let G be empty-yielding X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); :: thesis: cell (G,0,(width G)) = { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) }
A1: cell (G,0,(width G)) = (v_strip (G,0)) /\ (h_strip (G,(width G))) by GOBOARD5:def 3;
A2: h_strip (G,(width G)) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 <= s } by Th22;
A3: v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,1)) `1 } by Th18;
thus cell (G,0,(width G)) c= { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } c= cell (G,0,(width G))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (G,0,(width G)) or x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } )
assume A4: x in cell (G,0,(width G)) ; :: thesis: x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) }
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 A3;
x in h_strip (G,(width G)) by A1, A4, XBOOLE_0:def 4;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: (G * (1,(width G))) `2 <= s2 by A2;
s1 = s2 by A5, A7, SPPOL_2:1;
hence x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } 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 : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } or x in cell (G,0,(width G)) )
assume x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) } ; :: thesis: x in cell (G,0,(width G))
then A9: ex r, s being Real st
( x = |[r,s]| & r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s ) ;
then A10: x in h_strip (G,(width G)) by A2;
x in v_strip (G,0) by A3, A9;
hence x in cell (G,0,(width G)) by A1, A10, XBOOLE_0:def 4; :: thesis: verum