let X be set ; :: thesis: for A being FinSequence of bool X
for n1, n2, m1, m2 being Nat st n1 <= m1 & n2 <= m2 holds
ROUGH (A,m1,n2) c= ROUGH (A,n1,m2)
let A be FinSequence of bool X; :: thesis: for n1, n2, m1, m2 being Nat st n1 <= m1 & n2 <= m2 holds
ROUGH (A,m1,n2) c= ROUGH (A,n1,m2)
let n1, n2, m1, m2 be Nat; :: thesis: ( n1 <= m1 & n2 <= m2 implies ROUGH (A,m1,n2) c= ROUGH (A,n1,m2) )
assume Z1: n1 <= m1 ; :: thesis: ( not n2 <= m2 or ROUGH (A,m1,n2) c= ROUGH (A,n1,m2) )
assume Z2: n2 <= m2 ; :: thesis: ROUGH (A,m1,n2) c= ROUGH (A,n1,m2)
let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in ROUGH (A,m1,n2) or z in ROUGH (A,n1,m2) )
assume A1: z in ROUGH (A,m1,n2) ; :: thesis: z in ROUGH (A,n1,m2)
then z in { x where x is Element of X : ( m1 <= #occurrences (x,A) & #occurrences (x,A) <= n2 ) } by ROUGH2;
then consider a being Element of X such that
A2: ( z = a & m1 <= #occurrences (a,A) & #occurrences (a,A) <= n2 ) ;
( n1 <= #occurrences (a,A) & #occurrences (a,A) <= m2 ) by Z1, Z2, A2, XXREAL_0:2;
then z in { x where x is Element of X : ( n1 <= #occurrences (x,A) & #occurrences (x,A) <= m2 ) } by A2;
hence z in ROUGH (A,n1,m2) by A1, ROUGH2; :: thesis: verum