip G,0 ) by A2, A3;
hence x in cell G,0 ,0 by A1, XBOOLE_0:def 4; :: thesis: verum