let X be set ; :: thesis: for f being PartFunc of REAL ,REAL st f is_strongly_quasiconvex_on X holds
f is_quasiconvex_on X
let f be PartFunc of REAL ,REAL ; :: thesis: ( f is_strongly_quasiconvex_on X implies f is_quasiconvex_on X )
assume A1: f is_strongly_quasiconvex_on X ; :: thesis: f is_quasiconvex_on X
A2: for p being Real st 0 < p & p < 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)

proof

let p be Real; :: thesis: ( 0 < p & p < 1 implies for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) )
assume A3: ( 0 < p & p < 1 ) ; :: thesis: for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)

proof

let r, s be Real; :: thesis: ( r in X & s in X & (p * r) + ((1 - p) * s) in X implies f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) )
assume A4: ( r in X & s in X & (p * r) + ((1 - p) * s) in X ) ; :: thesis: f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)

now

per cases ( r <> s or r = s ) ;

suppose r <> s ; :: thesis: f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)

hence f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) by A1, A3, A4, Def4; :: thesis: verum

end;

suppose r = s ; :: thesis: f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s)

hence f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) ; :: thesis: verum

end;

end;

end;

hence f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) ; :: thesis: verum

end;

hence for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
f . ((p * r) + ((1 - p) * s)) <= max (f . r),(f . s) ; :: thesis: verum

end;

X c= dom f by A1, Def4;
hence f is_quasiconvex_on X by A2, Def2; :: thesis: verum