let G be V9() X_equal-in-line Y_equal-in-column Matrix of (TOP-REAL 2); cell (G,(len G),0) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) }
A1:
cell (G,(len G),0) = (v_strip (G,(len G))) /\ (h_strip (G,0))
by GOBOARD5:def 3;
A2:
h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (1,1)) `2 }
by Th21;
A3:
v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 <= r }
by Th19;
thus
cell (G,(len G),0) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) }
XBOOLE_0:def 10 { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) } c= cell (G,(len G),0)proof
let x be
object ;
TARSKI:def 3 ( not x in cell (G,(len G),0) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) } )
assume A4:
x in cell (
G,
(len G),
0)
;
x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) }
then
x in v_strip (
G,
(len G))
by A1, XBOOLE_0:def 4;
then consider r1,
s1 being
Real such that A5:
x = |[r1,s1]|
and A6:
(G * ((len G),1)) `1 <= r1
by A3;
x in h_strip (
G,
0)
by A1, A4, XBOOLE_0:def 4;
then consider r2,
s2 being
Real such that A7:
x = |[r2,s2]|
and A8:
s2 <= (G * (1,1)) `2
by A2;
s1 = s2
by A5, A7, SPPOL_2:1;
hence
x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) }
by A5, A6, A8;
verum
end;
let x be object ; TARSKI:def 3 ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) } or x in cell (G,(len G),0) )
assume
x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) }
; x in cell (G,(len G),0)
then A9:
ex r, s being Real st
( x = |[r,s]| & (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 )
;
then A10:
x in h_strip (G,0)
by A2;
x in v_strip (G,(len G))
by A3, A9;
hence
x in cell (G,(len G),0)
by A1, A10, XBOOLE_0:def 4; verum