let j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) holds h_strip (G,j) is closed
let G be Matrix of (TOP-REAL 2); :: thesis: h_strip (G,j) is closed

now :: thesis: ( ( 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 A1: j < 1 ; :: thesis: h_strip (G,j) is closed

A2: now :: thesis: ( j >= width G implies h_strip (G,j) is closed )

assume j >= width G ; :: thesis: h_strip (G,j) is closed
then h_strip (G,j) = { |[r,s]| where r, s is Real : (G * (1,j)) `2 <= s } by GOBOARD5:def 2;
hence h_strip (G,j) is closed by Th13; :: thesis: verum

end;

now :: thesis: ( j < width G implies h_strip (G,j) is closed )

assume j < width G ; :: thesis: h_strip (G,j) is closed
then h_strip (G,j) = { |[r,s]| where r, s is Real : s <= (G * (1,(j + 1))) `2 } by A1, GOBOARD5:def 2;
hence h_strip (G,j) is closed by Th12; :: thesis: verum

end;

hence h_strip (G,j) is closed by A2; :: thesis: verum

end;

case ( 1 <= j & j < width G ) ; :: thesis: 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 ; :: according to TARSKI:def 3 :: thesis: ( 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 } ; :: thesis: 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; :: thesis: 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 ; :: according to TARSKI:def 3 :: thesis: ( 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 ) } ; :: thesis: 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 } ; :: thesis: 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 ; :: according to TARSKI:def 3 :: thesis: ( 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 ) } ; :: thesis: 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 } ; :: thesis: 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; :: thesis: verum

end;

( P1 is closed & P2 is closed ) by Th12, Th13;
hence h_strip (G,j) is closed by A3, A4, TOPS_1:8; :: thesis: verum

end;

case j >= width G ; :: thesis: h_strip (G,j) is closed

then h_strip (G,j) = { |[r,s]| where r, s is Real : (G * (1,j)) `2 <= s } by GOBOARD5:def 2;
hence h_strip (G,j) is closed by Th13; :: thesis: verum

end;

end;

end;

hence h_strip (G,j) is closed ; :: thesis: verum