let i be Element of NAT ; :: thesis: for G being V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2) st 1 <= i & i < len G holds
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 V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: ( 1 <= i & i < len G implies 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: 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 ) }
A2: cell G,i,(width G) = (v_strip G,i) /\ (h_strip G,(width G)) by GOBOARD5:def 3;
A3: v_strip G,i = { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 ) } by A1, Th20;
A4: h_strip G,(width G) = { |[r,s]| where r, s is Real : (G * 1,(width G)) `2 <= s } by Th22;
thus 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= cell G,i,(width G)

proof

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