let i, j be Element of NAT ; :: thesis: for G being V21() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2) st 1 <= i & i < len G & 1 <= j & j < width G holds
cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) }
let G be V21() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } )
assume that
A1: ( 1 <= i & i < len G ) and
A2: ( 1 <= j & j < width G ) ; :: thesis: cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) }
A3: h_strip G,j = { |[r,s]| where r, s is Real : ( (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } by A2, Th23;
A4: cell G,i,j = (v_strip G,i) /\ (h_strip G,j) by GOBOARD5:def 3;
A5: 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;
thus cell G,i,j c= { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 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 : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } c= cell G,i,j

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in cell G,i,j or x in { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } )
assume A6: x in cell G,i,j ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) }
then x in v_strip G,i by A4, XBOOLE_0:def 4;
then consider r1, s1 being Real such that
A7: x = |[r1,s1]| and
A8: ( (G * i,1) `1 <= r1 & r1 <= (G * (i + 1),1) `1 ) by A5;
x in h_strip G,j by A4, A6, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A9: x = |[r2,s2]| and
A10: ( (G * 1,j) `2 <= s2 & s2 <= (G * 1,(j + 1)) `2 ) by A3;
s1 = s2 by A7, A9, 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,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } by A7, A8, A10; :: 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,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } or x in cell G,i,j )
assume x in { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } ; :: thesis: x in cell G,i,j
then A11: ex r, s being Real st
( x = |[r,s]| & (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) ;
then A12: x in h_strip G,j by A3;
x in v_strip G,i by A5, A11;
hence x in cell G,i,j by A4, A12, XBOOLE_0:def 4; :: thesis: verum