let G be Go-board; :: thesis: Int (v_strip G,0 ) = { |[r,s]| where r, s is Real : r < (G * 1,1) `1 }
0 <> width G by GOBOARD1:def 5;
then 1 <= width G by NAT_1:14;
then A1: v_strip G,0 = { |[r,s]| where r, s is Real : r <= (G * 1,1) `1 } by GOBOARD5:11;
thus Int (v_strip G,0 ) c= { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } c= Int (v_strip G,0 )
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Int (v_strip G,0 ) or x in { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } )
assume A2: x in Int (v_strip G,0 ) ; :: thesis: x in { |[r,s]| where r, s is Real : r < (G * 1,1) `1 }
then reconsider u = x as Point of (Euclid 2) by Lm5;
consider r1 being real number such that
A3: r1 > 0 and
A4: Ball u,r1 c= v_strip G,0 by A2, Th8;
reconsider r1 = r1 as Real by XREAL_0:def 1;
reconsider p = u as Point of (TOP-REAL 2) by Lm5;
set q = |[((p `1 ) + (r1 / 2)),((p `2 ) + 0 )]|;
A5: p = |[(p `1 ),(p `2 )]| by EUCLID:57;
A6: ( r1 / 2 > 0 & r1 / 2 < r1 ) by A3, XREAL_1:217, XREAL_1:218;
then |[((p `1 ) + (r1 / 2)),((p `2 ) + 0 )]| in Ball u,r1 by A5, Th10;
then |[((p `1 ) + (r1 / 2)),((p `2 ) + 0 )]| in v_strip G,0 by A4;
then ex r2, s2 being Real st
( |[((p `1 ) + (r1 / 2)),((p `2 ) + 0 )]| = |[r2,s2]| & r2 <= (G * 1,1) `1 ) by A1;
then A7: (p `1 ) + (r1 / 2) <= (G * 1,1) `1 by SPPOL_2:1;
p `1 < (p `1 ) + (r1 / 2) by A6, XREAL_1:31;
then p `1 < (G * 1,1) `1 by A7, XXREAL_0:2;
hence x in { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } by A5; :: thesis: verum
end;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } or x in Int (v_strip G,0 ) )
assume x in { |[r,s]| where r, s is Real : r < (G * 1,1) `1 } ; :: thesis: x in Int (v_strip G,0 )
then consider r, s being Real such that
A8: x = |[r,s]| and
A9: r < (G * 1,1) `1 ;
reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:13;
A10: u in Ball u,(((G * 1,1) `1 ) - r) by A9, Th4, XREAL_1:52;
A11: Ball u,(((G * 1,1) `1 ) - r) c= v_strip G,0
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Ball u,(((G * 1,1) `1 ) - r) or y in v_strip G,0 )
A12: Ball u,(((G * 1,1) `1 ) - r) = { v where v is Point of (Euclid 2) : dist u,v < ((G * 1,1) `1 ) - r } by METRIC_1:18;
assume y in Ball u,(((G * 1,1) `1 ) - r) ; :: thesis: y in v_strip G,0
then consider v being Point of (Euclid 2) such that
A13: v = y and
A14: dist u,v < ((G * 1,1) `1 ) - r by A12;
reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:13;
A15: q = |[(q `1 ),(q `2 )]| by EUCLID:57;
then A16: sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) < ((G * 1,1) `1 ) - r by A14, Th9;
A17: (r - (q `1 )) ^2 >= 0 by XREAL_1:65;
0 <= (s - (q `2 )) ^2 by XREAL_1:65;
then ((r - (q `1 )) ^2 ) + 0 <= ((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ) by XREAL_1:8;
then sqrt ((r - (q `1 )) ^2 ) <= sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by A17, SQUARE_1:94;
then sqrt ((r - (q `1 )) ^2 ) <= ((G * 1,1) `1 ) - r by A16, XXREAL_0:2;
then A18: abs (r - (q `1 )) <= ((G * 1,1) `1 ) - r by COMPLEX1:158;
per cases ( r <= q `1 or r >= q `1 ) ;
suppose r <= q `1 ; :: thesis: y in v_strip G,0
then A19: (q `1 ) - r >= 0 by XREAL_1:50;
abs (r - (q `1 )) = abs (- (r - (q `1 ))) by COMPLEX1:138
.= (q `1 ) - r by A19, ABSVALUE:def 1 ;
then q `1 <= (G * 1,1) `1 by A18, XREAL_1:11;
hence y in v_strip G,0 by A1, A13, A15; :: thesis: verum
end;
end;
end;
reconsider B = Ball u,(((G * 1,1) `1 ) - r) as Subset of (TOP-REAL 2) by TOPREAL3:13;
B is open by Th6;
hence x in Int (v_strip G,0 ) by A8, A10, A11, TOPS_1:54; :: thesis: verum