let G be Go-board; Int (cell (G,0,0)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) }
cell (G,0,0) = (v_strip (G,0)) /\ (h_strip (G,0))
by GOBOARD5:def 3;
then A1:
Int (cell (G,0,0)) = (Int (v_strip (G,0))) /\ (Int (h_strip (G,0)))
by TOPS_1:17;
A2:
Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 }
by Th15;
A3:
Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 }
by Th12;
thus
Int (cell (G,0,0)) c= { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) }
XBOOLE_0:def 10 { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } c= Int (cell (G,0,0))proof
let x be
object ;
TARSKI:def 3 ( not x in Int (cell (G,0,0)) or x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } )
assume A4:
x in Int (cell (G,0,0))
;
x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) }
then
x in Int (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 Int (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 : ( r < (G * (1,1)) `1 & 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 : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } or x in Int (cell (G,0,0)) )
assume
x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) }
; x in Int (cell (G,0,0))
then A9:
ex r, s being Real st
( x = |[r,s]| & r < (G * (1,1)) `1 & s < (G * (1,1)) `2 )
;
then A10:
x in Int (h_strip (G,0))
by A2;
x in Int (v_strip (G,0))
by A3, A9;
hence
x in Int (cell (G,0,0))
by A1, A10, XBOOLE_0:def 4; verum