let X be set ; :: thesis: for n, m being Nat

for A being FinSequence of bool X st n <= m holds

ROUGH (A,m) c= ROUGH (A,n)

let n, m be Nat; :: thesis: for A being FinSequence of bool X st n <= m holds

ROUGH (A,m) c= ROUGH (A,n)

let A be FinSequence of bool X; :: thesis: ( n <= m implies ROUGH (A,m) c= ROUGH (A,n) )

assume A1: n <= m ; :: thesis: ROUGH (A,m) c= ROUGH (A,n)

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in ROUGH (A,m) or z in ROUGH (A,n) )

assume A2: z in ROUGH (A,m) ; :: thesis: z in ROUGH (A,n)

then z in { x where x is Element of X : m <= #occurrences (x,A) } by Def24;

then consider a being Element of X such that

A3: ( z = a & m <= #occurrences (a,A) ) ;

n <= #occurrences (a,A) by A1, A3, XXREAL_0:2;

then z in { x where x is Element of X : n <= #occurrences (x,A) } by A3;

hence z in ROUGH (A,n) by A2, Def24; :: thesis: verum

for A being FinSequence of bool X st n <= m holds

ROUGH (A,m) c= ROUGH (A,n)

let n, m be Nat; :: thesis: for A being FinSequence of bool X st n <= m holds

ROUGH (A,m) c= ROUGH (A,n)

let A be FinSequence of bool X; :: thesis: ( n <= m implies ROUGH (A,m) c= ROUGH (A,n) )

assume A1: n <= m ; :: thesis: ROUGH (A,m) c= ROUGH (A,n)

let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in ROUGH (A,m) or z in ROUGH (A,n) )

assume A2: z in ROUGH (A,m) ; :: thesis: z in ROUGH (A,n)

then z in { x where x is Element of X : m <= #occurrences (x,A) } by Def24;

then consider a being Element of X such that

A3: ( z = a & m <= #occurrences (a,A) ) ;

n <= #occurrences (a,A) by A1, A3, XXREAL_0:2;

then z in { x where x is Element of X : n <= #occurrences (x,A) } by A3;

hence z in ROUGH (A,n) by A2, Def24; :: thesis: verum