let G be V9() X_equal-in-line Y_equal-in-column Matrix of (); :: thesis: cell (G,0,()) = { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) }
A1: cell (G,0,()) = (v_strip (G,0)) /\ (h_strip (G,())) by GOBOARD5:def 3;
A2: h_strip (G,()) = { |[r,s]| where r, s is Real : (G * (1,())) `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,()) c= { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) } c= cell (G,0,())
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (G,0,()) or x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) } )
assume A4: x in cell (G,0,()) ; :: thesis: x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) }
then x in v_strip (G,0) by ;
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,()) by ;
then consider r2, s2 being Real such that
A7: x = |[r2,s2]| and
A8: (G * (1,())) `2 <= s2 by A2;
s1 = s2 by ;
hence x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `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,())) `2 <= s ) } or x in cell (G,0,()) )
assume x in { |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) } ; :: thesis: x in cell (G,0,())
then A9: ex r, s being Real st
( x = |[r,s]| & r <= (G * (1,1)) `1 & (G * (1,())) `2 <= s ) ;
then A10: x in h_strip (G,()) by A2;
x in v_strip (G,0) by A3, A9;
hence x in cell (G,0,()) by ; :: thesis: verum