let j be Nat; for G being Matrix of (TOP-REAL 2) holds h_strip (G,j) is closed
let G be Matrix of (TOP-REAL 2); h_strip (G,j) is closed
now ( ( j < 1 & h_strip (G,j) is closed ) or ( 1 <= j & j < width G & h_strip (G,j) is closed ) or ( j >= width G & h_strip (G,j) is closed ) )per cases
( j < 1 or ( 1 <= j & j < width G ) or j >= width G )
;
case
( 1
<= j &
j < width G )
;
h_strip (G,j) is closed then A3:
h_strip (
G,
j)
= { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
by GOBOARD5:def 2;
reconsider P2 =
{ |[r1,s1]| where r1, s1 is Real : s1 <= (G * (1,(j + 1))) `2 } as
Subset of
(TOP-REAL 2) by Lm5;
reconsider P1 =
{ |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } as
Subset of
(TOP-REAL 2) by Lm3;
A4:
{ |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } = { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 }
proof
A5:
{ |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 } c= { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
proof
let x be
object ;
TARSKI:def 3 ( not x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 } or x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } )
assume A6:
x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 }
;
x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
then A7:
x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 }
by XBOOLE_0:def 4;
x in { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 }
by A6, XBOOLE_0:def 4;
then consider r2,
s2 being
Real such that A8:
|[r2,s2]| = x
and A9:
s2 <= (G * (1,(j + 1))) `2
;
consider r1,
s1 being
Real such that A10:
|[r1,s1]| = x
and A11:
(G * (1,j)) `2 <= s1
by A7;
s1 = s2
by A10, A8, SPPOL_2:1;
hence
x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
by A10, A11, A9;
verum
end;
A12:
{ |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } c= { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 }
proof
let x be
object ;
TARSKI:def 3 ( not x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } or x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } )
assume
x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
;
x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 }
then
ex
r,
s being
Real st
(
x = |[r,s]| &
(G * (1,j)) `2 <= s &
s <= (G * (1,(j + 1))) `2 )
;
hence
x in { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 }
;
verum
end;
{ |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } c= { |[r1,s1]| where r1, s1 is Real : s1 <= (G * (1,(j + 1))) `2 }
proof
let x be
object ;
TARSKI:def 3 ( not x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } or x in { |[r1,s1]| where r1, s1 is Real : s1 <= (G * (1,(j + 1))) `2 } )
assume
x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) }
;
x in { |[r1,s1]| where r1, s1 is Real : s1 <= (G * (1,(j + 1))) `2 }
then
ex
r,
s being
Real st
(
x = |[r,s]| &
(G * (1,j)) `2 <= s &
s <= (G * (1,(j + 1))) `2 )
;
hence
x in { |[r1,s1]| where r1, s1 is Real : s1 <= (G * (1,(j + 1))) `2 }
;
verum
end;
then
{ |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } c= { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 }
by A12, XBOOLE_1:19;
hence
{ |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } = { |[r1,s1]| where r1, s1 is Real : (G * (1,j)) `2 <= s1 } /\ { |[r2,s2]| where r2, s2 is Real : s2 <= (G * (1,(j + 1))) `2 }
by A5;
verum
end;
(
P1 is
closed &
P2 is
closed )
by Th12, Th13;
hence
h_strip (
G,
j) is
closed
by A3, A4, TOPS_1:8;
verum end; end; end;
hence
h_strip (G,j) is closed
; verum