hereby ( ( i >= width G implies { |[r,s]| where r, s is Real : (G * (1,i)) `2 <= s } is Subset of (TOP-REAL 2) ) & ( ( not 1 <= i or not i < width G ) & not i >= width G implies { |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } is Subset of (TOP-REAL 2) ) )
assume that
1
<= i
and
i < width G
;
{ |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } is Subset of (TOP-REAL 2)set A =
{ |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } ;
{ |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } c= the
carrier of
(TOP-REAL 2)
proof
let x be
object ;
TARSKI:def 3 ( not x in { |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } or x in the carrier of (TOP-REAL 2) )
assume
x in { |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) }
;
x in the carrier of (TOP-REAL 2)
then
ex
r,
s being
Real st
(
x = |[r,s]| &
(G * (1,i)) `2 <= s &
s <= (G * (1,(i + 1))) `2 )
;
hence
x in the
carrier of
(TOP-REAL 2)
;
verum
end; hence
{ |[r,s]| where r, s is Real : ( (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } is
Subset of
(TOP-REAL 2)
;
verum
end;
hereby ( ( not 1 <= i or not i < width G ) & not i >= width G implies { |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } is Subset of (TOP-REAL 2) )
assume
i >= width G
;
{ |[r,s]| where r, s is Real : (G * (1,i)) `2 <= s } is Subset of (TOP-REAL 2)set A =
{ |[r,s]| where r, s is Real : (G * (1,i)) `2 <= s } ;
{ |[r,s]| where r, s is Real : (G * (1,i)) `2 <= s } c= the
carrier of
(TOP-REAL 2)
hence
{ |[r,s]| where r, s is Real : (G * (1,i)) `2 <= s } is
Subset of
(TOP-REAL 2)
;
verum
end;
assume that
( not 1 <= i or not i < width G )
and
i < width G
; { |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } is Subset of (TOP-REAL 2)
set A = { |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } ;
{ |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } c= the carrier of (TOP-REAL 2)
hence
{ |[r,s]| where r, s is Real : s <= (G * (1,(i + 1))) `2 } is Subset of (TOP-REAL 2)
; verum