let G be Go-board; Int (cell (G,0,(width G))) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) }
cell (G,0,(width G)) = (v_strip (G,0)) /\ (h_strip (G,(width G)))
by GOBOARD5:def 3;
then A1:
Int (cell (G,0,(width G))) = (Int (v_strip (G,0))) /\ (Int (h_strip (G,(width G))))
by TOPS_1:17;
A2:
Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s }
by Th16;
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,(width G))) c= { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) }
XBOOLE_0:def 10 { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } c= Int (cell (G,0,(width G)))proof
let x be
object ;
TARSKI:def 3 ( not x in Int (cell (G,0,(width G))) or x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } )
assume A4:
x in Int (cell (G,0,(width G)))
;
x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) }
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,(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 A2;
s1 = s2
by A5, A7, SPPOL_2:1;
hence
x in { |[r,s]| where r, s is Real : ( r < (G * (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 : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } or x in Int (cell (G,0,(width G))) )
assume
x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) }
; x in Int (cell (G,0,(width G)))
then A9:
ex r, s being Real st
( x = |[r,s]| & r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s )
;
then A10:
x in Int (h_strip (G,(width G)))
by A2;
x in Int (v_strip (G,0))
by A3, A9;
hence
x in Int (cell (G,0,(width G)))
by A1, A10, XBOOLE_0:def 4; verum