let F be PartFunc of REAL,REAL; :: thesis: for X being set
for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )
let X be set ; :: thesis: for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )
let r be Real; :: thesis: ( 0 < r implies ( F is_convex_on X iff r (#) F is_convex_on X ) )
assume A1: 0 < r ; :: thesis: ( F is_convex_on X iff r (#) F is_convex_on X )
A2: dom F = dom (r (#) F) by VALUED_1:def 5;
thus ( F is_convex_on X implies r (#) F is_convex_on X ) :: thesis: ( r (#) F is_convex_on X implies F is_convex_on X )

proof

assume A3: F is_convex_on X ; :: thesis: r (#) F is_convex_on X
then A4: X c= dom F ;
thus X c= dom (r (#) F) by A2, A3; :: according to RFUNCT_3:def 12 :: thesis: 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
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s))
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
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s)) )
assume A5: ( 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
(r (#) F) . ((p * r) + ((1 - p) * s)) <= (p * ((r (#) F) . r)) + ((1 - p) * ((r (#) F) . s))
let x, y be Real; :: thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) )
assume that
A6: x in X and
A7: y in X and
A8: (p * x) + ((1 - p) * y) in X ; :: thesis: (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y))
F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A3, A5, A6, A7, A8;
then r * (F . ((p * x) + ((1 - p) * y))) <= r * ((p * (F . x)) + ((1 - p) * (F . y))) by A1, XREAL_1:64;
then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * (r * (F . x))) + (((1 - p) * r) * (F . y)) by A2, A4, A8, VALUED_1:def 5;
then (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * (r * (F . y))) by A2, A4, A6, VALUED_1:def 5;
hence (r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A2, A4, A7, VALUED_1:def 5; :: thesis: verum

end;

assume A9: r (#) F is_convex_on X ; :: thesis: F is_convex_on X
then A10: X c= dom (r (#) F) ;
hence X c= dom F by VALUED_1:def 5; :: according to RFUNCT_3:def 12 :: thesis: 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)) <= (p * (F . r)) + ((1 - p) * (F . s))
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)) <= (p * (F . r)) + ((1 - p) * (F . s)) )
assume A11: ( 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)) <= (p * (F . r)) + ((1 - p) * (F . s))
let x, y be Real; :: thesis: ( x in X & y in X & (p * x) + ((1 - p) * y) in X implies F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) )
assume that
A12: x in X and
A13: y in X and
A14: (p * x) + ((1 - p) * y) in X ; :: thesis: F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y))
(r (#) F) . ((p * x) + ((1 - p) * y)) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A9, A11, A12, A13, A14;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * ((r (#) F) . x)) + ((1 - p) * ((r (#) F) . y)) by A10, A14, VALUED_1:def 5;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * ((r (#) F) . y)) by A10, A12, VALUED_1:def 5;
then r * (F . ((p * x) + ((1 - p) * y))) <= (p * (r * (F . x))) + ((1 - p) * (r * (F . y))) by A10, A13, VALUED_1:def 5;
then (r * (F . ((p * x) + ((1 - p) * y)))) / r <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XREAL_1:72;
then F . ((p * x) + ((1 - p) * y)) <= (r * ((p * (F . x)) + ((1 - p) * (F . y)))) / r by A1, XCMPLX_1:89;
hence F . ((p * x) + ((1 - p) * y)) <= (p * (F . x)) + ((1 - p) * (F . y)) by A1, XCMPLX_1:89; :: thesis: verum