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

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 ) )end;

hence
h_strip (G,j) is closed
; :: thesis: verumper cases
( j < 1 or ( 1 <= j & j < width G ) or j >= width G )
;

end;

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

end;

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;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

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

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

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 }

hence h_strip (G,j) is closed by A3, A4, TOPS_1:8; :: thesis: verum

end;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

( P1 is closed & P2 is closed )
by Th12, Th13;
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 ) }

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;proof

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 }
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;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

proof

{ |[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 }
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;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

proof

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;
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;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

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

hence h_strip (G,j) is closed by A3, A4, TOPS_1:8; :: thesis: verum