let j be Element of NAT ; :: thesis: for G being V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2) st 1 <= j & j < width G holds
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 V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: ( 1 <= j & j < width G implies 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: 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 ) }
A2: cell G,0 ,j = (v_strip G,0 ) /\ (h_strip G,j) by GOBOARD5:def 3;
A3: v_strip G,0 = { |[r,s]| where r, s is Real : r <= (G * 1,1) `1 } by Th18;
A4: h_strip G,j = { |[r,s]| where r, s is Real : ( (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } by A1, Th23;
thus 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= cell G,0 ,j

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in 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 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 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 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 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 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 v_strip G,0 & x in h_strip G,j ) by A3, A4;
hence x in cell G,0 ,j by A2, XBOOLE_0:def 4; :: thesis: verum