let j be Nat; :: thesis: for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= j & j < width G holds
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 V9() X_equal-in-line Y_equal-in-column Matrix of (); :: thesis: ( 1 <= j & j < width G implies 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 ) } )
A1: cell (G,(len G),j) = (v_strip (G,(len G))) /\ (h_strip (G,j)) by GOBOARD5:def 3;
assume ( 1 <= j & j < width G ) ; :: thesis: 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: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by Th23;
A3: v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 <= r } by Th19;
thus 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= cell (G,(len G),j)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in 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 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 v_strip (G,(len G)) by ;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: (G * ((len G),1)) `1 <= r1 by A3;
x in h_strip (G,j) by ;
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 ;
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 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 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 h_strip (G,j) by A2;
x in v_strip (G,(len G)) by A3, A9;
hence x in cell (G,(len G),j) by ; :: thesis: verum