let i be Nat; :: thesis: for G being V9() X_equal-in-line Y_equal-in-column Matrix of () st 1 <= i & i < len G holds
cell (G,i,()) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) }

let G be V9() X_equal-in-line Y_equal-in-column Matrix of (); :: thesis: ( 1 <= i & i < len G implies cell (G,i,()) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) } )
A1: cell (G,i,()) = (v_strip (G,i)) /\ (h_strip (G,())) by GOBOARD5:def 3;
assume ( 1 <= i & i < len G ) ; :: thesis: cell (G,i,()) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) }
then A2: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by Th20;
A3: h_strip (G,()) = { |[r,s]| where r, s is Real : (G * (1,())) `2 <= s } by Th22;
thus cell (G,i,()) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `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,())) `2 <= s ) } c= cell (G,i,())
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (G,i,()) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) } )
assume A4: x in cell (G,i,()) ; :: thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) }
then x in v_strip (G,i) by ;
then consider r1, s1 being Real such that
A5: x = |[r1,s1]| and
A6: ( (G * (i,1)) `1 <= r1 & r1 <= (G * ((i + 1),1)) `1 ) by A2;
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 A3;
s1 = s2 by ;
hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 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 : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) } or x in cell (G,i,()) )
assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) } ; :: thesis: x in cell (G,i,())
then A9: ex r, s being Real st
( x = |[r,s]| & (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,())) `2 <= s ) ;
then A10: x in h_strip (G,()) by A3;
x in v_strip (G,i) by A2, A9;
hence x in cell (G,i,()) by ; :: thesis: verum