let G be Go-board; :: thesis: Int (h_strip G,0 ) = { |[r,s]| where r, s is Real : s < (G * 1,1) `2 }
0 <> len G by GOBOARD1:def 5;
then 1 <= len G by NAT_1:14;
then A1: h_strip G,0 = { |[r,s]| where r, s is Real : s <= (G * 1,1) `2 } by GOBOARD5:8;
thus Int (h_strip G,0 ) c= { |[r,s]| where r, s is Real : s < (G * 1,1) `2 } :: according to XBOOLE_0:def 10 :: thesis: { |[r,s]| where r, s is Real : s < (G * 1,1) `2 } c= Int (h_strip G,0 )
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Int (h_strip G,0 ) or x in { |[r,s]| where r, s is Real : s < (G * 1,1) `2 } )
assume A2: x in Int (h_strip G,0 ) ; :: thesis: x in { |[r,s]| where r, s is Real : s < (G * 1,1) `2 }
then reconsider u = x as Point of (Euclid 2) by Lm5;
consider s1 being real number such that
A3: s1 > 0 and
A4: Ball u,s1 c= h_strip G,0 by A2, Th8;
reconsider s1 = s1 as Real by XREAL_0:def 1;
reconsider p = u as Point of (TOP-REAL 2) by Lm5;
set q = |[((p `1 ) + 0 ),((p `2 ) + (s1 / 2))]|;
A5: p = |[(p `1 ),(p `2 )]| by EUCLID:57;
A6: ( s1 / 2 > 0 & s1 / 2 < s1 ) by A3, XREAL_1:217, XREAL_1:218;
then |[((p `1 ) + 0 ),((p `2 ) + (s1 / 2))]| in Ball u,s1 by A5, Th11;
then |[((p `1 ) + 0 ),((p `2 ) + (s1 / 2))]| in h_strip G,0 by A4;
then ex r2, s2 being Real st
( |[((p `1 ) + 0 ),((p `2 ) + (s1 / 2))]| = |[r2,s2]| & s2 <= (G * 1,1) `2 ) by A1;
then A7: (p `2 ) + (s1 / 2) <= (G * 1,1) `2 by SPPOL_2:1;
p `2 < (p `2 ) + (s1 / 2) by A6, XREAL_1:31;
then p `2 < (G * 1,1) `2 by A7, XXREAL_0:2;
hence x in { |[r,s]| where r, s is Real : s < (G * 1,1) `2 } 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 : s < (G * 1,1) `2 } or x in Int (h_strip G,0 ) )
assume x in { |[r,s]| where r, s is Real : s < (G * 1,1) `2 } ; :: thesis: x in Int (h_strip G,0 )
then consider r, s being Real such that
A8: x = |[r,s]| and
A9: s < (G * 1,1) `2 ;
reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:13;
A10: u in Ball u,(((G * 1,1) `2 ) - s) by A9, Th4, XREAL_1:52;
A11: Ball u,(((G * 1,1) `2 ) - s) c= h_strip G,0
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Ball u,(((G * 1,1) `2 ) - s) or y in h_strip G,0 )
A12: Ball u,(((G * 1,1) `2 ) - s) = { v where v is Point of (Euclid 2) : dist u,v < ((G * 1,1) `2 ) - s } by METRIC_1:18;
assume y in Ball u,(((G * 1,1) `2 ) - s) ; :: thesis: y in h_strip G,0
then consider v being Point of (Euclid 2) such that
A13: v = y and
A14: dist u,v < ((G * 1,1) `2 ) - s 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) `2 ) - s by A14, Th9;
A17: (s - (q `2 )) ^2 >= 0 by XREAL_1:65;
0 <= (r - (q `1 )) ^2 by XREAL_1:65;
then ((s - (q `2 )) ^2 ) + 0 <= ((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 ) by XREAL_1:8;
then sqrt ((s - (q `2 )) ^2 ) <= sqrt (((r - (q `1 )) ^2 ) + ((s - (q `2 )) ^2 )) by A17, SQUARE_1:94;
then sqrt ((s - (q `2 )) ^2 ) <= ((G * 1,1) `2 ) - s by A16, XXREAL_0:2;
then A18: abs (s - (q `2 )) <= ((G * 1,1) `2 ) - s by COMPLEX1:158;
per cases ( s <= q `2 or s >= q `2 ) ;
suppose s <= q `2 ; :: thesis: y in h_strip G,0
then A19: (q `2 ) - s >= 0 by XREAL_1:50;
abs (s - (q `2 )) = abs (- (s - (q `2 ))) by COMPLEX1:138
.= (q `2 ) - s by A19, ABSVALUE:def 1 ;
then q `2 <= (G * 1,1) `2 by A18, XREAL_1:11;
hence y in h_strip G,0 by A1, A13, A15; :: thesis: verum
end;
end;
end;
reconsider B = Ball u,(((G * 1,1) `2 ) - s) as Subset of (TOP-REAL 2) by TOPREAL3:13;
B is open by Th6;
hence x in Int (h_strip G,0 ) by A8, A10, A11, TOPS_1:54; :: thesis: verum