let i be Nat; 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); ( 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 ) } )
A1:
cell (G,i,(width G)) = (v_strip (G,i)) /\ (h_strip (G,(width G)))
by GOBOARD5:def 3;
assume
( 1 <= i & i < len G )
; 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 ) }
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,(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 ) }
XBOOLE_0:def 10 { |[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
object ;
TARSKI:def 3 ( 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 A4:
x in cell (
G,
i,
(width G))
;
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 A1, XBOOLE_0:def 4;
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,
(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 A3;
s1 = s2
by A5, A7, 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 A5, A6, A8;
verum
end;
let x be object ; TARSKI:def 3 ( 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 ) }
; x in cell (G,i,(width G))
then A9:
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 A10:
x in h_strip (G,(width G))
by A3;
x in v_strip (G,i)
by A2, A9;
hence
x in cell (G,i,(width G))
by A1, A10, XBOOLE_0:def 4; verum